A161588 The list of the k values in the common solutions to the 2 equations 11*k+1=A^2, 15*k+1=B^2.

0, 13, 2184, 364728, 60907405, 10171171920, 1698524803248, 283643470970509, 47366761127271768, 7909965464783414760, 1320916865857702993165, 220585206632771616443808, 36836408590807002243122784, 6151459649458136602985061133, 1027256925050918005696262086440

Offset: 1

Author

Paul Weisenhorn, Jun 14 2009

Keywords

• nonn
• easy

Comments

The 2 equations are equivalent to the Pell equation x^2-165*y^2=1,

with x=(165*k+13)/2 and y=A*B/2, case C=11 in A160682.

Links

• Index entries for linear recurrences with constant coefficients, signature (168,-168,1).

Crossrefs

Cf. A160682, A085260 (sequence of A), A126816 (sequence of B).

Programs

Maple

t:=0: for n from 0 to 1000000 do a:=sqrt(11*n+1): b:=sqrt(15*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)=168*(k(t+2)-k(t+1))+k(t).

k(t)=((13+w)*((167+13*w)/2)^(t-1)+(13-w)*((167-13*w)/2)^(t-1))/330 where w=sqrt(165).

k(t) = floor of ((13+w)*((167+13*w)/2)^(t-1))/330;

G.f.: -13*x^2/((x-1)*(x^2-167*x+1)).

Extensions

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