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Triangular numbers that are 7 times triangular numbers.
11

%I #45 Sep 08 2022 08:45:07

%S 0,21,105,5460,26796,1386945,6806205,352278696,1728749400,89477401965,

%T 439095541521,22726907820540,111528538797060,5772545109015321,

%U 28327809758911845,1466203730782071120,7195152150224811696,372409975073537049285,1827540318347343259065

%N Triangular numbers that are 7 times triangular numbers.

%H Colin Barker, <a href="/A077400/b077400.txt">Table of n, a(n) for n = 0..831</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/2102.13494">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_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,254,-254,-1,1).

%F Let b(n) = A077399(n) then a(n) = 7*b(n).

%F G.f.: -21*x*(x^2+4*x+1) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)). - _Colin Barker_, Jul 02 2013

%F From __Vladimir Pletser_, Feb 21 2021: (Start)

%F a(n) = 254*a(n - 2) - a (n - 4) + 126.

%F a(n) = a(n - 1) + 254*(a(n - 2) - a(n - 3)) - (a (n - 4) - a(n - 5)). (End)

%p f := gfun:-rectoproc({a(-2) = 21, a(-1) = 0, a(0) = 0, a(1) = 21, a(n) = 254*a(n-2)-a(n-4)+126}, a(n), remember): map(f, [`$`(0 .. 1000)])[] # _Vladimir Pletser_, Feb 21 2021

%t LinearRecurrence[{1,254,-254,-1,1},{0,21,105,5460,26796},20] (* _Harvey P. Dale_, Oct 28 2013 *)

%o (PARI) concat(0, Vec(-21*x*(x^2+4*x+1)/((x-1)*(x^2-16*x+1)*(x^2+16*x+1)) + O(x^100))) \\ _Colin Barker_, May 15 2015

%o (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q,30); [0] cat Coefficients(R!(-21*x*(x^2+4*x+1) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)))); // _G. C. Greubel_, Jan 18 2018

%Y Cf. A077398, A077399, A077401.

%K easy,nonn

%O 0,2

%A Bruce Corrigan (scentman(AT)myfamily.com), Nov 05 2002