Original question and givens
Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=-2 and initial_values=[10, 11, 12], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).
G001 — Use order r=3, coefficients=[1, 2, 3] in oldest-to-newest order, constant=-2 and initial_values=[10, 11, 12], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Give (order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope).
Step-by-step derivation
S001 · implies
Before
source operation=recurrence-specification-qualification;original snapshot and mathematics retained
After
explicit ordered source output fields=[order, coefficients, constant, initial_values, complete_initials, total_rule, index_scope]
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=3;oldest-to-newest coefficients=[1, 2, 3];constant=-2;ALL initial a_0..a_(r-1)=[10, 11, 12]
After
for EVERY n>=3,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=3;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=3 ORIGINAL coefficient0 times a_0 := mul(1, 10)
After
recurrence ORIGINAL n=3 ORIGINAL coefficient0 times a_0 = 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, S002
S004 · calculation
Before
recurrence ORIGINAL n=3 partial sum after coefficient0 := sum(-2, 10)
After
recurrence ORIGINAL n=3 partial sum after coefficient0 = 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, S003
S005 · calculation
Before
recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 := mul(2, 11)
After
recurrence ORIGINAL n=3 ORIGINAL coefficient1 times a_1 = 22
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=3 partial sum after coefficient1 := sum(8, 22)
After
recurrence ORIGINAL n=3 partial sum after coefficient1 = 30
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=3 ORIGINAL coefficient2 times a_2 := mul(3, 12)
After
recurrence ORIGINAL n=3 ORIGINAL coefficient2 times a_2 = 36
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=3 partial sum after coefficient2 := sum(30, 36)
After
recurrence ORIGINAL n=3 partial sum after coefficient2 = 66
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 · implies
Before
recurrence ORIGINAL:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[10, 11, 12];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 10, 10], [1, 1, 2, 11, 22], [2, 2, 3, 12, 36]]
After
new a_3=66;next state=[11, 12, 66];drop ONLY oldest state entry then append new scalar;prefix now=[10, 11, 12, 66]
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, S008
S010 · implies
Before
recurrence ORIGINAL:complete ORIGINAL initial state=[10, 11, 12];actual generated prefix=[10, 11, 12, 66]
After
ANY two sequences with SAME r initials and SAME total rule agree at n=3 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, S009
Result
(3, [1, 2, 3], -2, [10, 11, 12], Yes, Yes, rule_applies_for_n_ge_3)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.