A160682 The list of the A values in the common solutions to 13*k+1 = A^2 and 17*k+1 = B^2.

1, 14, 209, 3121, 46606, 695969, 10392929, 155197966, 2317576561, 34608450449, 516809180174, 7717529252161, 115246129602241, 1720974414781454, 25699370092119569, 383769576967012081, 5730844284413061646, 85578894689228912609, 1277952576054020627489

Offset

1,2

Comments

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

Index entries for the Pell equation

Index entries for linear recurrences with constant coefficients, signature (15,-1).

Formula

a(n) = 15*a(n-1)-a(n-2).

G.f.: (1-x)*x/(1-15*x+x^2).

a(n) = (2^(-1-n)*((15-sqrt(221))^n*(13+sqrt(221))+(-13+sqrt(221))*(15+sqrt(221))^n))/sqrt(221). - Colin Barker, Jul 25 2016

Mathematica

LinearRecurrence[{15, -1}, {1, 14}, 20] (* Harvey P. Dale, Oct 08 2012 *)
CoefficientList[Series[(1 - x)/(1 - 15 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)

Prog

(Magma) I:=[1, 14]; [n le 2 select I[n] else 15*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 12 2014
(PARI) a(n) = round((2^(-1-n)*((15-sqrt(221))^n*(13+sqrt(221))+(-13+sqrt(221))*(15+sqrt(221))^n))/sqrt(221)) \\ Colin Barker, Jul 25 2016

Crossrefs

Cf. similar sequences listed in A238379.

Sequence in context: A063071 A251963 A192007 * A097261 A158555 A097183

Adjacent sequences: A160679 A160680 A160681 * A160683 A160684 A160685

Keyword

nonn,easy

Author

Paul Weisenhorn, May 23 2009

Extensions

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

First formula corrected by Harvey P. Dale, Oct 08 2012

Status

approved