A161583 The list of the k values in the common solutions to the 2 equations 15*k+1=A^2, 19*k+1=B^2.

0, 17, 4896, 1405152, 403273745, 115738159680, 33216448554432, 9533004996962321, 2735939217679631712, 785205022469057339040, 225351105509401776672785, 64674982076175840847750272, 18561494504756956921527655296, 5327084247883170460637589319697

Offset: 1

Author

Paul Weisenhorn, Jun 14 2009

Keywords

• nonn

Comments

The 2 equations are equivalent to the Pell equation x^2-285*y^2=1,

with x=(285*k+17)/2 and y=A*B/2, case C=15 in A160682.

Links

• Index entries for linear recurrences with constant coefficients, signature (288, -288, 1).

Crossrefs

Cf. A160682, A161595 (sequence of A), A161599 (sequence of B)

Programs

Maple

t:=0: for n from 0 to 1000000 do a:=sqrt(15*n+1): b:=sqrt(19*n+1):
if (trunc(a)=a) and (trunc(b)=b) then t:=t+1: print(t,n,a,b): end if: end do:

Formula

k(t+3)=288*(k(t+2)-k(t+1))+k(t).

k(t)=((17+w)*((287+17*w)/2)^(t-1)+(17-w)*((287-17*w)/2)^(t-1))/570 where w=sqrt(285).

k(t) = floor of ((17+w)*((287+17*w)/2)^(t-1))/570;

G.f.: -17*x^2/((x-1)*(x^2-287*x+1)).

Extensions

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