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Indices of centered square numbers (A001844) which are also pentagonal numbers (A000326).
3

%I #5 Jun 13 2015 00:55:24

%S 1,2,9,22,119,300,1651,4172,22989,58102,320189,809250,4459651,

%T 11271392,62114919,156990232,865149209,2186591850,12049974001,

%U 30455295662,167834486799,424187547412,2337632841179,5908170368100,32559025289701,82290197605982

%N Indices of centered square numbers (A001844) which are also pentagonal numbers (A000326).

%C Also positive integers y in the solutions to 3*x^2 - 4*y^2 - x + 4*y - 2 = 0, the corresponding values of x being A254709.

%H Colin Barker, <a href="/A254710/b254710.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,14,-14,-1,1).

%F a(n) = a(n-1)+14*a(n-2)-14*a(n-3)-a(n-4)+a(n-5).

%F G.f.: x*(x^3+7*x^2-x-1) / ((x-1)*(x^2-4*x+1)*(x^2+4*x+1)).

%e 9 is in the sequence because the 9th centered square number is 145, which is also the 10th pentagonal number.

%o (PARI) Vec(x*(x^3+7*x^2-x-1)/((x-1)*(x^2-4*x+1)*(x^2+4*x+1)) + O(x^100))

%Y Cf. A000326, A001844, A254709, A254711.

%K nonn,easy

%O 1,2

%A _Colin Barker_, Feb 06 2015