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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.)