%I #7 Dec 17 2020 18:56:53

%S 417912,9706632,3050311681782,70865417283102,22269721626195937752,

%T 517374380230514907672,162586828187971503638961822,

%U 3777247909935632832763236342,1187014240408376459988712771009992,27576939095353370682323270116205112

%N Pentagonal numbers (A000326) that are the sum of twelve consecutive pentagonal numbers.

%H Colin Barker, <a href="/A259404/b259404.txt">Table of n, a(n) for n = 1..290</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,7300802,-7300802,-1,1).

%F G.f.: -6*x*(377*x^4+7980*x^3-131798379*x^2+1548120*x+69652) / ((x-1)*(x^2-2702*x+1)*(x^2+2702*x+1))

%e 417912 is in the sequence because P(528) = 417912 = 32340 + 32782 + 33227 + 33675 + 34126 + 34580 + 35037 + 35497 + 35960 + 36426 + 36895 + 37367 = P(147)+ ... +P(158).

%t Select[Total/@Partition[PolygonalNumber[5,Range[5*10^6]],12,1],IntegerQ[ (1+Sqrt[ 1+24#])/6]&] (* The program generates the first four terms of the sequence. To generate more, increase the Range constant but the program will take a long time to run. *) (* _Harvey P. Dale_, Dec 17 2020 *)

%o (PARI) Vec(-6*x*(377*x^4+7980*x^3-131798379*x^2+1548120*x+69652) / ((x-1)*(x^2-2702*x+1)*(x^2+2702*x+1)) + O(x^20))

%Y Cf. A000326, A133301, A257714, A257715, A259402, A259403.

%K nonn,easy

%O 1,1

%A _Colin Barker_, Jun 26 2015