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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

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Author

Paul Weisenhorn, May 23 2009

Keywords

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

Crossrefs

Cf. similar sequences listed in A238379.

Programs

  • 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
  • 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 *)
  • 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

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

Extensions

Edited, extended by R. J. Mathar, Sep 02 2009
First formula corrected by Harvey P. Dale, Oct 08 2012

A161595 The list of the A values in the common solutions to the 2 equations 15*k+1=A^2, 19*k+1=B^2.

Original entry on oeis.org

1, 16, 271, 4591, 77776, 1317601, 22321441, 378146896, 6406175791, 108526841551, 1838550130576, 31146825378241, 527657481299521, 8939030356713616, 151435858582831951, 2565470565551429551, 43461563755791470416, 736281113282903567521, 12473317362053569177441
Offset: 1

Views

Author

Paul Weisenhorn, Jun 14 2009

Keywords

Comments

The 2 equations are equivalent to the Pell equation x^2- 285*y^2=1,
with x=(285*k+17)/2 and y=A*B/2, case C=15 in A160682.
Also: the first differences of A078366.
Positive values of x (or y) satisfying x^2 - 17xy + y^2 + 15 = 0. - Colin Barker, Feb 14 2014

Links

Crossrefs

Cf. A078366, A160682, A161599 (sequence of B), A161583 (sequence of k).
Cf. similar sequences listed in A238379.

Programs

  • Maple
    t:=0: for a from 1 to 1000000 do b:=sqrt((19*a^2-4)/15):
if (trunc(b)=b) then t:=t+1: n:=(a^2-1)/15: print(t,a,b,n): end if: end do:
  • Mathematica
    Rest[CoefficientList[Series[x (1-x)/(1-17x+x^2),{x,0,40}],x]] (* or *) LinearRecurrence[{17,-1},{1,16},20] (* Harvey P. Dale, Oct 12 2012 *)
  • PARI
    Vec(x*(1-x)/(1-17*x+x^2) + O(x^100)) \\ Colin Barker, Feb 14 2014

Formula

a(t+2) = 17*a(t+1)-a(t).
a(t) = ((285+15*w)*((17+w)/2)^(t-1)+(285-15*w)*((17-w)/2)^(t-1))/570, where w=sqrt(285).
a(t) = ceiling of ((285+15*w)*((17+w)/2)^(t-1))/570.
G.f.: x*(1-x)/(1-17*x+x^2).
a(n) = 17*a(n-1)-a(n-2). - Colin Barker, Feb 14 2014

Extensions

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

A161583 The list of the k values in the common solutions to the 2 equations 15*k+1=A^2, 19*k+1=B^2.

Original entry on oeis.org

0, 17, 4896, 1405152, 403273745, 115738159680, 33216448554432, 9533004996962321, 2735939217679631712, 785205022469057339040, 225351105509401776672785, 64674982076175840847750272, 18561494504756956921527655296, 5327084247883170460637589319697
Offset: 1

Views

Author

Paul Weisenhorn, Jun 14 2009

Keywords

Comments

The 2 equations are equivalent to the Pell equation x^2-285*y^2=1,
with x=(285*k+17)/2 and y=A*B/2, case C=15 in A160682.

Links

Crossrefs

Cf. A160682, A161595 (sequence of A), A161599 (sequence of B)

Programs

  • Maple
    t:=0: for n from 0 to 1000000 do a:=sqrt(15*n+1): b:=sqrt(19*n+1):
if (trunc(a)=a) and (trunc(b)=b) then t:=t+1: print(t,n,a,b): end if: end do:

Formula

k(t+3)=288*(k(t+2)-k(t+1))+k(t).
k(t)=((17+w)*((287+17*w)/2)^(t-1)+(17-w)*((287-17*w)/2)^(t-1))/570 where w=sqrt(285).
k(t) = floor of ((17+w)*((287+17*w)/2)^(t-1))/570;
G.f.: -17*x^2/((x-1)*(x^2-287*x+1)).

Extensions

Edited, extended by R. J. Mathar, Sep 02 2009
Showing 1-3 of 3 results.

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