A077262 Second member of the Diophantine pair (m,k) that satisfies 5*(m^2 + m) = k^2 + k; a(n) = k.

0, 5, 14, 99, 260, 1785, 4674, 32039, 83880, 574925, 1505174, 10316619, 27009260, 185124225, 484661514, 3321919439, 8696898000, 59609425685, 156059502494, 1069647742899, 2800374146900, 19194049946505, 50250675141714, 344423251294199, 901711778403960

Offset

0,2

Comments

The first member of the (m,k) pairs are in A077259.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

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 (1,18,-18,-1,1).

Formula

a(n) = (-1 + sqrt(8*b(n) + 1))/2 where b(n) are the entries in A077261.

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

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

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]

From Vladimir Pletser, Jul 26 2020: (Start)

Can be defined for negative n by setting a(-n) = - a(n-1) - 1 for all n in Z.

a(n) = a(n-1) + 18*a(n-2) - 18*a(n-3) - a(n-4) + a(n-5). (End)

Example

a(3) = (-1 + sqrt(8*4950 + 1))/2 = (-1 + sqrt(39601))/2 = (199 - 1)/2 = 99.

Maple

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

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A077259, A077260, A077261.

Sequence in context: A024167 A316421 A317424 * A184439 A370006 A316233

Adjacent sequences: A077259 A077260 A077261 * A077263 A077264 A077265

Keyword

easy,nonn

Author

Bruce Corrigan (scentman(AT)myfamily.com), Nov 01 2002

Status

approved