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A131557 Triangular numbers that are the sums of five consecutive triangular numbers.

%I A131557 #42 Mar 22 2024 18:48:14
%S A131557 55,2485,17020,799480,5479705,257429395,1764447310,82891465030,
%T A131557 568146553435,26690794309585,182941425758080,8594352876220660,
%U A131557 58906570947547645,2767354935348742255,18967732903684582930,891079694829418784770,6107551088415488155135
%N A131557 Triangular numbers that are the sums of five consecutive triangular numbers.
%H A131557 Alois P. Heinz, <a href="/A131557/b131557.txt">Table of n, a(n) for n = 1..250</a>
%H A131557 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,322,-322,-1,1).
%F A131557 The subsequences with odd indices and even indices satisfy the same recurrence relations: a(n+2) = 322*a(n+1) - a(n) - 680 and a(n+1) = 161*a(n) - 340 + 9*sqrt(320*a(n)^2 - 1360*a(n) - 175).
%F A131557 G.f.: -5*x*(11+486*x-635*x^2+2*x^4) / ( (x-1)*(x^2+18*x+1)*(x^2-18*x+1) ).
%F A131557 8*a(n) = 17 + 45*A007805(n) + 18*(-1)^n*A049629(n). - _R. J. Mathar_, Apr 28 2020
%e A131557 a(1) = 55 = 3+6+10+15+21.
%p A131557 a:= n-> `if`(n<2, [0, 55][n+1], (<<0|1|0>, <0|0|1>, <1|-323|323>>^iquo(n-2, 2, 'r'). `if`(r=0, <<2485, 799480, 257429395>>, <<17020, 5479705, 1764447310>>))[1, 1]): seq(a(n), n=1..20); # _Alois P. Heinz_, Sep 25 2008, revised Dec 15 2011
%t A131557 LinearRecurrence[{1, 322, -322, -1, 1}, {55, 2485, 17020, 799480, 5479705}, 20] (* _Jean-François Alcover_, Oct 05 2019 *)
%Y A131557 Cf. A129803.
%K A131557 nonn,easy
%O A131557 1,1
%A A131557 _Richard Choulet_, Oct 06 2007
%E A131557 More terms from _Alois P. Heinz_, Sep 25 2008
%E A131557 a(6) and a(8) corrected by _Harvey P. Dale_, Oct 02 2011
%E A131557 a(10), a(12), a(14) corrected at the suggestion of _Harvey P. Dale_ by _D. S. McNeil_, Oct 02 2011