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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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A126473 Number of strings over a 5 symbol alphabet with adjacent symbols differing by three or less.

Original entry on oeis.org

1, 5, 23, 107, 497, 2309, 10727, 49835, 231521, 1075589, 4996919, 23214443, 107848529, 501037445, 2327695367, 10813893803, 50238661313, 233396326661, 1084301290583, 5037394142315, 23402480441009, 108722104190981, 505095858086951, 2346549744920747
Offset: 0

Views

Author

R. H. Hardin, Dec 27 2006

Keywords

Comments

[Empirical] a(base,n) = a(base-1,n) + 7^(n-1) for base >= 3n-2; a(base,n) = a(base-1,n) + 7^(n-1)-2 when base = 3n-3.
From Johannes W. Meijer, Aug 01 2010: (Start)
The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 or 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
For the side squares the 512 red kings lead to 47 different red king sequences, see the cross-references for some examples.
The sequence above corresponds to four A[5] vectors with the decimal [binary] values 367 [1,0,1,1,0,1,1,1,1], 463 [1,1,1,0,0,1,1,1,1], 487 [1,1,1,1,0,0,1,1,1] and 493 [1,1,1,1,0,1,1,0,1]. These vectors lead for the corner squares to A179596 and for the central square to A179597.
This sequence belongs to a family of sequences with g.f. (1+x)/(1-4*x-k*x^2). Red king sequences that are members of this family are A003947 (k=0), A015448 (k=1), A123347 (k=2), A126473 (k=3; this sequence) and A086347 (k=4). Other members of this family are A000351 (k=5), A001834 (k=-1), A111567 (k=-2), A048473 (k=-3) and A053220 (k=-4)
Inverse binomial transform of A154244. (End)
Equals the INVERT transform of A055099: (1, 4, 14, 50, 178, ...). - Gary W. Adamson, Aug 14 2010
Number of one-sided n-step walks taking steps from {E, W, N, NE, NW}. - Shanzhen Gao, May 10 2011
For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,2,3,4} containing no subwords 00 and 11. - Milan Janjic, Jan 31 2015

Links

Crossrefs

Cf. 5 symbol differing by two or less A126392, one or less A057960.
Cf. Red king sequences side squares [numerical value A[5]]: A086347 [495], A179598 [239], A126473 [367], A123347 [335], A179602 [95], A154964 [31], A015448 [327], A152187 [27], A003947 [325], A108981 [11], A007483 [2]. - Johannes W. Meijer, Aug 01 2010
Cf. A055099.

Programs

  • Maple
    with(LinearAlgebra): nmax:=19; m:=2; A[5]:= [1,0,1,1,0,1,1,1,1]: A:=Matrix([[0,1,0,1,1,0,0,0,0],[1,0,1,1,1,1,0,0,0],[0,1,0,0,1,1,0,0,0],[1,1,0,0,1,0,1,1,0],A[5],[0,1,1,0,1,0,0,1,1],[0,0,0,1,1,0,0,1,0],[0,0,0,1,1,1,1,0,1],[0,0,0,0,1,1,0,1,0]]): 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); # Johannes W. Meijer, Aug 01 2010
# second Maple program:
a:= n-> (M-> M[1,2]+M[2,2])(<<0|1>, <3|4>>^n):
seq(a(n), n=0..24);  # Alois P. Heinz, Jun 28 2021
  • Mathematica
    LinearRecurrence[{4, 3}, {1, 5}, 24] (* Jean-François Alcover, Dec 10 2024 *)
  • PARI
    a(n)=([0,1; 3,4]^n*[1;5])[1,1] \\ Charles R Greathouse IV, May 10 2016

Formula

From Johannes W. Meijer, Aug 01 2010: (Start)
G.f.: (1+x)/(1-4*x-3*x^2).
a(n) = 4*a(n-1) + 3*a(n-2) with a(0) = 1 and a(1) = 5.
a(n) = ((1+3/sqrt(7))/2)*(A)^(-n) + ((1-3/sqrt(7))/2)*(B)^(-n) with A = (-2 + sqrt(7))/3 and B = (-2-sqrt(7))/3.
Lim_{k->oo} a(n+k)/a(k) = (-1)^(n+1)*A000244(n)/(A015530(n)*sqrt(7)-A108851(n))
(End)
a(n) = A015330(n)+A015330(n+1). - R. J. Mathar, May 09 2023

Extensions

Edited by Johannes W. Meijer, Aug 10 2010

A152187 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=1, a(1)=5.

Original entry on oeis.org

1, 5, 20, 85, 355, 1490, 6245, 26185, 109780, 460265, 1929695, 8090410, 33919705, 142211165, 596232020, 2499751885, 10480415755, 43940006690, 184222098845, 772366329985, 3238209484180, 13576460102465, 56920427728295
Offset: 0

Views

Author

Philippe Deléham, Nov 28 2008

Keywords

Comments

Unsigned version of A152185.
From Johannes W. Meijer, Aug 01 2010: (Start)
The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 and 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
The sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the corner squares to A015523 and for the central square to A179606.
This sequence belongs to a family of sequences with g.f. (1+2*x)/(1 - 3*x - k*x^2). Red king sequences that are members of this family are A007483 (k=2), A108981 (k=4), A152187 (k=5; this sequence), A154964 (k=6), A179602 (k=7) and A179598 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A036563 (k=-2), A054486 (k=-1), A084244 (k=0), A108300 (k=1) and A000351 (k=10).
Inverse binomial transform of A015449 (without the first leading 1).
(End)

Links

Crossrefs

Cf. A015523, A072263, A072264, A179606, A197189.

Programs

  • Mathematica
    LinearRecurrence[{3,5},{1,5},40] (* Harvey P. Dale, May 03 2013 *)

Formula

G.f.: (1+2*x)/(1 - 3*x - 5*x^2).
Lim_{k->infinity} a(n+k)/a(k) = (A072263(n) + A015523(n)*sqrt(29))/2. - Johannes W. Meijer, Aug 01 2010
G.f.: G(0)*(1+2*x)/(2-3*x), where G(k) = 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013

A179599 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 4*x)/(1 - 3*x - 8*x^2).

Original entry on oeis.org

1, 7, 29, 143, 661, 3127, 14669, 69023, 324421, 1525447, 7171709, 33718703, 158529781, 745338967, 3504255149, 16475477183, 77460472741, 364185235687, 1712239488989, 8050200352463, 37848516969301, 177947153727607
Offset: 0

Views

Author

Johannes W. Meijer, Jul 28 2010

Keywords

Comments

The a(n) represent the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
The sequence above corresponds to 10 red king vectors, i.e., A[5] vectors, with decimal values 239, 351, 375, 381, 431, 471, 477, 491, 494 and 501. These vectors lead for the corner squares to A015525 and for the side squares to A179598.

Links

Crossrefs

Cf. A179597 (central square).

Programs

  • Maple
    with(LinearAlgebra): nmax:=22; m:=5; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:= [1,1,0,0,1,0,1,1,0]: A[5]:= [1,0,1,1,1,1,1,0,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: A:=Matrix([A[1],A[2],A[3],A[4],A[5],A[6],A[7],A[8],A[9]]): 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);

Formula

G.f.: (1+4*x)/(1 - 3*x - 8*x^2).
a(n) = 3*a(n-1) + 8*a(n-2) with a(0) = 1 and a(1) = 7.
a(n) = ((11+4*A)*A^(-n-1) + (11+4*B)*B^(-n-1))/41 with A = (-3+sqrt(41))/16 and B = (-3-sqrt(41))/16.
Showing 1-3 of 3 results.

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