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Numbers n such that 1+2n+3n^2 is a triangular number.
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%I #28 Jun 13 2015 00:52:08

%S 0,1,46,135,4540,13261,444906,1299475,43596280,127335321,4271990566,

%T 12477562015,418611479220,1222673742181,41019652973026,

%U 119809549171755,4019507379877360,11740113145089841,393870703575008286,1150411278669632695,38595309442970934700

%N Numbers n such that 1+2n+3n^2 is a triangular number.

%C The y solution to the generalized Pell equation x^2 + x = 2 + 4*y + 6*y^2. - _T. D. Noe_, Apr 28 2011

%C 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 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 Colin Barker, <a href="/A122513/b122513.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,98,-98,-1,1).

%F 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 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 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 Corresponding values of triangular numbers tri = m(m+1)/2 and m's are

%e tri = 1, 6, 6441, 54946, 61843881, 527588886, 593824936321

%e m = 1, 3, 113, 331, 11121, 32483, 1089793.

%p ivs:=[0,1,46,135,4540]:

%p rec:= a(n) = a(n-1)+98*a(n-2)-98*a(n-3)-a(n-4)+a(n-5):

%p f:= gfun:-rectoproc({rec, seq(a(i)=ivs[i],i=1..5)},a(n),remember):

%p seq(f(n),n=1..100); # _Robert Israel_, Dec 15 2014

%t 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 LinearRecurrence[{1, 98, -98, -1, 1}, {1, 46, 135, 4540, 13261}, 30] (* _T. D. Noe_, Apr 28 2011 *)

%o (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 Cf. A000217 (triangular numbers), A086285 (numbers n such that 1+2n+3n^2 is prime).

%Y Cf. A000326, A000384, A245783, A249164.

%K nonn,easy

%O 1,3

%A _Zak Seidov_, Oct 20 2006

%E a(8) and a(9) from _Robert G. Wilson v_, Jan 08 2007

%E a(10) and a(11) from _Donovan Johnson_, Apr 28 2011

%E Extended by _T. D. Noe_, Apr 28 2011