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A253470 Indices of centered triangular numbers (A005448) which are also centered pentagonal numbers (A005891). 2

%I

%S 1,5,36,280,2201,17325,136396,1073840,8454321,66560725,524031476,

%T 4125691080,32481497161,255726286205,2013328792476,15850904053600,

%U 124793903636321,982500325036965,7735208696659396,60899169248238200,479458145289246201,3774765993065731405

%N Indices of centered triangular numbers (A005448) which are also centered pentagonal numbers (A005891).

%C Also indices of pentagonal numbers (A000326) which are also centered pentagonal numbers (A005891).

%C Also positive integers x in the solutions to 3*x^2 - 5*y^2 - 3*x + 5*y = 0, the corresponding values of y being A182432.

%H Colin Barker, <a href="/A253470/b253470.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-9,1).

%F a(n) = 9*a(n-1)-9*a(n-2)+a(n-3).

%F G.f.: x*(4*x-1) / ((x-1)*(x^2-8*x+1)).

%F a(n) = (6-(4-sqrt(15))^n*(3+sqrt(15))+(-3+sqrt(15))*(4+sqrt(15))^n)/12. - _Colin Barker_, Mar 03 2016

%e 5 is in the sequence because the 5th centered triangular number is 31, which is also the 4th centered pentagonal number.

%o (PARI) Vec(x*(4*x-1)/((x-1)*(x^2-8*x+1)) + O(x^100))

%Y Cf. A005448, A005891, A182432, A253654.

%K nonn,easy

%O 1,2

%A _Colin Barker_, Jan 01 2015

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Last modified January 25 06:09 EST 2021. Contains 340416 sequences. (Running on oeis4.)