Step-by-step derivation
S001 · implies
Before
source operation=recurrence-initial-value-uniqueness;original snapshot and mathematics retained
After
explicit ordered source output fields=[order, initial_count, target_n, target_value, unique_prefix, certificate]
Conditions: none
Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
recurrence ORIGINAL:ORIGINAL order=4;oldest-to-newest coefficients=[1, 2, 3, 4];constant=0;ALL initial a_0..a_(r-1)=[2, 3, 4, 5]
After
for EVERY n>=4,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=4;forward rule is total and single-valued
Conditions: none
Fix all r initials and the oldest-to-newest coefficient roles.Given the preceding r values,the next value is uniquely determined.This is a forward definition,not a closed form or convergence claim.
Step identifiers and dependencies
Rule: recurrence_complete_initial_state_and_total_single_value_rule · Uses: G001, S001
S003 · calculation
Before
recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 := mul(1, 2)
After
recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 = 2
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(0, 2)
After
recurrence ORIGINAL n=4 partial sum after coefficient0 = 2
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 := mul(2, 3)
After
recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 = 6
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · calculation
Before
recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(2, 6)
After
recurrence ORIGINAL n=4 partial sum after coefficient1 = 8
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 := mul(3, 4)
After
recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 = 12
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(8, 12)
After
recurrence ORIGINAL n=4 partial sum after coefficient2 = 20
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 := mul(4, 5)
After
recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 = 20
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
recurrence ORIGINAL n=4 partial sum after coefficient3 := sum(20, 20)
After
recurrence ORIGINAL n=4 partial sum after coefficient3 = 40
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · implies
Before
recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[2, 3, 4, 5];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 2, 2], [1, 1, 2, 3, 6], [2, 2, 3, 4, 12], [3, 3, 4, 5, 20]]
After
new a_4=40;next state=[3, 4, 5, 40];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 4, 5, 40]
Conditions: none
Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update.
Step identifiers and dependencies
Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S010
S012 · calculation
Before
recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 := mul(1, 3)
After
recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 = 3
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S011
S013 · calculation
Before
recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(0, 3)
After
recurrence ORIGINAL n=5 partial sum after coefficient0 = 3
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S012
S014 · calculation
Before
recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 := mul(2, 4)
After
recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 = 8
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S013
S015 · calculation
Before
recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(3, 8)
After
recurrence ORIGINAL n=5 partial sum after coefficient1 = 11
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S014
S016 · calculation
Before
recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 := mul(3, 5)
After
recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 = 15
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S015
S017 · calculation
Before
recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(11, 15)
After
recurrence ORIGINAL n=5 partial sum after coefficient2 = 26
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S016
S018 · calculation
Before
recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 := mul(4, 40)
After
recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 = 160
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S017
S019 · calculation
Before
recurrence ORIGINAL n=5 partial sum after coefficient3 := sum(26, 160)
After
recurrence ORIGINAL n=5 partial sum after coefficient3 = 186
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S018
S020 · implies
Before
recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[3, 4, 5, 40];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 3, 3], [1, 2, 2, 4, 8], [2, 3, 3, 5, 15], [3, 4, 4, 40, 160]]
After
new a_5=186;next state=[4, 5, 40, 186];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 4, 5, 40, 186]
Conditions: none
Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update.
Step identifiers and dependencies
Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S019
S021 · calculation
Before
recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 := mul(1, 4)
After
recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 = 4
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S020
S022 · calculation
Before
recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(0, 4)
After
recurrence ORIGINAL n=6 partial sum after coefficient0 = 4
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S021
S023 · calculation
Before
recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 := mul(2, 5)
After
recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 = 10
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S022
S024 · calculation
Before
recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(4, 10)
After
recurrence ORIGINAL n=6 partial sum after coefficient1 = 14
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S023
S025 · calculation
Before
recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 := mul(3, 40)
After
recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 = 120
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S024
S026 · calculation
Before
recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(14, 120)
After
recurrence ORIGINAL n=6 partial sum after coefficient2 = 134
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S025
S027 · calculation
Before
recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 := mul(4, 186)
After
recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 = 744
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S026
S028 · calculation
Before
recurrence ORIGINAL n=6 partial sum after coefficient3 := sum(134, 744)
After
recurrence ORIGINAL n=6 partial sum after coefficient3 = 878
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S027
S029 · implies
Before
recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[4, 5, 40, 186];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 4, 4], [1, 3, 2, 5, 10], [2, 4, 3, 40, 120], [3, 5, 4, 186, 744]]
After
new a_6=878;next state=[5, 40, 186, 878];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 4, 5, 40, 186, 878]
Conditions: none
Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update.
