A076140 Triangular numbers T(k) that are three times another triangular number: T(k) such that T(k) = 3*T(m) for some m.

0, 3, 45, 630, 8778, 122265, 1702935, 23718828, 330360660, 4601330415, 64088265153, 892634381730, 12432793079070, 173166468725253, 2411897769074475, 33593402298317400, 467895734407369128, 6516946879404850395, 90769360577260536405, 1264254101202242659278

Offset: 0

Author

Bruce Corrigan (scentman(AT)myfamily.com), Oct 31 2002

Keywords

• easy
• nonn

Comments

This is a subsequence of A045943. - Michel Marcus, Apr 26 2014

Examples

			a(3) = 630 because 630 = T(35) and 630/3 = 210 = T(20).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Vladimir Pletser, Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers, arXiv:2101.00998 [math.NT], 2021.
• Vladimir Pletser, Closed Form Equations for Triangular Numbers Multiple of Other Triangular Numbers, arXiv:2102.12392 [math.GM], 2021.
• Vladimir Pletser, Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations, arXiv:2102.13494 [math.NT], 2021.
• Vladimir Pletser, Using Pell equation solutions to find all triangular numbers multiple of other triangular numbers, 2022.
• Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Crossrefs

Subsequence of A000217.

The m values are in A061278 and the k values are in A001571.

Cf. A076139, A045899, A123480, A217855.

Cf. A045943.

Programs

Mathematica

Join[{0}, CoefficientList[Series[3/(1 - 15x + 15x^2 - x^3), {x, 0, 20}], x]]  (* Harvey P. Dale, Apr 02 2011 *)
triNums = Accumulate[Range[0, 9999]]; Select[triNums, MemberQ[triNums, #/3] &] (* Alonso del Arte, Mar 24 2020 *)

PARI

concat(0, Vec(-3*x/((x-1)*(x^2-14*x+1)) + O(x^100))) \\ Colin Barker, May 15 2015

Formula

a(n) = (3/288)*(-24 + (12 - 6*sqrt(3))*(7 - 4*sqrt(3))^n + (12 + 6*sqrt(3))*(7 + 4*sqrt(3))^n).

From Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002: (Start)

a(0) = 0, a(1) = 3, a(2) = 45; a(n) = 15*(a(n-1) -a (n-2)) + a(n-3) for n >= 3.

G.f.: (3*x)/(1 - 15*x + 15*x^2 - x^3). (End)

a(n) = 3*A076139(n) = 3/2*A217855(n) = 3/4*A123480(n) = 3/8*A045899(n). - Peter Bala, Dec 31 2012

a(0) = 0, a(n) = 14 * a(n - 1) - a(n - 2) + 3 for n > 0. - Vladimir Pletser, Mar 23 2020

a(n) = ((2+sqrt(3))*(7+4*sqrt(3))^n + ((2-sqrt(3))*(7-4*sqrt(3))^n))/16 - 1/4 = ((2+sqrt(3))^(2n+1) + ((2-sqrt(3))^(2n+1)))/16 - 1/4. - Vladimir Pletser, Jan 15 2021

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002