A103644 Expansion of g.f. (3x+1)/((1-3*x)*(1+5*x+9*x^2)).

1, 1, 4, 25, 1, 256, 169, 1225, 5476, 961, 64009, 25600, 358801, 1164241, 515524, 15642025, 3243601, 101284096, 239228089, 216825625, 3736387876, 287336401, 27697946329, 47210598400

Offset: 0

Author

Creighton Dement, Feb 11 2005

Keywords

• nonn
• easy

Comments

A floretion-generated sequence of squares.

This sequence is also related to several other sequences of squares.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Creighton Dement, Online Floretion Multiplier. [broken link]
• Index entries for linear recurrences with constant coefficients, signature (-2,6,27).

Crossrefs

Cf. A103645.

Programs

Maple

A103644 := proc(n)
    6*3^n+5*(-1)^n*A190970(n+1)+18*(-1)^(n+1)*A190970(n) ;
    %/11 ;
end proc:
seq(A103644(n),n=0..20) ; # R. J. Mathar, Mar 23 2023

Mathematica

CoefficientList[Series[(3x+1)/(1+2x-6x^2-27x^3),{x,0,30}],x] (* or *) LinearRecurrence[{-2,6,27},{1,1,4},30] (* Harvey P. Dale, Dec 13 2017 *)

Formula

a(n+3) = -2a(n+2) + 6a(n+1) + 27a(n), a(0) = 1, a(1) = 1, a(2) = 4.

a(n) = (1/11)*(2*3^n-(-5/2-(I*sqrt(11))/2)^n-(-5/2+(I*sqrt(11))/2)^n). [Creighton Dement, May 24 2009]

11*a(n) = 6*3^n + 5*b(n) + 18*b(n-1) where b(n) = (-1)^n*A190970(n+1). - R. J. Mathar, Mar 23 2023

Extensions

Corrected by T. D. Noe, Nov 07 2006