A103644 - OEIS

%I #29 Mar 07 2024 14:13:43

%S 1,1,4,25,1,256,169,1225,5476,961,64009,25600,358801,1164241,515524,

%T 15642025,3243601,101284096,239228089,216825625,3736387876,287336401,

%U 27697946329,47210598400

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

%C A floretion-generated sequence of squares.

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

%H Harvey P. Dale, <a href="/A103644/b103644.txt">Table of n, a(n) for n = 0..1000</a>

%H Creighton Dement, <a href="http://fumba.eu/sitelayout/Floretion.php">Online Floretion Multiplier</a>. [broken link]

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-2,6,27).

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

%F 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]

%F 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

%p A103644 := proc(n)

%p     6*3^n+5*(-1)^n*A190970(n+1)+18*(-1)^(n+1)*A190970(n) ;

%p     %/11 ;

%p end proc:

%p seq(A103644(n),n=0..20) ; # _R. J. Mathar_, Mar 23 2023

%t 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 *)

%Y Cf. A103645.

%K nonn,easy

%O 0,3

%A _Creighton Dement_, Feb 11 2005

%E Corrected by _T. D. Noe_, Nov 07 2006