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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A175658 Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 2*Pell(n+1)+2*Pell(n)-2^n, with Pell = A000129.

Original entry on oeis.org

1, 4, 10, 26, 66, 166, 414, 1026, 2530, 6214, 15214, 37154, 90546, 220294, 535230, 1298946, 3149506, 7630726, 18476494, 44714786, 108168210, 261575494, 632367774, 1528408194, 3693378466, 8923553734, 21557263150, 52071634466
Offset: 0

Views

Author

Johannes W. Meijer, Aug 06 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 bishop on the eight side and corner squares but on the central square the bishop turns into a raging elephant, see A175654.
The sequence above corresponds to 24 A[5] vectors with decimal values 23, 29, 53, 83, 86, 89, 92, 113, 116, 149, 209, 212, 275, 278, 281, 284, 305, 308, 338, 344, 368, 401, 404 and 464. These vectors lead for the side squares to A000079 and for the corner squares to 2*A094723 (a(n)=2*Pell(n+1)-2^n).
From Clark Kimberling, Aug 23 2017 (Start)
p-INVERT of (1,1,1,....), where p(S) = 1-S-2*S^2+2*S^3.
Suppose s = (c(0), c(1), c(2),...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453). See A291000 for a guide to related sequences. (End)

Links

Crossrefs

Cf. A175654, A175655 (central square).
Cf. A000129 (Pell(n)), A078057 (Pell(n)+Pell(n+1)), A094723 (Pell(n+2)-2^n).

Programs

  • Magma
    I:=[1,4,10]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 21 2013
  • Maple
    nmax:=27; m:=5; A[5]:= [0,0,0,0,1,0,1,1,1]: A:=Matrix([[0,0,0,0,1,0,0,0,1], [0,0,0,1,0,1,0,0,0], [0,0,0,0,1,0,1,0,0], [0,1,0,0,0,0,0,1,0], A[5], [0,1,0,0,0,0,0,1,0], [0,0,1,0,1,0,0,0,0], [0,0,0,1,0,1,0,0,0], [1,0,0,0,1,0,0,0,0]]): 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[{4,-3,-2},{1,4,10},30] (* Harvey P. Dale, Jun 18 2013 *)
CoefficientList[Series[(1 - 3 x^2) / (1 - 4 x + 3 x^2 + 2 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)
  • PARI
    Vec((1 - 3*x^2) / ((1 - 2*x)*(1 - 2*x - x^2)) + O(x^30)) \\ Colin Barker, Aug 29 2017

Formula

G.f.: ( 1-3*x^2 ) / ( (2*x-1)*(x^2+2*x-1) ).
a(n) = 4*a(n-1)-3*a(n-2)-2*a(n-3) with a(0)=1, a(1)=4 and a(2)=10.
Limit_{n->oo} a(n+1)/a(n) = 1+sqrt(2).
a(n) = (1-sqrt(2))^(1+n) + (1+sqrt(2))^(1+n) - 2^n. - Colin Barker, Aug 29 2017

A175659 Eight bishops and one elephant on a 3 X 3 chessboard: a(n)= (3^(n+1)-Jacobsthal(n+1))-(3^n-Jacobsthal(n)), with Jacobsthal=A001045.

Original entry on oeis.org

1, 6, 16, 52, 156, 476, 1436, 4332, 13036, 39196, 117756, 353612, 1061516, 3185916, 9560476, 28686892, 86071596, 258236636, 774753596, 2324348172, 6973219276, 20920007356, 62760721116, 188283561452, 564853480556
Offset: 0

Views

Author

Johannes W. Meijer, Aug 06 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 bishop on the eight side and corner squares but on the central square the bishop turns into a raging elephant, see A175654.
The sequence above corresponds to 4 A[5] vectors with decimal values 343, 349, 373 and 469. These vectors lead for the side squares to A000079 and for the corner squares to A093833 (a(n)=3^n-Jacobsthal(n)).

Links

Crossrefs

Cf. A000079, A001045, A093833, A175654, A175655 (central square).

Programs

  • Magma
    I:=[1, 6, 16]; [n le 3 select I[n] else 4*Self(n-1)-Self(n-2)-6*Self(n-3): n in [1..35]]; // Vincenzo Librandi, Jul 21 2013
  • Maple
    nmax:=24; m:=5; A[5]:= [1,0,1,0,1,0,1,1,1]: A:=Matrix([[0,0,0,0,1,0,0,0,1], [0,0,0,1,0,1,0,0,0], [0,0,0,0,1,0,1,0,0], [0,1,0,0,0,0,0,1,0], A[5], [0,1,0,0,0,0,0,1,0], [0,0,1,0,1,0,0,0,0], [0,0,0,1,0,1,0,0,0], [1,0,0,0,1,0,0,0,0]]): 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
    CoefficientList[Series[(1 + 2 x - 7 x^2) / (1 - 4 x + x^2 + 6 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)

Formula

G.f.: (1+2*x-7*x^2)/(1-4*x+x^2+6*x^3).
a(n) = 4*a(n-1)-a(n-2)-6*a(n-3) with a(0)=1, a(1)=6 and a(2)=16.
a(n) = (-2*(-1)^n)/3-2^n/3+2*3^n. [Colin Barker, Oct 07 2012]

A175661 Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 2^(n+2)-3*F(n+1), with F(n) = A000045(n).

Original entry on oeis.org

1, 5, 10, 23, 49, 104, 217, 449, 922, 1883, 3829, 7760, 15685, 31637, 63706, 128111, 257353, 516536, 1036033, 2076857, 4161466, 8335475, 16691245, 33415328, 66883789, 133853549, 267846202, 535917479, 1072199137, 2144987528
Offset: 0

Views

Author

Johannes W. Meijer, Aug 06 2010, Aug 10 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 bishop on the eight side and corner squares but on the central square the bishop turns into a raging elephant, see A175654.
The sequence above corresponds to four A[5] vectors with decimal values 171, 174, 234 and 426. These vectors lead for the side squares to A000079 and for the corner squares to A175660 (a(n)=2^(n+2)-3*F(n+2)).

Links

Crossrefs

Cf. A175655 (central square), A000045.
Cf. A027973 (2^(n+2)+F(n)-F(n+4)), A099036 (2^n-F(n)), A167821 (2^(n+1)-2*F(n+2)), A175657 (3*2^n-2*F(n+1)), A175660 (2^(n+2)-3*F(n+2)), A179610 (convolution of (-4)^n and F(n+1)).

Programs

  • Magma
    I:=[1,5,10]; [n le 3 select I[n] else 3*Self(n-1)-Self(n-2)-2*Self(n-3): n in [1..35]]; // Vincenzo Librandi, Jul 21 2013
  • Maple
    nmax:=29; m:=5; A[5]:= [0,1,0,1,0,1,0,1,1]: A:=Matrix([[0,0,0,0,1,0,0,0,1], [0,0,0,1,0,1,0,0,0], [0,0,0,0,1,0,1,0,0], [0,1,0,0,0,0,0,1,0], A[5], [0,1,0,0,0,0,0,1,0], [0,0,1,0,1,0,0,0,0], [0,0,0,1,0,1,0,0,0], [1,0,0,0,1,0,0,0,0]]): 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
    CoefficientList[Series[(1 + 2 x - 4 x^2) / (1 - 3 x + x^2 + 2 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)
LinearRecurrence[{3,-1,-2},{1,5,10},30] (* Harvey P. Dale, Apr 15 2019 *)

Formula

G.f.: (1 + 2*x - 4*x^2)/(1 - 3*x + x^2 + 2*x^3).
a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) with a(0)=1, a(1)=5 and a(2)=10.
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