A114775 - OEIS

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A114775

Expansion of x^2*(1+x^2)*(1 - x^4 + x^7)/((1 - x^4 + x^6)*(1 - x^4 - x^6)).

0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 3, 4, 5, 7, 5, 7, 9, 12, 9, 12, 16, 21, 16, 21, 28, 37, 28, 37, 49, 65, 49, 65, 86, 114, 86, 114, 151, 200, 151, 200, 265, 351, 265, 351, 465, 616, 465, 616, 816, 1081, 816, 1081, 1432, 1897, 1432, 1897, 2513

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,14

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1,0,0,0,1).

FORMULA

Let M = {{0, 1, 0}, {0, 0, 1}, {-I, -1, 0}}, v(0) = {0, 1, 1}, and v(n) = M*v(n-1) then {a(n), a(n+1)} = {abs(Re( v(n)((1)) )), abs(Im( v(n)((1)) ))}.

G.f.: x^2*(1+x^2)*(1 - x^4 + x^7)/((1 - x^4 + x^6)*(1 - x^4 - x^6)). - Colin Barker, Jan 01 2013

MATHEMATICA

M = {{0, 1, 0}, {0, 0, 1}, {-I, -1, 0}}; v[0] = {0, 1, 1};

v[n_]:= v[n] = M.v[n-1];

Flatten[Table[{Abs[Re[v[n][[1]]]], Abs[Im[v[n][[1]]]]}, {n, 0, 40}]]

PROG

(Magma) R<x>:=PowerSeriesRing(Integers(), 82);

[0, 0] cat Coefficients(R!( x^2*(1+x^2)*(1-x^4+x^7)/(1-2*x^4+x^8-x^12) )); // G. C. Greubel, Jul 08 2021

(Sage)

def A114775_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P( x^2*(1+x^2)*(1-x^4+x^7)/(1-2*x^4+x^8-x^12) ).list()