A161584 The list of the k values in the common solutions to the 2 equations 13*k+1=A^2, 17*k+1=B^2.

0, 15, 3360, 749280, 167086095, 37259449920, 8308690246080, 1852800665425935, 413166239699737440, 92134218652376023200, 20545517593240153436175, 4581558289073901840243840, 1021666952945886870220940160

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 of A160682.

Links

• Index entries for linear recurrences with constant coefficients, signature (224, -224, 1).

Crossrefs

Cf. A160682 (sequence of A), A161591 (sequence of B)

Programs

Maple

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

Mathematica

LinearRecurrence[{224,-224,1},{0,15,3360},20] (* Harvey P. Dale, Nov 22 2013 *)

Formula

k(t+3)=224*(k(t+2)-k(t+1))+k(t).

k(t)=((15+w)*((223+15*w)/2)^(t-1)+(15-w)*((223-15*w)/2)^(t-1))/442 where w=sqrt(221).

k(t) = floor of ((15+w)*((223+15*w)/2)^(t-1))/442;

G.f.: -15*x^2/((x-1)*(x^2-223*x+1)).

a(1)=0, a(2)=15, a(3)=3360, a(n)=224*a(n-1)-224*a(n-2)+a(n-3). - Harvey P. Dale, Nov 22 2013

Extensions

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