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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A077423 Chebyshev sequence U(n,12)=S(n,24) with Diophantine property.

Original entry on oeis.org

1, 24, 575, 13776, 330049, 7907400, 189447551, 4538833824, 108742564225, 2605282707576, 62418042417599, 1495427735314800, 35827847605137601, 858372914787987624, 20565122107306565375, 492704557660569581376
Offset: 0

Views

Author

Wolfdieter Lang, Nov 29 2002

Keywords

Comments

b(n)^2 - 143*a(n)^2 = 1 with the companion sequence b(n)=A077424(n+1).
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 24's along the main diagonal, and i's along the subdiagonal and the superdiagonal (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,...,23}. - Milan Janjic, Jan 25 2015

Links

Crossrefs

Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077912 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), this sequence (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).
Cf. A323182.

Programs

  • GAP
    a:=[1,24];; for n in [3..20] do a[n]:=24*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 22 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( 1/(1-24*x+x^2) )); // G. C. Greubel, Dec 22 2019
  • Maple
    seq( simplify(ChebyshevU(n, 12)), n=0..20); # G. C. Greubel, Dec 22 2019
  • Mathematica
    Table[GegenbauerC[n, 1, 12], {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)
ChebyshevU[Range[21] -1, 12] (* G. C. Greubel, Dec 22 2019 *)
  • PARI
    vector(21, n, polchebyshev(n-1, 2, 12) ) \\ G. C. Greubel, Dec 22 2019
  • Sage
    [lucas_number1(n,24,1) for n in range(1,20)] # Zerinvary Lajos, Jun 25 2008

Formula

a(n) = 24*a(n-1) - a(n-2), a(-1) = 0, a(0) = 1.
a(n) = S(n, 24) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the 2nd kind. See A049310.
a(n) = (ap^(n+1) - am^(n+1))/(ap - am) with ap= 12+sqrt(143) and am = 12-sqrt(143).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*24^(n-2*k).
a(n) = sqrt((A077424(n+1)^2 - 1)/143).
G.f.: 1/(1-24*x+x^2). - Philippe Deléham, Nov 18 2008
a(n) = Sum_{k=0..n} A101950(n,k)*23^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/11*(11 + sqrt(143)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 1/24*(11 + sqrt(143)). - Peter Bala, Dec 23 2012

A323118 a(n) = U_{n}(n) where U_{n}(x) is a Chebyshev polynomial of the second kind.

Original entry on oeis.org

1, 2, 15, 204, 3905, 96030, 2883167, 102213944, 4178507265, 193501094490, 10011386405999, 572335117886532, 35827847605137601, 2437406399741075126, 179059769134174484415, 14127079203550978667760, 1191321539697176278429697, 106935795565608726499866930
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2019

Keywords

Links

Crossrefs

Main diagonal of A323182.
Cf. A115066, A318192, A323117, A349073, A349074, A349075, A349076.

Programs

  • Mathematica
    Table[ChebyshevU[n, n], {n, 0, 20}] (* Vaclav Kotesovec, Jan 05 2019 *)
  • PARI
    a(n) = polchebyshev(n, 2, n);
  • PARI
    a(n) = sum(k=0, n\2, (n^2-1)^k*n^(n-2*k)*binomial(n+1, 2*k+1));
  • PARI
    a(n) = sum(k=0, n, (2*n-2)^k*binomial(n+1+k, 2*k+1)); \\ Seiichi Manyama, Mar 03 2021

Formula

a(n) = Sum_{k=0..floor(n/2)} (n^2-1)^k*n^(n-2*k) * binomial(n+1,2*k+1).
a(n) ~ 2^n * n^n. - Vaclav Kotesovec, Jan 05 2019
a(n) = Sum_{k=0..n} (2*n-2)^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} (2*n-2)^k * binomial(n+1+k,2*k+1). - Seiichi Manyama, Mar 03 2021

A322836 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{n}(x), evaluated at x=k.

Original entry on oeis.org

1, 1, 0, 1, 1, -1, 1, 2, 1, 0, 1, 3, 7, 1, 1, 1, 4, 17, 26, 1, 0, 1, 5, 31, 99, 97, 1, -1, 1, 6, 49, 244, 577, 362, 1, 0, 1, 7, 71, 485, 1921, 3363, 1351, 1, 1, 1, 8, 97, 846, 4801, 15124, 19601, 5042, 1, 0, 1, 9, 127, 1351, 10081, 47525, 119071, 114243, 18817, 1, -1
Offset: 0

Views

Author

Seiichi Manyama, Dec 28 2018

Keywords

Examples

			Square array begins:
   1, 1,    1,     1,      1,      1,       1, ...
   0, 1,    2,     3,      4,      5,       6, ...
  -1, 1,    7,    17,     31,     49,      71, ...
   0, 1,   26,    99,    244,    485,     846, ...
   1, 1,   97,   577,   1921,   4801,   10081, ...
   0, 1,  362,  3363,  15124,  47525,  120126, ...
  -1, 1, 1351, 19601, 119071, 470449, 1431431, ...

Links

Crossrefs

Mirror of A101124.
Columns 0-20 give A056594, A000012, A001075, A001541, A001091, A001079, A023038, A011943(n+1), A001081, A023039, A001085, A077422, A077424, A097308, A097310, A068203, A322888, A056771, A322889, A078986, A322890.
Rows 0-10 give A000012, A001477, A056220, A144129, A144130, A243131, A243132, A243133, A243134, A243135, A243136.
Main diagonal gives A115066.
Cf. A323182 (Chebyshev polynomial of the second kind).

Programs

  • Mathematica
    Table[ChebyshevT[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Dec 28 2018 *)
  • PARI
    T(n,k) = polchebyshev(n,1,k);
matrix(7, 7, n, k, T(n-1,k-1)) \\ Michel Marcus, Dec 28 2018
  • PARI
    T(n, k) = round(cos(n*acos(k)));\\ Seiichi Manyama, Mar 05 2021
  • PARI
    T(n, k) = if(n==0, 1, n*sum(j=0, n, (2*k-2)^j*binomial(n+j, 2*j)/(n+j))); \\ Seiichi Manyama, Mar 05 2021

Formula

A(0,k) = 1, A(1,k) = k and A(n,k) = 2 * k * A(n-1,k) - A(n-2,k) for n > 1.
A(n,k) = cos(n*arccos(k)). - Seiichi Manyama, Mar 05 2021
A(n,k) = n * Sum_{j=0..n} (2*k-2)^j * binomial(n+j,2*j)/(n+j) for n > 0. - Seiichi Manyama, Mar 05 2021
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