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A122513
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Numbers n such that 1+2n+3n^2 is a triangular number.
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3
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0, 1, 46, 135, 4540, 13261, 444906, 1299475, 43596280, 127335321, 4271990566, 12477562015, 418611479220, 1222673742181, 41019652973026, 119809549171755, 4019507379877360, 11740113145089841, 393870703575008286, 1150411278669632695, 38595309442970934700
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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):
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000217 (triangular numbers), A086285 (numbers n such that 1+2n+3n^2 is prime).
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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