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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.

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A055099 Expansion of g.f.: (1 + x)/(1 - 3*x - 2*x^2).

Original entry on oeis.org

1, 4, 14, 50, 178, 634, 2258, 8042, 28642, 102010, 363314, 1293962, 4608514, 16413466, 58457426, 208199210, 741512482, 2640935866, 9405832562, 33499369418, 119309773378, 424928058970, 1513403723666, 5390067288938, 19197009314146, 68371162520314, 243507506189234
Offset: 0

Views

Author

Wolfdieter Lang, Apr 26 2000

Keywords

Comments

Row sums of triangle A054458.
a(n) = term (1,1) in M^n, M = the 3 X 3 matrix [1,1,1; 1,1,1; 2,2,1]. - Gary W. Adamson, Mar 12 2009
Equals the INVERT transform of A001333: (1, 3, 7, 17, 41, 99, ...). - Gary W. Adamson, Aug 14 2010
a(n) is the number of one sided n-step walks taking steps from {(0,1), (-1,0), (1,0), (1,1)}. - Shanzhen Gao, May 13 2011
Number of quaternary words of length n on {0,1,2,3} containing no subwords 03 or 30. - Philippe Deléham, Apr 27 2012
Pisano period lengths: 1, 1, 4, 1, 24, 4, 48, 1, 12, 24, 30, 4, 12, 48, 24, 2, 272, 12, 18, 24, ... - R. J. Mathar, Aug 10 2012
a(n) = A007481(2*n+1) - A007481(2*n) = A007481(2*(n+1)) - A007481(2*n+1). - Reinhard Zumkeller, Oct 25 2015
Number of length-n words on a,b,c,d avoiding aa and ab. For n >= 1, the number of such words ending with a or the number of those ending with b is A007482(n-1), and the number of those ending with c or the number of those ending with d is a(n-1). - Jianing Song, Jun 01 2022

Examples

			a(3) = 50 because among the 4^3 = 64 quaternary words of length 3 only 14 namely 003, 030, 031, 032, 033, 103, 130, 203, 230, 300, 301, 302, 303, 330 contain the subwords 03 or 30. - _Philippe Deléham_, Apr 27 2012

References

  • I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (Problem 2.4.6).

Links

Crossrefs

Column k=2 of A255256.
Cf. A001333, A002203, A007481, A007482, A054458, A104934, A202396.

Programs

  • Haskell
    a055099 n = a007481 (2 * n + 1) - a007481 (2 * n)
-- Reinhard Zumkeller, Oct 25 2015
  • Magma
    I:=[1,4]; [n le 2 select I[n] else 3*Self(n-1) + 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jun 27 2021
  • Maple
    a := proc(n) option remember; `if`(n < 2, [1, 4][n+1], (3*a(n-1) + 2*a(n-2))) end:
seq(a(n), n=0..23); # Peter Luschny, Jan 06 2019
  • Mathematica
    max = 24; cv = ContinuedFraction[ Sqrt[2], max] // Convergents // Numerator; Series[ 1/(1 - cv.x^Range[max]), {x, 0, max}] // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Jun 21 2013, after Gary W. Adamson *)
LinearRecurrence[{3, 2}, {1, 4}, 24] (* Jean-François Alcover, Sep 23 2017 *)
  • Sage
    [(i*sqrt(2))^(n-1)*( i*sqrt(2)*chebyshev_U(n, -3*i/(2*sqrt(2))) + chebyshev_U(n-1, -3*i/(2*sqrt(2))) ) for n in (0..40)] # G. C. Greubel, Jun 27 2021

Formula

a(n) = a*c^n - b*d^n, a := (5 + sqrt(17))/(2*sqrt(17)), b := (5 - sqrt(17))/(2*sqrt(17)), c := (3 + sqrt(17))/2, d := (3 - sqrt(17))/2.
a(n) = Sum_{m=0..n} A054458(n, m).
a(n) = F32(n) + F32(n-1) with F32(n) = A007482(n), n >= 1, a(0) = 1.
a(n) = A007482(n) + A007482(n-1) = 2*A007482(n) - A104934(n). - R. J. Mathar, Jul 23 2010
a(n) = 3*a(n-1) + 2*a(n-2) with a(0) = 1, a(1) = 4. - Vincenzo Librandi, Dec 08 2010
a(n) = (Sum_{k = 0..n} A202396(n,k)*3^k)/2^n. - Philippe Deléham, Feb 05 2012
a(n) = (i*sqrt(2))^(n-1)*( i*sqrt(2)*ChebyshevU(n, -3*i/(2*sqrt(2))) + ChebyshevU(n-1, -3*i/(2*sqrt(2))) ). - G. C. Greubel, Jun 27 2021
a(n) = 2*a(n-1) + 2*A007482(n-1), n >= 1. - Jianing Song, Jun 01 2022
E.g.f.: exp(3*x/2)*(17*cosh(sqrt(17)*x/2) + 5*sqrt(17)*sinh(sqrt(17)*x/2))/17. - Stefano Spezia, May 24 2024

Extensions

Edited by N. J. A. Sloane, Jun 08 2010

A180035 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1+x)/(1-5*x-3*x^2).

Original entry on oeis.org

1, 6, 33, 183, 1014, 5619, 31137, 172542, 956121, 5298231, 29359518, 162692283, 901539969, 4995776694, 27683503377, 153404846967, 850074744966, 4710588265731, 26103165563553, 144647592614958, 801547459765449
Offset: 0

Views

Author

Johannes W. Meijer, Aug 09 2010

Keywords

Comments

The a(n) represent the number of n-move routes of a fairy chess piece starting in the corner and side squares (m = 1, 3, 7, 9; 2, 4, 6, 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a white queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen, see A180032.
The sequence above corresponds to 56 red queen vectors, i.e. A[5] vector, with decimal values between 7 and 448. The central squares lead for these vectors to A180036.
For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,2,3,5} containing no subwords 00, 11 and 22. - Milan Janjic, Jan 31 2015

Links

Programs

  • Magma
    I:=[1,6]; [n le 2 select I[n] else 5*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011
  • Maple
    with(LinearAlgebra): nmax:=20; m:=1; A[5]:= [0,0,0,0,0,0,1,1,1]: A:=Matrix([[0,1,1,1,1,0,1,0,1], [1,0,1,1,1,1,0,1,0], [1,1,0,0,1,1,1,0,1], [1,1,0,0,1,1,1,1,0], A[5], [0,1,1,1,1,0,0,1,1], [1,0,1,1,1,0,0,1,1], [0,1,0,1,1,1,1,0,1], [1,0,1,0,1,1,1,1,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);
  • Mathematica
    LinearRecurrence[{5,3},{1,6},50] (* Vincenzo Librandi, Nov 15 2011 *)

Formula

G.f.: (1+x)/(1-5*x-3*x^2).
a(n) = 5*a(n-1) + 3*a(n-2) with a(0) = 1 and a(1) = 6.
a(n) = ((7+A)*A^(-n-1)+(7+B)*B^(-n-1))/37 with A = (-5+sqrt(37))/6 and B = (-5-sqrt(37))/6.
a(n) = Sum_{k, 0<=k<=n} A202396(n,k)*2^k. - Philippe Deléham, Dec 21 2011
a(n) = A015536(n+1)+A015536(n). - R. J. Mathar, Aug 04 2019
Showing 1-2 of 2 results.

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

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