A161582 The list of the k values in the common solutions to the 2 equations 5*k+1=A^2, 9*k+1=B^2.

0, 7, 336, 15792, 741895, 34853280, 1637362272, 76921173511, 3613657792752, 169764995085840, 7975341111241735, 374671267233275712, 17601574218852716736, 826899317018844410887, 38846666325666834594960, 1824966417989322381552240, 85734574979172485098360327

Offset: 1

Author

Paul Weisenhorn, Jun 14 2009

Keywords

• nonn
• easy

Comments

The 2 equations are equivalent to the Pell equation x^2-45*y^2=1, with x=(45*k+7)/2 and y= A*B/2, case C=5 in A160682.

Links

• Index entries for linear recurrences with constant coefficients, signature (48,-48,1).

Crossrefs

Cf. A160682, A049685 (sequence of A), A033890 (sequence of B).

Programs

Maple

t:=0: for n from 0 to 1000000 do a:=sqrt(5*n+1); b:=sqrt(9*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[{48,-48,1},{0,7,336},30] (* or *) Rest[CoefficientList[ Series[ -7x^2/((x-1)(x^2-47x+1)),{x,0,30}],x]] (* Harvey P. Dale, Mar 21 2013 *)

Formula

k(t+3) = 48*(k(t+2)-k(t+1))+k(t).

With w = sqrt(5),

k(t) = ((7+3*w)*((47+21*w)/2)^(t-1)+(7-3*w)*((47-21*w)/2)^(t-1))/90.

k(t) = floor((7+3*w)*((47+21*w)/2)^(t-1)/90) = 7*|A156093(t-1)|.

G.f.: -7*x^2/((x-1)*(x^2-47*x+1)).

a(1)=0, a(2)=7, a(3)=336, a(n) = 48*a(n-1)-48*a(n-2)+a(n-3). - Harvey P. Dale, Mar 21 2013

Extensions

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