Step identifiers and dependencies
Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S028
S030 · calculation
Before
recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 := mul(1, 5)
After
recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 = 5
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S029
S031 · calculation
Before
recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(0, 5)
After
recurrence ORIGINAL n=7 partial sum after coefficient0 = 5
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S030
S032 · calculation
Before
recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 := mul(2, 40)
After
recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 = 80
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S031
S033 · calculation
Before
recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(5, 80)
After
recurrence ORIGINAL n=7 partial sum after coefficient1 = 85
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S032
S034 · calculation
Before
recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 := mul(3, 186)
After
recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 = 558
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S033
S035 · calculation
Before
recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(85, 558)
After
recurrence ORIGINAL n=7 partial sum after coefficient2 = 643
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S034
S036 · calculation
Before
recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 := mul(4, 878)
After
recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 = 3512
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S035
S037 · calculation
Before
recurrence ORIGINAL n=7 partial sum after coefficient3 := sum(643, 3512)
After
recurrence ORIGINAL n=7 partial sum after coefficient3 = 4155
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S036
S038 · implies
Before
recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[5, 40, 186, 878];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 5, 5], [1, 4, 2, 40, 80], [2, 5, 3, 186, 558], [3, 6, 4, 878, 3512]]
After
new a_7=4155;next state=[40, 186, 878, 4155];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 4, 5, 40, 186, 878, 4155]
Conditions: none
Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update.
Step identifiers and dependencies
Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S037
S039 · calculation
Before
recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 := mul(1, 40)
After
recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 = 40
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S038
S040 · calculation
Before
recurrence ORIGINAL n=8 partial sum after coefficient0 := sum(0, 40)
After
recurrence ORIGINAL n=8 partial sum after coefficient0 = 40
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S039
S041 · calculation
Before
recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 := mul(2, 186)
After
recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 = 372
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S040
S042 · calculation
Before
recurrence ORIGINAL n=8 partial sum after coefficient1 := sum(40, 372)
After
recurrence ORIGINAL n=8 partial sum after coefficient1 = 412
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S041
S043 · calculation
Before
recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 := mul(3, 878)
After
recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 = 2634
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S042
S044 · calculation
Before
recurrence ORIGINAL n=8 partial sum after coefficient2 := sum(412, 2634)
After
recurrence ORIGINAL n=8 partial sum after coefficient2 = 3046
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S043
S045 · calculation
Before
recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 := mul(4, 4155)
After
recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 = 16620
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S044
S046 · calculation
Before
recurrence ORIGINAL n=8 partial sum after coefficient3 := sum(3046, 16620)
After
recurrence ORIGINAL n=8 partial sum after coefficient3 = 19666
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S045
S047 · implies
Before
recurrence ORIGINAL:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[40, 186, 878, 4155];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 40, 40], [1, 5, 2, 186, 372], [2, 6, 3, 878, 2634], [3, 7, 4, 4155, 16620]]
After
new a_8=19666;next state=[186, 878, 4155, 19666];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 4, 5, 40, 186, 878, 4155, 19666]
Conditions: none
Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update.
Step identifiers and dependencies
Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S046
S048 · calculation
Before
recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5 := mul(1, 186)
After
recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5 = 186
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S047
S049 · calculation
Before
recurrence ORIGINAL n=9 partial sum after coefficient0 := sum(0, 186)
After
recurrence ORIGINAL n=9 partial sum after coefficient0 = 186
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S048
S050 · calculation
Before
recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6 := mul(2, 878)
After
recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6 = 1756
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S049
S051 · calculation
Before
recurrence ORIGINAL n=9 partial sum after coefficient1 := sum(186, 1756)
After
recurrence ORIGINAL n=9 partial sum after coefficient1 = 1942
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S050
S052 · calculation
Before
recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7 := mul(3, 4155)
After
recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7 = 12465
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S051
S053 · calculation
Before
recurrence ORIGINAL n=9 partial sum after coefficient2 := sum(1942, 12465)
After
recurrence ORIGINAL n=9 partial sum after coefficient2 = 14407
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S052
S054 · calculation
Before
recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8 := mul(4, 19666)
After
recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8 = 78664
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S053
S055 · calculation
Before
recurrence ORIGINAL n=9 partial sum after coefficient3 := sum(14407, 78664)
After
recurrence ORIGINAL n=9 partial sum after coefficient3 = 93071
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S054
S056 · implies
Before
recurrence ORIGINAL:ORIGINAL target n=9;previous state a_(n-r)..a_(n-1)=[186, 878, 4155, 19666];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 1, 186, 186], [1, 6, 2, 878, 1756], [2, 7, 3, 4155, 12465], [3, 8, 4, 19666, 78664]]
After
new a_9=93071;next state=[878, 4155, 19666, 93071];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 4, 5, 40, 186, 878, 4155, 19666, 93071]
Conditions: none
Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update.
Step identifiers and dependencies
Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S055
S057 · implies
Before
recurrence ORIGINAL:complete ORIGINAL initial state=[2, 3, 4, 5];actual generated prefix=[2, 3, 4, 5, 40, 186, 878, 4155, 19666, 93071]
After
ANY two sequences with SAME r initials and SAME total rule agree at n=4 by substitution,then at every later n by induction;this determines the requested prefix only
Conditions: none
Induct from equality of all r initials:the same single-valued rule gives equality of the next term,and hence every later prefix term.Finite generation proves no closed form,asymptotic behavior or analyticity.
Step identifiers and dependencies
Rule: recurrence_forward_uniqueness_induction_not_analytic_claim · Uses: G001, S056