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%I #20 Feb 27 2022 11:37:27
%S 120,276,1176,28920,126756,306936,1345620,33362196,146264856,
%T 354192420,1552832856,38499933816,168789505620,408737734296,
%U 1791967758756,44428890250020,194782943209176,471682991173716,2067929240760120,51270900848577816,224779347673872036
%N Triangular numbers (A000217) that are the sum of eight consecutive triangular numbers.
%H Colin Barker, <a href="/A257712/b257712.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1154,-1154,0,0,-1,1).
%F G.f.: -12*x*(3*x^8+7*x^6+13*x^5-3387*x^4+2312*x^3+75*x^2+13*x+10) / ((x-1)*(x^2-6*x+1)*(x^2+6*x+1)*(x^4+34*x^2+1)).
%e 120 is in the sequence because T(15) = 120 = 1+3+6+10+15+21+28+36 = T(1)+ ... +T(8).
%t LinearRecurrence[{1, 0, 0, 1154, -1154, 0, 0, -1, 1}, {120, 276, 1176, 28920, 126756, 306936, 1345620, 33362196, 146264856}, 30] (* _Vincenzo Librandi_, Jun 27 2015 *)
%t Select[Total/@Partition[Accumulate[Range[5*10^6]],8,1],OddQ[ Sqrt[ 1+8#]]&] (* The program generates the first 16 terms of the sequence *) (* _Harvey P. Dale_, Feb 27 2022 *)
%o (PARI) Vec(-12*x*(3*x^8+7*x^6+13*x^5-3387*x^4+2312*x^3+75*x^2+13*x+10) / ((x-1)*(x^2-6*x+1)*(x^2+6*x+1)*(x^4+34*x^2+1)) + O(x^100))
%Y Cf. A000217, A001110, A129803, A131557, A257711, A257713, A259413, A259414, A259415.
%K nonn,easy
%O 1,1
%A _Colin Barker_, May 05 2015
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