An oldest-to-newest order-two recurrence trace

Goal and scope

Align states, coefficients and all initial values, generate a5 and explain uniqueness.

A finite exact integer trace, without guessing a closed form.

Givens and symbols

• work

• G1 — Order r=2, initial_values=(a0,a1)=(1,3), coefficients=(2,1), constant=1; for all n≥2 use a_n=1+2a_(n-2)+a_(n-1).

• A_n — latest-two-term state
• F — shift-and-append update

Rules and sources

• mko_definition · cited — An order-r recurrence needs all r initial values and coefficients aligned oldest to newest. Shift the state left and append the new term. A finite prefix proves no closed form, convergence or asymptotics. · mko-recurrence-relation
• work · prerequisite — Order-r states are ordered oldest to newest; the single-valued integer update is total; induction from all initial values proves uniqueness.

Main derivation

A_1=(a_0,a_1)=(1,3)

F(u,v)=(v,1+2u+v)

a_2=1+2(1)+3

a_2=6,\quad A_2=(3,6)

a_3=1+2(3)+6

a_3=13,\quad A_3=(6,13)

a_4=1+2(6)+13

a_4=26,\quad A_4=(13,26)

a_5=1+2(13)+26

a_5=53,\quad A_5=(26,53)

a_0=b_0,\ a_1=b_1\quad\text{and same recurrence}

(a_{n-2},a_{n-1})=(b_{n-2},b_{n-1})\implies a_n=b_n

a_0=b_0=1,\ a_1=b_1=3

a_n=b_n\quad\text{for every }n\ge0

a_0=1\text{ only},\quad a_1=t

a_2=3+t\text{ depends on }t

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — Completing a finite trace asserts no convergence or asymptotics.
• S07 · premise_audit · G1, S06, work — Uniqueness does not prove a closed form or growth rate.
• S06 · premise_audit · G1, S05, work — The induction base requires both initial values.
• S05 · premise_audit · G1, S04, work — Every intermediate value is an explicitly small integer.
• S04 · premise_audit · G1, S03, work — The constant 1 appears at every update.
• S03 · premise_audit · G1, S02, work — The coefficient 2 multiplies3, not 6.
• S02 · premise_audit · G1, S01, work — The state is not merely the new scalar6.
• S01 · premise_audit · G1, work — Do not reverse coefficients to u+2v.

Error controls and negative controls

Coefficient alignment

Pair(2,1) with a_(n-2),a_(n-1).

State width

Keep two terms in every state, shifting before appending.

• a2=1+1+2 times 3=8. → reject — The oldest term gets coefficient 2, yielding6.
• a0=1 alone gives uniqueness. → reject — a1=t leaves a2=3+t variable.
• Six increasing terms prove divergence. → reject — A finite prefix cannot prove an infinite-tail property.

Result and limitations

The prefix is 1,3,6,13,26,53; two initial values and the total rule determine a unique sequence.

Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 124f407c6762352d6b2a2e969d2d845466e4482a03a8f31e13b95650c4a212a5
• original source SHA-256: 22806b93c67d3eddec49366fb95886552a6c5b9b138df147ef955ed07c293e53
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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