A161591 The list of the B values in the common solutions to the 2 equations 13*k + 1 = A^2, 17*k + 1 = B^2.

1, 16, 239, 3569, 53296, 795871, 11884769, 177475664, 2650250191, 39576277201, 590993907824, 8825332340159, 131788991194561, 1968009535578256, 29388354042479279, 438857301101610929, 6553471162481684656, 97863210136123658911, 1461394680879373199009

Offset: 1

Author

Paul Weisenhorn, Jun 14 2009

Keywords

• nonn

Comments

The 2 equations are equivalent to the Pell equation x^2 - 221*y^2 = 1, with x = (221*k+15)/2 and y = A*B/2, case C=13 in A160682.

Links

• Andersen, K., Carbone, L. and Penta, D., 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 (15,-1).

Crossrefs

Cf. A160682 (sequence of A), A161584 (sequence of k).

Programs

Maple

t:=0: for b from 1 to 1000000 do a:=sqrt((13*b^2+4)/17):
if (trunc(a)=a) then t:=t+1: n:=(b^2-1)/17: print(t,a,b,n): end if: end do:

Mathematica

LinearRecurrence[{15,-1},{1,16},30] (* Harvey P. Dale, Dec 04 2015 *)

Sage

[(lucas_number2(n,15,1)-lucas_number2(n-1,15,1))/13 for n in range(1, 20)] # Zerinvary Lajos, Nov 10 2009

Formula

B(t+2) = 15*B(t+1) - B(t).

B(t) = ((221+17*w)*((15+w)/2)^(t-1) + (221-17*w)*((15-w)/2)^(t-1))/442 where w=sqrt(221).

B(t) = floor of ((221+17*w)*((15+w)/2)^(t-1))/442 = A078364(t-2) + A078364(t-1).

G.f.: x*(1+x)/(1-15*x+x^2).

Extensions

Edited, extended by R. J. Mathar, Sep 02 2009