A129803 Triangular numbers that are the sum of three consecutive triangular numbers.

10, 136, 1891, 26335, 366796, 5108806, 71156485, 991081981, 13803991246, 192264795460, 2677903145191, 37298379237211, 519499406175760, 7235693307223426, 100780206894952201, 1403687203222107385, 19550840638214551186, 272308081731781609216

Offset: 1

Author

Zak Seidov, May 18 2007

Keywords

• nonn
• easy

Comments

Indices m: 4, 16, 61, 229, 856, 3196, 11929, with recurrence m(i) = 5(m(i-1) - m(i-2)) + m(i-3) (see A133161).

If first term is omitted, same sequence as A128862. - R. J. Mathar, Jun 13 2008

Examples

			With tr(k) = k(k+1)/2 = A000217(k):
10 = tr(4) = tr(1) + tr(2) + tr(3) = 1 + 3 + 6,
136 = tr(16) = tr(8) + tr(9) + tr(10) = 36 + 45 + 55,
1891 = tr(61) = tr(34) + tr(35) + tr(36) = 595 + 630 + 666,
26335 = tr(229) = tr(131) + tr(132) + tr(133) = 8646 + 8778 + 8911,
366796 = tr(856) = tr(493) + tr(494) + tr(495) = 121771 + 122265 + 122760.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..850
• Index entries for linear recurrences with constant coefficients, signature (15, -15, 1).

Crossrefs

Cf. A000217, A128862, A133161.

Programs

Mathematica

LinearRecurrence[{15,-15,1},{10,136,1891},20] (* Harvey P. Dale, Oct 31 2024 *)

PARI

Vec((10*z - 14*z^2 + z^3)/((1-z)*(1 - 14*z + z^2)) + O(z^30)) \\ Michel Marcus, Sep 16 2015

Formula

a(n) = tr(m) = tr(k) + tr(k+1) + tr(k+2), where tr(k) = k(k+1)/2 = A000217(k).

From Richard Choulet, Oct 06 2007: (Start)

a(n+2) = 14*a(n+1) - a(n) - 3.

a(n+1) = 7*a(n) - 3/2 + 1/2*sqrt(192*a(n)^2 - 96*a(n) - 15).

G.f.: x*(10-14*x+x^2) / ((1-x)*(1-14*x+x^2)). (End)

a(n) = (4-3*(7-4*sqrt(3))^n*(-2+sqrt(3))+3*(2+sqrt(3))*(7+4*sqrt(3))^n)/16. - Colin Barker, Mar 05 2016