A128862 -id:A128862 - cp's OEIS frontend

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

Zak Seidov, May 18 2007

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

			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.

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

Cf. A000217, A128862, A133161.

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

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

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

A133161 Indices of the triangular numbers which are also centered triangular number.

Original entry on oeis.org

1, 4, 16, 61, 229, 856, 3196, 11929, 44521, 166156, 620104, 2314261, 8636941, 32233504, 120297076, 448954801, 1675522129, 6253133716, 23337012736, 87094917229, 325042656181, 1213075707496, 4527260173804, 16895964987721

Offset: 1

Richard Choulet, Oct 09 2007

Also, indices of the triangular numbers which are sums of three consecutive triangular numbers (see A129803).

A. Kozikowska, Topological classes of statically determinate beams with arbitrary number of supports, JOURNAL OF THEORETICAL AND APPLIED MECHANICS 49, 4, pp. 1079-1100, Warsaw 2011; (see Eq. 5.18). - _N. J. A. Sloane_, Dec 17 2011.
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Cf. A001834, A102871, A128862, A129803, A001571.

LinearRecurrence[{5,-5,1},{1,4,16},30] (* Harvey P. Dale, Aug 29 2017 *)

a(n+2)=4*a(n+1)-a(n)+1.
a(n+1)=2*a(n)+0.5+0.5*(12*a(n)^2+12*a(n)-15)^0.5.
G.f.: x*(1-x+x^2)/(1-x)/(1-4*x+x^2). - R. J. Mathar, Oct 24 2007
a(n)-a(n-1)= A005320(n-1). - R. J. Mathar, Mar 14 2016

A307491 Numbers that are both centered triangular and tetrahedral.

Original entry on oeis.org

1, 4, 10, 4960, 428536

Offset: 1

Ilya Gutkovskiy, Apr 10 2019

If it exists, a(6) > 10^29. - Bert Dobbelaere, Apr 12 2019

Eric Weisstein's World of Mathematics, Centered Triangular Number
Eric Weisstein's World of Mathematics, Tetrahedral Number

Intersection of A000292 and A005448.

Cf. A003556, A027568, A128862, A131750, A307492.

cp's OEIS Frontend

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

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A133161 Indices of the triangular numbers which are also centered triangular number.

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A307491 Numbers that are both centered triangular and tetrahedral.

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