cp's OEIS Frontend

This summarizes the case C=13 of common solutions to C*k+1=A^2, (C+4)*k+1=B^2.

The 2 equations are equivalent to the Pell equation x^2-C*(C+4)*y^2=1,

with x=(C*(C+4)*k+C+2)/2; y=A*B/2 and with smallest values x(1) = (C+2)/2, y(1)=1/2.

Generic recurrences are:

A(j+2)=(C+2)*A(j+1)-A(j) with A(1)=1; A(2)=C+1.

B(j+2)=(C+2)*B(j+1)-B(j) with B(1)=1; B(2)=C+3.

k(j+3)=(C+1)*(C+3)*( k(j+2)-k(j+1) )+k(j) with k(1)=0; k(2)=C+2; k(3)=(C+1)*(C+2)*(C+3).

x(j+2)=(C^2+4*C+2)*x(j+1)-x(j) with x(1)=(C+2)/2; x(2)=(C^2+4*C+1)*(C+2)/2;

Binet-type of solutions of these 2nd order recurrences are:

R=C^2+4*C; S=C*sqrt(R); T=(C+2); U=sqrt(R); V=(C+4)*sqrt(R);

A(j)=((R+S)*(T+U)^(j-1)+(R-S)*(T-U)^(j-1))/(R*2^j);

B(j)=((R+V)*(T+U)^(j-1)+(R-V)*(T-U)^(j-1))/(R*2^j);

x(j)+sqrt(R)*y(j)=((T+U)*(C^2*4*C+2+(C+2)*sqrt(R))^(j-1))/2^j;

k(j)=(((T+U)*(R+2+T*U)^(j-1)+(T-U)*(R+2-T*U)^(j-1))/2^j-T)/R. [Paul Weisenhorn, May 24 2009]

.C -A----- -B----- -k-----

01 A001519 A002878 A058038

02 A001653 A002315 A045899/2

03 A004253 A030221 A160695

04 A001653 A002315 A078522/4

05 A049685 A033890 A161582

06 A070997 A057080 A159683/2

07 A070998 A057081 A161585

08 A072256 A054320 A045502/4

09 A078922 A097783 A161586

10 A077417 A077416 A159681/2

11 A085260 A126816 A161588

12 A001570 A028230 A059989/4

13 A160682 A161591 A161584

14 A157456 A159678 A159679/2

15 A161595 A161599 A161583

16 A007805 A049629 A157459/4

For n>=2, a(n) equals the permanent of the (2n-2)X(2n-2) tridiagonal matrix with sqrt(13)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. [John M. Campbell, Jul 08 2011]

Positive values of x (or y) satisfying x^2 - 15xy + y^2 + 13 = 0. - Colin Barker, Feb 11 2014

Naturally, all terms are triangular numbers.

Numbers m such that k*m+1 and 8*m+1 are both squares:

k=1: A006454;

k=3: this sequence;

k=4: A029549;

k=5: 0, 3, 231, 4560, 333336, 6575751, ...

k=6: A200999;

k=7: A157879.

Numbers m such that 3*m+1 and k*m+1 are both squares:

k=1: A045899;

k=2: A045502;

k=4: A059989;

k=5: A159683;

k=6: 8*A029546;

k=7: A160695;

k=8: this sequence.