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 A122513 Numbers n such that 1+2n+3n^2 is a triangular number. 3
 0, 1, 46, 135, 4540, 13261, 444906, 1299475, 43596280, 127335321, 4271990566, 12477562015, 418611479220, 1222673742181, 41019652973026, 119809549171755, 4019507379877360, 11740113145089841, 393870703575008286, 1150411278669632695, 38595309442970934700 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The y solution to the generalized Pell equation x^2 + x = 2 + 4*y + 6*y^2. - T. D. Noe, Apr 28 2011 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 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 LINKS Colin Barker, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (1,98,-98,-1,1). FORMULA a(n) = a(n-1)+98*a(n-2)-98*a(n-3)-a(n-4)+a(n-5). - Colin Barker, Dec 15 2014 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 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 EXAMPLE Corresponding values of triangular numbers tri = m(m+1)/2 and m's are tri = 1, 6, 6441, 54946, 61843881, 527588886, 593824936321 m = 1, 3, 113, 331, 11121, 32483, 1089793. MAPLE ivs:=[0, 1, 46, 135, 4540]: rec:= a(n) = a(n-1)+98*a(n-2)-98*a(n-3)-a(n-4)+a(n-5): f:= gfun:-rectoproc({rec, seq(a(i)=ivs[i], i=1..5)}, a(n), remember): seq(f(n), n=1..100); # Robert Israel, Dec 15 2014 MATHEMATICA 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 *) LinearRecurrence[{1, 98, -98, -1, 1}, {1, 46, 135, 4540, 13261}, 30] (* T. D. Noe, Apr 28 2011 *) PROG (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 CROSSREFS Cf. A000217 (triangular numbers), A086285 (numbers n such that 1+2n+3n^2 is prime). Cf. A000326, A000384, A245783, A249164. Sequence in context: A044678 A074866 A229207 * A252687 A053019 A044378 Adjacent sequences:  A122510 A122511 A122512 * A122514 A122515 A122516 KEYWORD nonn,easy AUTHOR Zak Seidov, Oct 20 2006 EXTENSIONS a(8) and a(9) from Robert G. Wilson v, Jan 08 2007 a(10) and a(11) from Donovan Johnson, Apr 28 2011 Extended by T. D. Noe, Apr 28 2011 STATUS approved

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Last modified June 22 15:08 EDT 2021. Contains 345383 sequences. (Running on oeis4.)