A077401 Second member of Diophantine pair (m,k) that satisfies 7*(m^2 + m) = k^2 + k; a(n) = k.

0, 6, 14, 104, 231, 1665, 3689, 26543, 58800, 423030, 937118, 6741944, 14935095, 107448081, 238024409, 1712427359, 3793455456, 27291389670, 60457262894, 434949807368, 963522750855, 6931905528225, 15355906750793, 110475538644239, 244730985261840

Offset

0,2

Links

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

Vladimir Pletser, Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers, arXiv:2101.00998 [math.NT], 2021.

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 (1,16,-16,-1,1).

Formula

G.f.: x*(6 + 8*x - 6*x^2 - x^3)/((1-x)*(1 - 16*x^2 + x^4)).

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]

From Vladimir Pletser, Jul 26 2020: (Start)

Let b(n) be A077400(n); then a(n) = (-1 + sqrt(8*b(n) + 1))/2.

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

a(n) = a(n-1) + 16*a(n-2) - 16*a(n-3) - a(n-4) + a(n-5). (End)

Maple

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

Mathematica

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 *)

Prog

(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))

Crossrefs

Cf. A077399, A077400. The m values are in A077398.

Sequence in context: A213681 A308489 A301425 * A364530 A263695 A158965

Adjacent sequences: A077398 A077399 A077400 * A077402 A077403 A077404

Keyword

nonn,easy

Author

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

Status

approved