cp's OEIS Frontend

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.

Previous Showing 11-12 of 12 results.

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.

Original entry on oeis.org

0, 11, 1320, 157080, 18691211, 2224097040, 264648856560, 31490989833611, 3747163141343160, 445880922830002440, 53056082653628947211, 6313227954859014715680, 751221070545569122218720, 89388994166967866529312011, 10636539084798630547865910600
Offset: 1

Views

Author

Paul Weisenhorn, Jun 14 2009

Keywords

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

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

A269028 a(n) = 40*a(n - 1) - a(n - 2) for n>1, a(0) = 1, a(1) = 1.

Original entry on oeis.org

1, 1, 39, 1559, 62321, 2491281, 99588919, 3981065479, 159143030241, 6361740144161, 254310462736199, 10166056769303799, 406387960309415761, 16245352355607326641, 649407706263983649879, 25960062898203738668519, 1037753108221885563090881
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 18 2016

Keywords

Comments

In general, the ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - b(n - 2) with n>1 and b(0)=1, b(1)=1, is (1 - (k - 1)*x)/(1 - k*x +x^2). This recurrence gives the closed form b(n) = (2^( -n - 1)*((k - 2)*(k - sqrt(k^2 - 4))^n + sqrt(k^2 - 4)*(k - sqrt(k^2 - 4))^n - (k - 2)*(sqrt(k^2 - 4) + k)^n + sqrt(k^2 - 4)*(sqrt(k^2 - 4) + k)^n))/sqrt(k^2 - 4).

Links

Crossrefs

Cf. A001519, A001835, A001653, A049685, A070997, A070998, A072256, A078922, A160682, A007805, A075839, A157014, A159664, A159668, A157877, A238379, A097315.

Programs

  • Magma
    [n le 2 select 1 else 40*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 19 2016
  • Mathematica
    Table[Cosh[n Log[20 + Sqrt[399]]] - Sqrt[19/21] Sinh[n Log[20 + Sqrt[399]]], {n, 0, 17}]
Table[(2^(-n - 2) (38 (40 - 2 Sqrt[399])^n + 2 Sqrt[399] (40 - 2 Sqrt[399])^n - 38 (40 + 2 Sqrt[399])^n + 2 Sqrt[399] (40 + 2 Sqrt[399])^n))/Sqrt[399], {n, 0, 17}]
LinearRecurrence[{40, -1}, {1, 1}, 17]

Formula

G.f.: (1 - 39*x)/(1 - 40*x + x^2).
a(n) = cosh(n*log(20 + sqrt(399))) - sqrt(19/21)*sinh(n*log(20 + sqrt(399))).
a(n) = (2^(-n - 2)*(38*(40 - 2*sqrt(399))^n + 2*sqrt(399)*(40 - 2*sqrt(399))^n - 38*(40 + 2*sqrt(399))^n + 2*sqrt(399)*(40 + 2*sqrt(399))^n))/sqrt(399).
Sum_{n>=0} 1/a(n) = 2.0262989201139499769986...
Previous Showing 11-12 of 12 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.