A161586 The list of the k values in the common solutions to the 2 equations 9*k+1=A^2, 13*k+1=B^2.

0, 1320, 157080, 18691211, 2224097040, 264648856560, 31490989833611, 3747163141343160, 445880922830002440, 53056082653628947211, 6313227954859014715680, 751221070545569122218720, 89388994166967866529312011, 10636539084798630547865910600

Offset

1,2

Comments

The 2 equations are equivalent to the Pell equation x^2-117*y^2=1,

with x=(117*k+11)/2 and y=A*B/2, case C=9 in A160682.

Links

Table of n, a(n) for n=1..14.

Index entries for linear recurrences with constant coefficients, signature (120,-120,1).

Formula

k(t+3)=120*(k(t+2)-k(t+1))+k(t).

k(t)=((11+w)*((119+11*w)/2)^(t-1)+(11-w)*((119-11*w)/2)^(t-1))/234 where w=sqrt(117).

k(t) = floor of ((11+w)*((119+11*w)/2)^(t-1))/234;

G.f.: -11*x^2/((x-1)*(x^2-119*x+1)).

Maple

t:=0: for n from 0 to 1000000 do a:=sqrt(9*n+1): b:=sqrt(13*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[{120, -120, 1}, {0, 1320, 157080, 18691211}, 20] (* Harvey P. Dale, Apr 01 2024 *)

Crossrefs

Cf. A160682, A078922 (sequence of A), A097783 (sequence of B).

Sequence in context: A069737 A185464 A323802 * A350612 A013641 A295449

Adjacent sequences: A161583 A161584 A161585 * A161587 A161588 A161589

Keyword

nonn,easy

Author

Paul Weisenhorn, Jun 14 2009

Extensions

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

Status

approved