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A258444 9-gonal numbers (A001106) that are the sum of twelve consecutive 9-gonal numbers.

%I A258444 #18 Sep 08 2022 08:46:12
%S A258444 1349094322576,1910746510353532612000,2706224588156555124000697809136,
%T A258444 3832874471762384783002138104903925699456,
%U A258444 5428568929785331587316097630206410288870519307600,7688579639781530489126233275115806835015504771403279234656
%N A258444 9-gonal numbers (A001106) that are the sum of twelve consecutive 9-gonal numbers.
%H A258444 Colin Barker, <a href="/A258444/b258444.txt">Table of n, a(n) for n = 1..108</a>
%H A258444 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1416317955,-1416317955,1).
%F A258444 a(n) = 1416317955*a(n-1) - 1416317955*a(n-2) + a(n-3).
%F A258444 G.f.: -16*x*(76*x^2-106213627505*x+84318395161) / ((x-1)*(x^2-1416317954*x+1)).
%F A258444 a(n) = (55406+2523*(708158977+408855776*sqrt(3))^(-n)*(43-24*sqrt(3)+(43+24*sqrt(3))*(708158977+408855776*sqrt(3))^(2*n)))/224. - _Colin Barker_, Mar 07 2016
%e A258444 1349094322576 is in the sequence because A001106(620851) = 1349094322576 = 112417626816 + 112418881350 + 112420135891 + 112421390439 + 112422644994 + 112423899556 + 112425154125 + 112426408701 + 112427663284 + 112428917874 + 112430172471 + 112431427075 = A001106(179219) + ... + A001106(179230).
%t A258444 CoefficientList[Series[16 (76 x^2 - 106213627505 x + 84318395161)/((1 - x) (x^2 - 1416317954 x + 1)), {x, 0, 33}], x] (* _Vincenzo Librandi_, May 31 2015 *)
%t A258444 LinearRecurrence[{1416317955,-1416317955,1},{1349094322576,1910746510353532612000,2706224588156555124000697809136},10] (* _Harvey P. Dale_, Jan 19 2016 *)
%o A258444 (PARI) Vec(-16*x*(76*x^2-106213627505*x+84318395161) / ((x-1)*(x^2-1416317954*x+1)) + O(x^20))
%o A258444 (Magma) I:=[1349094322576,1910746510353532612000, 2706224588156555124000697809136]; [n le 3 select I[n] else 1416317955*Self(n-1)-1416317955*Self(n-2)+Self(n-3): n in [1..10]]; // _Vincenzo Librandi_, May 31 2015
%Y A258444 Cf. A001106, A258441, A258442, A258443.
%K A258444 nonn,easy
%O A258444 1,1
%A A258444 _Colin Barker_, May 30 2015