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

1,1

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

Links

Colin Barker, Table of n, a(n) for n = 1..850

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

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

Example

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.

Prog

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

Crossrefs

Cf. A000217, A128862, A133161.

Sequence in context: A240917 A240654 A128862 * A065024 A026244 A371405

Adjacent sequences: A129800 A129801 A129802 * A129804 A129805 A129806

Keyword

nonn,easy

Author

Zak Seidov, May 18 2007

Status

approved