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%I #47 Feb 16 2024 06:37:37
%S 0,5,14,99,260,1785,4674,32039,83880,574925,1505174,10316619,27009260,
%T 185124225,484661514,3321919439,8696898000,59609425685,156059502494,
%U 1069647742899,2800374146900,19194049946505,50250675141714,344423251294199,901711778403960
%N Second member of the Diophantine pair (m,k) that satisfies 5*(m^2 + m) = k^2 + k; a(n) = k.
%C The first member of the (m,k) pairs are in A077259.
%H Colin Barker, <a href="/A077262/b077262.txt">Table of n, a(n) for n = 0..1000</a>
%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 (1,18,-18,-1,1).
%F a(n) = (-1 + sqrt(8*b(n) + 1))/2 where b(n) are the entries in A077261.
%F a(n) = (sqrt(5*A000045(A007310(n+1))^2 - 4) - 1)/2. - _Vladeta Jovovic_, Nov 02 2002. - Definition corrected by _R. J. Mathar_, Sep 16 2009
%F G.f.: (x*(x^3+5*x^2-9*x-5))/((x-1)*(x^2-4*x-1)*(x^2+4*x-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
%F a(n) = 18*a(n-2) - a(n-4) + 8. - _Vladimir Pletser_, Mar 23 2020 ; a(-2) = -6, a(-1) = -1, a(0) = 0, a(1) = 5. [Edited by _Vladimir Pletser_, Jul 26 2020]
%F From _Vladimir Pletser_, Jul 26 2020: (Start)
%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) + 18*a(n-2) - 18*a(n-3) - a(n-4) + a(n-5). (End)
%e a(3) = (-1 + sqrt(8*4950 + 1))/2 = (-1 + sqrt(39601))/2 = (199 - 1)/2 = 99.
%p f := gfun:-rectoproc({a(-2) = -6, a(-1) = -1, a(0) = 0, a(1) = 5, a(n) = 18*a(n - 2) - a(n - 4) + 8}, a(n), remember); map(f, [$ (0 .. 40)])[]; #_Vladimir Pletser_, Jul 26 2020
%t CoefficientList[Series[(x (x^3 + 5 x^2 - 9 x - 5))/((x - 1) (x^2 - 4 x - 1) (x^2 + 4 x - 1)), {x, 0, 24}], x] (* _Michael De Vlieger_, Apr 21 2021 *)
%o (PARI) concat(0, Vec(x*(x^3+5*x^2-9*x-5)/((x-1)*(x^2-4*x-1)*(x^2+4*x-1)) + O(x^100))) \\ _Colin Barker_, May 15 2015
%Y Cf. A077259, A077260, A077261.
%K easy,nonn
%O 0,2
%A Bruce Corrigan (scentman(AT)myfamily.com), Nov 01 2002
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