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

%I #8 Aug 08 2017 16:15:12

%S 1,2,10,25,137,346,1906,4817,26545,67090,369722,934441,5149561,

%T 13015082,71724130,181276705,998988257,2524858786,13914111466,

%U 35166746297,193798572265,489809589370,2699265900242,6822167504881,37595924031121,95020535478962

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

%C Also positive integers x in the solutions to 3*x^2 - 4*y^2 - x + 4*y - 2 = 0, the corresponding values of y being A254710.

%H Colin Barker, <a href="/A254709/b254709.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^4+x^3-6*x^2+x+1) / ((x-1)*(x^2-4*x+1)*(x^2+4*x+1)).

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

%t LinearRecurrence[{1,14,-14,-1,1},{1,2,10,25,137},30] (* _Harvey P. Dale_, Aug 08 2017 *)

%o (PARI) Vec(-x*(x^4+x^3-6*x^2+x+1)/((x-1)*(x^2-4*x+1)*(x^2+4*x+1)) + O(x^100))

%Y Cf. A000326, A001844, A254710, A254711.

%K nonn,easy

%O 1,2

%A _Colin Barker_, Feb 06 2015