A159678 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2-equation problem 7*n(j) + 1 = a(j)*a(j) and 9*n(j) + 1 = b(j)*b(j) with positive integer numbers.

1, 17, 271, 4319, 68833, 1097009, 17483311, 278635967, 4440692161, 70772438609, 1127918325583, 17975920770719, 286486814005921, 4565813103324017, 72766522839178351, 1159698552323529599, 18482410314337295233, 294558866477073194129, 4694459453318833810831

Offset

1,2

Comments

The sequence a(j) is A157456, the sequence n(j) is A159679, the sequence b(j) the sequence given here.

Numbers k such that 7*k^2 + 2 is a square. - Colin Barker, Mar 17 2014

Links

Colin Barker, Table of n, a(n) for n = 1..800

K. Andersen, L. Carbone, and D. Penta, Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.

Index entries for linear recurrences with constant coefficients, signature (16,-1).

Formula

The b(j) recurrence (this sequence) is b(1)=1, b(2)=17, b(t+2) = 16*b(t+1) - b(t).

From R. J. Mathar, Oct 31 2011: (Start)

G.f.: x*(1+x) / ( 1-16*x+x^2 ).

a(n) = A077412(n-1) + A077412(n-2). (End)

a(n) = 16*a(n-1) - a(n-2), with a(1)=1, a(2)=17. - Harvey P. Dale, Dec 25 2011

a(n) = ( (3-sqrt(7))*(8+3*sqrt(7))^n - (3+sqrt(7))*(8-3*sqrt(7))^n )/(2*sqrt(7)). - Colin Barker, Jul 25 2016

Maple

for a from 1 by 2 to 100000 do b:=sqrt((9*a*a-2)/7): if (trunc(b)=b) then
n:=(a*a-1)/7: La:=[op(La), a]:Lb:=[op(Lb), b]:Ln:=[op(Ln), n]: end if: end do:
# Second program
seq(simplify(ChebyshevU(n-1, 8) + ChebyshevU(n-2, 8)), n=1..30); # G. C. Greubel, Sep 27 2022

Mathematica

Rest[CoefficientList[Series[x (1+x)/(1-16x+x^2), {x, 0, 30}], x]] (* or *) LinearRecurrence[{16, -1}, {1, 17}, 30] (* Harvey P. Dale, Dec 25 2011 *)

Prog

(Sage) [(lucas_number2(n, 16, 1)-lucas_number2(n-1, 16, 1))/14 for n in range(1, 20)] # Zerinvary Lajos, Nov 10 2009
(PARI) Vec(x*(1+x)/(1-16*x+x^2) + O(x^30)) \\ Michel Marcus, Jan 03 2016
(PARI) a(n) = round((-(8-3*sqrt(7))^n*(3+sqrt(7))-(-3+sqrt(7))*(8+3*sqrt(7))^n)/(2*sqrt(7))) \\ Colin Barker, Jul 25 2016
(Magma) [n le 2 select 17^(n-1) else 16*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 03 2018

Crossrefs

Cf. A077412, A157456, A159679, A266698.

Sequence in context: A259849 A090380 A142898 * A162803 A097830 A163093

Adjacent sequences: A159675 A159676 A159677 * A159679 A159680 A159681

Keyword

nonn,easy

Author

Paul Weisenhorn, Apr 19 2009

Extensions

More terms from Zerinvary Lajos, Nov 10 2009

Status

approved