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A076140 Triangular numbers T(k) that are three times another triangular number: T(k) such that T(k) = 3*T(m) for some m. 16

%I #78 Apr 29 2022 03:22:25

%S 0,3,45,630,8778,122265,1702935,23718828,330360660,4601330415,

%T 64088265153,892634381730,12432793079070,173166468725253,

%U 2411897769074475,33593402298317400,467895734407369128,6516946879404850395,90769360577260536405,1264254101202242659278

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

%C This is a subsequence of A045943. - _Michel Marcus_, Apr 26 2014

%H Vincenzo Librandi, <a href="/A076140/b076140.txt">Table of n, a(n) for n = 0..200</a>

%H Vladimir Pletser, <a href="https://arxiv.org/abs/2101.00998">Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers</a>, arXiv:2101.00998 [math.NT], 2021.

%H Vladimir Pletser, <a href="https://arxiv.org/abs/2102.12392">Closed Form Equations for Triangular Numbers Multiple of Other Triangular Numbers</a>, arXiv:2102.12392 [math.GM], 2021.

%H Vladimir Pletser, <a href="https://arxiv.org/abs/2102.13494">Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations</a>, arXiv:2102.13494 [math.NT], 2021.

%H Vladimir Pletser, <a href="https://www.researchgate.net/profile/Vladimir-Pletser/publication/359808848_USING_PELL_EQUATION_SOLUTIONS_TO_FIND_ALL_TRIANGULAR_NUMBERS_MULTIPLE_OF_OTHER_TRIANGULAR_NUMBERS/">Using Pell equation solutions to find all triangular numbers multiple of other triangular numbers</a>, 2022.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-15,1).

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

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

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

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

%F a(n) = 3*A076139(n) = 3/2*A217855(n) = 3/4*A123480(n) = 3/8*A045899(n). - _Peter Bala_, Dec 31 2012

%F a(0) = 0, a(n) = 14 * a(n - 1) - a(n - 2) + 3 for n > 0. - _Vladimir Pletser_, Mar 23 2020

%F 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

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

%t Join[{0}, CoefficientList[Series[3/(1 - 15x + 15x^2 - x^3), {x, 0, 20}], x]] (* _Harvey P. Dale_, Apr 02 2011 *)

%t triNums = Accumulate[Range[0, 9999]]; Select[triNums, MemberQ[triNums, #/3] &] (* _Alonso del Arte_, Mar 24 2020 *)

%o (PARI) concat(0, Vec(-3*x/((x-1)*(x^2-14*x+1)) + O(x^100))) \\ _Colin Barker_, May 15 2015

%Y Subsequence of A000217.

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

%Y Cf. A076139, A045899, A123480, A217855.

%Y Cf. A045943.

%K easy,nonn

%O 0,2

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

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

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)