A145934 Expansion of 1/(1-x*(1-6*x)).

1, 1, -5, -11, 19, 85, -29, -539, -365, 2869, 5059, -12155, -42509, 30421, 285475, 102949, -1609901, -2227595, 7431811, 20797381, -23793485, -148577771, -5816861, 885649765, 920550931, -4393347659, -9916653245, 16443432709

Offset: 0

Author

Philippe Deléham, Oct 25 2008

Keywords

• sign
• easy

Comments

Row sums of Riordan array (1, x(1-6x)).

For positive n, a(n) equals the determinant of the n X n tridiagonal matrix with 1's along the main diagonal, 3's along the superdiagonal, and 2's along the subdiagonal (see Mathematica code below). - John M. Campbell, Jul 08 2011

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..2569
• Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 17.
• Index entries for linear recurrences with constant coefficients, signature (1,-6).

Crossrefs

Cf. A010892, A107920, A106852, A106853, A106854.

Programs

Magma

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 14 2018

Mathematica

Table[Det[Array[KroneckerDelta[#1, #2] + KroneckerDelta[#1, #2 + 1]*2 + KroneckerDelta[#1, #2 - 1]*3 &, {n, n}]], {n, 1, 40}] (* John M. Campbell, Jul 08 2011 *)
LinearRecurrence[{1,-6}, {1,1}, 30] (* G. C. Greubel, Jan 14 2018 *)

PARI

Vec(1/(1-x*(1-6*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

Sage

[lucas_number1(n,1,6) for n in range(1, 29)] # Zerinvary Lajos, Apr 22 2009

Formula

a(n) = Sum_{k=0..n} A109466(n,k)*6^(n-k).

a(n) = a(n-1) - 6*a(n-2); a(0)=1, a(1)=1. - Philippe Deléham, Oct 25 2008