This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A122513 #28 Jun 13 2015 00:52:08
%S A122513 0,1,46,135,4540,13261,444906,1299475,43596280,127335321,4271990566,
%T A122513 12477562015,418611479220,1222673742181,41019652973026,
%U A122513 119809549171755,4019507379877360,11740113145089841,393870703575008286,1150411278669632695,38595309442970934700
%N A122513 Numbers n such that 1+2n+3n^2 is a triangular number.
%C A122513 The y solution to the generalized Pell equation x^2 + x = 2 + 4*y + 6*y^2. - _T. D. Noe_, Apr 28 2011
%C A122513 Also numbers n such that the sum of the pentagonal numbers P(n) and P(n+1) is equal to a hexagonal number. - _Colin Barker_, Dec 15 2014
%C A122513 Also numbers n such that the sum of the pentagonal numbers P(n) and P(n+1) is equal to a triangular number. - _Colin Barker_, Dec 15 2014
%H A122513 Colin Barker, <a href="/A122513/b122513.txt">Table of n, a(n) for n = 1..1000</a>
%H A122513 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,98,-98,-1,1).
%F A122513 a(n) = a(n-1)+98*a(n-2)-98*a(n-3)-a(n-4)+a(n-5). - _Colin Barker_, Dec 15 2014
%F A122513 G.f.: x^2*(5*x^3+9*x^2-45*x-1) / ((x-1)*(x^2-10*x+1)*(x^2+10*x+1)). - _Colin Barker_, Dec 15 2014
%F A122513 a(n) = (-(5/8)*sqrt(6)-3/2)*(5-2*sqrt(6))^n+(-3/2+(5/8)*sqrt(6))*(5+2*sqrt(6))^n-1/3+(-(1/3)*sqrt(6)-5/6)*(-5+2*sqrt(6))^n+((1/3)*sqrt(6)-5/6)*(-5-2*sqrt(6))^n. - _Robert Israel_, Dec 15 2014
%e A122513 Corresponding values of triangular numbers tri = m(m+1)/2 and m's are
%e A122513 tri = 1, 6, 6441, 54946, 61843881, 527588886, 593824936321
%e A122513 m = 1, 3, 113, 331, 11121, 32483, 1089793.
%p A122513 ivs:=[0,1,46,135,4540]:
%p A122513 rec:= a(n) = a(n-1)+98*a(n-2)-98*a(n-3)-a(n-4)+a(n-5):
%p A122513 f:= gfun:-rectoproc({rec, seq(a(i)=ivs[i],i=1..5)},a(n),remember):
%p A122513 seq(f(n),n=1..100); # _Robert Israel_, Dec 15 2014
%t A122513 triQ[n_] := IntegerQ[ Sqrt[8n + 1]]; lst = {}; Do[ If[ triQ[1 + 2n + 3n^2], AppendTo[lst, n]; Print@n], {n, 0, 65000000}] (* _Robert G. Wilson v_, Jan 08 2007 *)
%t A122513 LinearRecurrence[{1, 98, -98, -1, 1}, {1, 46, 135, 4540, 13261}, 30] (* _T. D. Noe_, Apr 28 2011 *)
%o A122513 (PARI) concat(0, Vec(x^2*(5*x^3+9*x^2-45*x-1)/((x-1)*(x^2-10*x+1)*(x^2+10*x+1)) + O(x^100))) \\ _Colin Barker_, Dec 15 2014
%Y A122513 Cf. A000217 (triangular numbers), A086285 (numbers n such that 1+2n+3n^2 is prime).
%Y A122513 Cf. A000326, A000384, A245783, A249164.
%K A122513 nonn,easy
%O A122513 1,3
%A A122513 _Zak Seidov_, Oct 20 2006
%E A122513 a(8) and a(9) from _Robert G. Wilson v_, Jan 08 2007
%E A122513 a(10) and a(11) from _Donovan Johnson_, Apr 28 2011
%E A122513 Extended by _T. D. Noe_, Apr 28 2011