A113254 Corresponds to m = 8 in a family of 4th-order linear recurrence sequences given by a(m,n) = m^4*a(n-4) + (2*m)^2*a(n-3) - 4*a(m-1), a(m,0) = -1, a(m,1) = 4, a(m,2) = -13 + 6*(m-1) + 3*(m-1)^2, a(m,3) = (-8+m^2)^2.

-1, 4, 176, 3136, -15616, 123904, 1028096, 4734976, -51183616, 975437824, 1521483776, 205520896, 39241908224, 4227925540864, -10627091267584, 53396107165696, 1029499365883904, 10479050187341824, -71775363146973184, 769363745204862976

Offset

0,2

Comments

Conjecture: a(m, 2*n+1) is a perfect square for all m,n (see A113249).

This was proved by an autonomous AI agent, see the Lean file. The proof uses an auxiliary sequence Y (a second-order recurrence) and shows the odd-indexed terms satisfy a(2n+1) = Y(n)^2 by induction over a three-term window. - Ralf Stephan, May 25 2026

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Google Deepmind, AlphaProof Nexus: A113254 Lean file

Index entries for linear recurrences with constant coefficients, signature (-4,0,256,4096).

Formula

G.f.: (-1+192*x^2+4096*x^3) / ((8*x+1)*(1-8*x)*(64*x^2+4*x+1)).

a(n) = -4*a(n-1) + 256*a(n-3) + 4096*a(n-4) for n > 3. - Colin Barker, May 20 2019

Mathematica

LinearRecurrence[{-4, 0, 256, 4096}, {-1, 4, 176, 3136}, 25] (* Paolo Xausa, Jun 10 2024 *)

Prog

(PARI) Vec(-(1 - 192*x^2 - 4096*x^3) / ((1 - 8*x)*(1 + 8*x)*(1 + 4*x + 64*x^2)) + O(x^25)) \\ Colin Barker, May 20 2019

Crossrefs

Cf. A000302, A097948, A056450, A113249, A113250, A113251, A113252, A113253, A113255, A113256.

Sequence in context: A006433 A251695 A343610 * A210780 A127606 A041945

Adjacent sequences: A113251 A113252 A113253 * A113255 A113256 A113257

Keyword

easy,sign,changed

Author

Creighton Dement, Nov 18 2005

Status

approved