A161585 The list of the k values in the common solutions to the 2 equations 7*k+1=A^2, 11*k+1=B^2.

0, 9, 720, 56880, 4492809, 354875040, 28030635360, 2214065318409, 174883129518960, 13813553166679440, 1091095817038156809, 86182755992847708480, 6807346627617930813120, 537694200825823686528009, 42471034518612453304899600, 3354674032769557987400540400

Offset: 1

Author

Paul Weisenhorn, Jun 14 2009

Keywords

• nonn
• easy

Comments

The 2 equations are equivalent to the Pell equation x^2-77*y^2=1,

with x=(77*k+9)/2 and y= A*B/2, case C=7 in A160682.

Links

• Index entries for linear recurrences with constant coefficients, signature (80,-80,1)

Crossrefs

Cf. A160682, A070998 (sequence of A), A057081 (sequence of B)

Programs

Maple

t:=0: for n from 0 to 1000000 do a:=sqrt(7*n+1): b:=sqrt(11*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[{80,-80,1},{0,9,720},20] (* Harvey P. Dale, Jun 07 2023 *)

Formula

k(t+3)=80*(k(t+2)-k(t+1))+k(t).

k(t)=((9+w)*((79+9*w)/2)^(t-1)+(9-w)*((79-9*w)/2)^(t-1))/154 where w=sqrt(77).

k(t) = floor of ((9+w)*((79+9*w)/2)^(t-1))/154.

G.f.: -9*x^2/((x-1)*(x^2-79*x+1)).

Extensions

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