A251963 Numbers n such that the sum of the triangular numbers T(n) and T(n+1) is equal to an octagonal number N(m) for some m.

0, 14, 208, 2910, 40544, 564718, 7865520, 109552574, 1525870528, 21252634830, 296011017104, 4122901604638, 57424611447840, 799821658665134, 11140078609864048, 155161278879431550, 2161117825702177664, 30100488280951055758, 419245718107612602960

Offset

1,2

Comments

Also nonnegative integers x in the solutions to 2*x^2-6*y^2+4*x+4*y+2+2 = 0, the corresponding values of y being A046184.

Links

Colin Barker, Table of n, a(n) for n = 1..874

Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Formula

a(n) = 15*a(n-1)-15*a(n-2)+a(n-3).

G.f.: 2*x^2*(x-7) / ((x-1)*(x^2-14*x+1)).

a(n) = (-6-(7-4*sqrt(3))^n*(3+2*sqrt(3))+(-3+2*sqrt(3))*(7+4*sqrt(3))^n)/6. - Colin Barker, Mar 05 2016

a(n) = 14*a(n-1) - a(n-2) + 12. - Vincenzo Librandi, Mar 05 2016

Example

14 is in the sequence because T(14)+T(15) = 105+120 = 225 = N(9).

Mathematica

RecurrenceTable[{a[1] == 0, a[2] == 14, a[n] == 14 a[n-1]- a[n-2] + 12}, a, {n, 20}] (* Vincenzo Librandi, Mar 05 2016 *)

Prog

(PARI) concat(0, Vec(2*x^2*(x-7) / ((x-1)*(x^2-14*x+1)) + O(x^100)))
(Magma) I:=[0, 14]; [n le 2 select I[n] else 14*Self(n-1)-Self(n-2)+12: n in [1..20]]; // Vincenzo Librandi, Mar 05 2016

Crossrefs

Cf. A000217, A000567, A046184.

Sequence in context: A333949 A002961 A063071 * A192007 A160682 A097261

Adjacent sequences: A251960 A251961 A251962 * A251964 A251965 A251966

Keyword

nonn,easy

Author

Colin Barker, Dec 11 2014

Status

approved