%I #42 Feb 16 2024 06:38:07
%S 0,6,14,104,231,1665,3689,26543,58800,423030,937118,6741944,14935095,
%T 107448081,238024409,1712427359,3793455456,27291389670,60457262894,
%U 434949807368,963522750855,6931905528225,15355906750793,110475538644239,244730985261840
%N Second member of Diophantine pair (m,k) that satisfies 7*(m^2 + m) = k^2 + k; a(n) = k.
%H Colin Barker, <a href="/A077401/b077401.txt">Table of n, a(n) for n = 0..1000</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 (1,16,-16,-1,1).
%F G.f.: x*(6 + 8*x - 6*x^2 - x^3)/((1-x)*(1 - 16*x^2 + x^4)).
%F a(n) = 16*a(n-2) - a(n-4) + 7, n >= 2 with a(-2)=-7, a(-1)=-1, a(0)=0, a(1)=6. [Corrected by _Vladimir Pletser_, Feb 29 2020, Jul 26 2020]
%F From _Vladimir Pletser_, Jul 26 2020: (Start)
%F Let b(n) be A077400(n); then a(n) = (-1 + sqrt(8*b(n) + 1))/2.
%F Can be defined for negative n by setting a(-n) = - a(n-1) - 1 for all n in Z.
%F a(n) = a(n-1) + 16*a(n-2) - 16*a(n-3) - a(n-4) + a(n-5). (End)
%p f := gfun:-rectoproc({a(-2) = -7, a(-1) = -1, a(0) = 0, a(1) = 6, a(n) = 16*a(n - 2) - a(n - 4) + 7}, a(n), remember); map(f, [$ (0 .. 40)])[]; # _Vladimir Pletser_, Jul 26 2020
%t CoefficientList[Series[x (6 + 8 x - 6 x^2 - x^3)/((1 - x) (1 - 16 x^2 + x^4)), {x, 0, 24}], x] (* _Michael De Vlieger_, Apr 21 2021 *)
%o (PARI) a(n)=if(n<0,0,polcoeff(x*(6+8*x-6*x^2-x^3)/((1-x)*(1-16*x^2+x^4))+x*O(x^n),n))
%Y Cf. A077399, A077400. The m values are in A077398.
%K nonn,easy
%O 0,2
%A Bruce Corrigan (scentman(AT)myfamily.com), Nov 05 2002