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%I #8 Apr 29 2019 22:09:24
%S 3,208,4387,240992,5063555,278105520,5843339043,320933530048,
%T 6743208193027,370357015570832,7781656411415075,427391675035211040,
%U 8980024755564804483,493209622633617970288,10362940786265372959267,569163477127520102502272
%N Numbers n such that the sum of the pentagonal numbers P(n), P(n+1) and P(n+2) is equal to the sum of the octagonal numbers N(m), N(m+1) and N(m+2) for some m.
%C Also nonnegative integers y in the solutions to 18*x^2-9*y^2+24*x-15*y+6 = 0, the corresponding values of x being A251863.
%H Colin Barker, <a href="/A251864/b251864.txt">Table of n, a(n) for n = 1..653</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1154,-1154,-1,1).
%F a(n) = a(n-1)+1154*a(n-2)-1154*a(n-3)-a(n-4)+a(n-5).
%F G.f: -x*(35*x^3+717*x^2+205*x+3) / ((x-1)*(x^2-34*x+1)*(x^2+34*x+1)).
%e 3 is in the sequence because P(3)+P(4)+P(5) = 12+22+35 = 69 = 8+21+40 = N(2)+N(3)+N(4).
%t LinearRecurrence[{1,1154,-1154,-1,1},{3,208,4387,240992,5063555},20] (* _Harvey P. Dale_, Apr 29 2019 *)
%o (PARI) Vec(-x*(35*x^3+717*x^2+205*x+3)/((x-1)*(x^2-34*x+1)*(x^2+34*x+1)) + O(x^100))
%Y Cf. A000326, A000567, A251863.
%K nonn,easy
%O 1,1
%A _Colin Barker_, Dec 10 2014
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