bibkey: hanna2014a246056a246423
authors: Paul D. Hanna
year: 2014
title: "OEIS A246056 and A246423, two binomial-square generating functions"
doi: null
url: https://oeis.org/A246056
claim: "A246056 conjectures the ternary-digit residue rule modulo three; A246423 conjectures residue two exactly at powers of two of exponent at least two modulo four."
strata_touched:
  - D5/S1/Recurrence/Invariants/HannaLegendreCentralBinomialTransform
  - D5/S1/Recurrence/Invariants/HannaLegendreCentralBinomialModThree
  - D5/S1/Recurrence/Invariants/HannaLegendreCentralBinomialModFour
license: citation-only
triage: anchor

Verified locator

Source: https://oeis.org/A246056

Sibling source: https://oeis.org/A246423

Both entries are unpaginated. The quoted locations are Name and Comments.

Hanna’s two binomial-square series

A246056, Name:

G.f.: Sum_{n>=0} x^n / (1-2x)^(2n+1) * [Sum_{k=0..n} C(n,k)^2 * 2^k * x^k] * [Sum_{k=0..n} C(n,k)^2 * 3^k * x^k].

A246056, Comments:

Conjecture: a(n) == 1 (mod 3) when n = 2*A005836(k) for k >= 0, and a(n) == 0 (mod 3) otherwise, where A005836 gives numbers whose base-3 representation contains no 2.

A246423, Name:

G.f.: Sum_{n>=0} x^n / (1-3x)^(2n+1) * [Sum_{k=0..n} C(n,k)^2 * 2^k * x^k] * [Sum_{k=0..n} C(n,k)^2 * 3^k * x^k].

A246423, Comments:

a(n) == 2 (mod 4) iff n = 2^k for k>=2, and a(n) == 0 (mod 4) elsewhere except at a(0)=1 (conjecture).

Both sequences have offset zero. The rational formal power series are defined coefficientwise: the summand indexed by n has order at least n, so the coefficient at m uses only indices n ≤ m. The integer lift proves integrality. The empty base-three digit list of zero contains no two and includes the constant term in the first conjecture. The second conjecture treats its constant term separately.

The exact integer coefficient transform is a binomial-square telescoping identity. It gives zero at odd degrees and signed central binomial coefficients at even degrees. The ternary Lucas recurrence gives the first residue rule; the binary Lucas recurrence and a first-moment correction give the second. These proofs do not assert the full Kummer valuation formula or a general Legendre-series API.

Literature-screen completeness and independent priority are ASSUMED-UNVERIFIED.