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, 11, 1320, 157080, 18691211, 2224097040, 264648856560, 31490989833611, 3747163141343160, 445880922830002440, 53056082653628947211, 6313227954859014715680, 751221070545569122218720, 89388994166967866529312011, 10636539084798630547865910600

Offset: 1

Author

Paul Weisenhorn, Jun 14 2009

Keywords

• nonn
• easy

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

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

Crossrefs

Cf. A004190, A078922 (sequence of A), A097783 (sequence of B), A085550, A160682.

Programs

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, 11, 1320}, 20] (* Harvey P. Dale, Apr 01 2024, corrected by Amiram Eldar, Dec 02 2024 *)

Formula

a(n) = 120*(a(n-1) - a(n-2)) + a(n-3).

a(n) = ((11+w)*((119+11*w)/2)^(n-1) + (11-w)*((119-11*w)/2)^(n-1) - 22)/234 where w = sqrt(117). [corrected by Amiram Eldar, Dec 02 2024]

a(n) = floor((11+w)*((119+11*w)/2)^(n-1) - 21)/234. [corrected by Amiram Eldar, Dec 02 2024]

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

From Amiram Eldar, Dec 02 2024: (Start)

a(n) == 0 (mod 11).

a(n) = A004190(n-2)*A004190(n-1), for n >= 2.

Sum_{n>=2} 1/a(n) = ((sqrt(13)-3)/2)^2 = A085550^2. (End)

Extensions

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

Missing term a(2) = 11 inserted by Amiram Eldar, Dec 02 2024