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

Author

Paul Weisenhorn, Apr 19 2009

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A077412, A157456, A159679, A266698.

Programs

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

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 *)

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

Sage

[(lucas_number2(n,16,1)-lucas_number2(n-1,16,1))/14 for n in range(1, 20)] # Zerinvary Lajos, Nov 10 2009

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

Extensions

More terms from Zerinvary Lajos, Nov 10 2009