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.

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A092499 Chebyshev polynomials S(n-1,21) with Diophantine property.

Original entry on oeis.org

0, 1, 21, 440, 9219, 193159, 4047120, 84796361, 1776676461, 37225409320, 779956919259, 16341869895119, 342399310878240, 7174043658547921, 150312517518628101, 3149388824232642200, 65986852791366858099
Offset: 0

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Author

Rainer Rosenthal, Apr 05 2004

Keywords

Comments

Sequence R_21: Starts with 0,1,21 and satisfies A*C=B^2-1 for successive A,B,C.
The natural numbers a(n)=n satisfy the recurrence a(n-1)*a(n+1)=a(n)^2-1. Let R_r denote the sequence starting with 0,1,r and with this recurrence. We see that R_2 = "the natural numbers" and we find R_3 = A001906. These R_r form a "family" of sequences, which coincides with the m-family (r=m-2, n -> n+1) provided by Wolfdieter Lang (see A078368). This sequence R_21 is strongly related to A041833, which gives the denominators in the continued fraction of sqrt(437).
All positive integer solutions of Pell equation b(n)^2 - 437*a(n)^2 = +4 together with b(n)=A097777(n), n>=0.
For n>=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 21's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,20}. - Milan Janjic, Jan 25 2015

Examples

			a(3)=440 because a(1)*440 = a(2)^2-1.

Links

Crossrefs

Cf. R_3=A001906, R_4=A001353, R_5=A004254, R_6=A001109, R_7=A004187, R_8=A001090, R_9=A018913, R_10=A004189, R_11=A004190, R_12=A004191, R_13=A078362, R_14=A007655, R_15=A078364, R_16=A077412, R_17=A078366, R_18=A049660, R_19=A078368, R_20=A075843, R_21=this, sequence, R_22=A077421. See also A041219 and A041917.

Programs

  • Mathematica
    LinearRecurrence[{21,-1},{0,1},30] (* Harvey P. Dale, Apr 23 2015 *)
  • Sage
    [lucas_number1(n,21,1) for n in range(0,20)] # Zerinvary Lajos, Jun 25 2008

Formula

a(0)=0, a(1)=1, a(2)=21 and a(n-1)*a(n+1) = a(n)^2-1
a(n) = S(n-1, 21)=U(n-1, 21/2) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x)= 0 = U(-1, x).
a(n) = S(2*n-1, sqrt(23))/sqrt(23), n>=1.
a(n) = 21*a(n-1)-a(n-2), n >= 1; a(0)=0, a(1)=1.
a(n) = (ap^n-am^n)/(ap-am) with ap := (21+sqrt(437))/2 and am := (21-sqrt(437))/2.
G.f.: x/(1-21*x+x^2).
a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*20^k. - Philippe Deléham, Feb 10 2012
Product {n >= 1} (1 + 1/a(n)) = 1/19*(19 + sqrt(437)). - Peter Bala, Dec 23 2012
Product {n >= 2} (1 - 1/a(n)) = 1/42*(19 + sqrt(437)). - Peter Bala, Dec 23 2012

Extensions

Extension, Chebyshev and Pell comments from Wolfdieter Lang, Aug 31 2004
Corrected by T. D. Noe, Nov 07 2006

A092444 a(n+1) = 11*a(n) - a(n-1) - 3, a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 7, 73, 793, 8647, 94321, 1028881, 11223367, 122428153, 1335486313, 14567921287, 158911647841, 1733460204961, 18909150606727, 206267196469033, 2250030010552633, 24544062919609927, 267734662105156561
Offset: 0

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Author

N. J. A. Sloane, Sep 19 2008, based on emails from M. F. Hasler and Jim Nastos on Apr 25 2008

Keywords

Comments

The old entry with this sequence number was a duplicate of A039963.
The simultaneous equations (p+1)(p+2) == -1 (mod q), (q+1)(q+2) == -1 (mod p), where p and q are odd, have solutions {3, 3}, {3, 21}, {7, 73}, {21, 507}, {73, 793}, {793, 8647} and suggested this recurrence.

Links

Programs

  • Mathematica
    nxt[{a_,b_}]:={b,11b-a-3}; NestList[nxt,{1,1},20][[;;,1]] (* or *) LinearRecurrence[{12,-12,1},{1,1,7},20] (* Harvey P. Dale, Jul 06 2025 *)

Formula

a(n) = 2*(A004190(n)-10*A004190(n-1))/3+1/3. G.f.: (1-11x+7x^2)/((1-x)(1-11x+x^2)). [From R. J. Mathar, Sep 20 2008]

Extensions

More terms from R. J. Mathar, Sep 20 2008

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
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