A155116 -id:A155116 - cp's OEIS frontend

A123620 Expansion of (1 + x + x^2) / (1 - 3x - 3x^2).

1, 4, 16, 60, 228, 864, 3276, 12420, 47088, 178524, 676836, 2566080, 9728748, 36884484, 139839696, 530172540, 2010036708, 7620627744, 28891993356, 109537863300, 415289569968, 1574482299804, 5969315609316, 22631393727360, 85802128010028, 325300565212164

Offset: 0

N. J. A. Sloane, Nov 20 2006

From Johannes W. Meijer, Aug 14 2010: (Start)
A berserker sequence, see A180141. For the corner squares 16 A[5] vectors with decimal values between 3 and 384 lead to this sequence. These vectors lead for the side squares to A180142 and for the central square to A155116.
This sequence belongs to a family of sequences with GF(x) = (1+x+k*x^2)/(1-3*x+(k-4)*x^2). Berserker sequences that are members of this family are 4*A055099(n) (k=2; with leading 1 added), A123620 (k=1; this sequence), A000302 (k=0), 4*A179606 (k=-1; with leading 1 added) and A180141 (k=-2). Some other members of this family are 4*A003688 (k=3; with leading 1 added), 4*A003946 (k=4; with leading 1 added), 4*A002878 (k=5; with leading 1 added) and 4*A033484 (k=6; with leading 1 added).
(End)
a(n) is the number of length n sequences on an alphabet of 4 letters that do not contain more than 2 consecutive equal letters. For example, a(3)=60 because we count all 4^3=64 words except: aaa, bbb, ccc, ddd. - Geoffrey Critzer, Mar 12 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, arXiv:math/0112281 [math.CO], 2001.
A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 1-14.
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 205
Index entries for linear recurrences with constant coefficients, signature (3,3).

Column 4 in A265584.

[1] cat [Round(((2^(1-n)*(-(3-Sqrt(21))^(1+n) + (3+Sqrt(21))^(1+n))))/(3*Sqrt(21))): n in [1..50]]; // G. C. Greubel, Oct 26 2017

nn=25;CoefficientList[Series[(1-z^(m+1))/(1-r z +(r-1)z^(m+1))/.{r->4,m->2},{z,0,nn}],z] (* Geoffrey Critzer, Mar 12 2014 *)
CoefficientList[Series[(1 + x + x^2)/(1 - 3 x - 3 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 14 2014 *)
LinearRecurrence[{3,3},{1,4,16},30] (* Harvey P. Dale, Jul 14 2023 *)

my(x='x+O('x^50)); Vec((1+x+x^2)/(1-3*x-3*x^2)) \\ G. C. Greubel, Oct 16 2017

a(0)=1, a(1)=4, a(2)=16, a(n)=3*a(n-1)+3*a(n-2) for n>2. - Philippe Deléham, Sep 18 2009

a(n) = ((2^(1-n)*(-(3-sqrt(21))^(1+n) + (3+sqrt(21))^(1+n)))) / (3*sqrt(21)) for n>0. - Colin Barker, Oct 17 2017

A180142 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - x^2)/(1 - 3x - 3x^2).

Original entry on oeis.org

1, 4, 14, 54, 204, 774, 2934, 11124, 42174, 159894, 606204, 2298294, 8713494, 33035364, 125246574, 474845814, 1800277164, 6825368934, 25876938294, 98106921684, 371951579934, 1410175504854, 5346381254364, 20269670277654, 76848154596054, 291353474621124

Offset: 0

Johannes W. Meijer, Aug 13 2010

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 rook on the eight side and corner squares but on the central square the rook goes berserk and turns into a berserker, see A180140.
The sequence above corresponds to 16 A[5] vectors with decimal values between 3 and 384. These vectors lead for the corner squares to A123620 and for the central square to A155116.
This sequence appears among the members of a family of sequences with g.f. (1 + x - k*x^2)/(1 - 3*x + (k-4)*x^2). Berserker sequences that are members of this family are 4*A007482 (k=2; with leading 1 added), A180142 (k=1; this sequence), A000302 (k=0), A180140 (k=-1) and 4*A154964 (k=-2; n>=1 and a(0)=1). Some other members of this family are 2*A180148 (k=3; with leading 1 added), 4*A025192 (k=4; with leading 1 added), 2*A005248 (k=5; with leading 1 added) and A123932 (k=6).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,3).

Cf. A180141 (corner squares), A180140 (side squares), A180147 (central square).

with(LinearAlgebra): nmax:=23; m:=2; A[5]:=[0,0,0,0,0,0,0,1,1]: A:= Matrix([[0,1,1,1,0,0,1,0,0], [1,0,1,0,1,0,0,1,0], [1,1,0,0,0,1,0,0,1], [1,0,0,0,1,1,1,0,0], A[5], [0,0,1,1,1,0,0,0,1], [1,0,0,1,0,0,0,1,1], [0,1,0,0,1,0,1,0,1], [0,0,1,0,0,1,1,1,0]]): 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);
# second Maple program:
a:= n-> ceil((<<0|1>, <3|3>>^n. <<2/3, 4>>)[1,1]):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 14 2021

LinearRecurrence[{3, 3}, {1, 4, 14}, 26] (* Jean-François Alcover, Jan 18 2025 *)

G.f.: (1 + x - x^2)/(1 - 3*x - 3*x^2).
a(n) = 3*a(n-1) + 3*a(n-2) for n >= 2 with a(0)=1, a(1)=4 and a(2)=14.
a(n) = (6-2*A)*A^(-n-1)/21 + (6-2*B)*B^(-n-1)/21 with A=(-3+sqrt(21))/6 and B=(-3-sqrt(21))/6.
Lim_{k->infinity} a(2*n+k)/a(k) = 2*A000244(n)/(A003501(n) - A004254(n)*sqrt(21)) for n >= 1.
Lim_{k->infinity} a(2*n-1+k)/a(k) = 2*A000244(n)/(A004253(n)*sqrt(21) - 3*A030221(n-1)) for n >= 1.

A370173 Riordan array (1-x-x^2, x*(1+x)).

Original entry on oeis.org

1, -1, 1, -1, 0, 1, 0, -2, 1, 1, 0, -1, -2, 2, 1, 0, 0, -3, -1, 3, 1, 0, 0, -1, -5, 1, 4, 1, 0, 0, 0, -4, -6, 4, 5, 1, 0, 0, 0, -1, -9, -5, 8, 6, 1, 0, 0, 0, 0, -5, -15, -1, 13, 7, 1, 0, 0, 0, 0, -1, -14, -20, 7, 19, 8, 1, 0, 0, 0, 0, 0, -6, -29, -21, 20, 26, 9, 1

Offset: 0

Philippe Deléham, Feb 27 2024

Triangle T(n,k) read by rows : matrix product of A155112*A130595.

Triangle T(n,k), read by rows, given by [-1, 2, -1/2, -1/2, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

			Triangle T(n,k) begins:
  1;
 -1,  1;
 -1,  0,  1;
  0, -2,  1,  1;
  0, -1, -2,  2, 1;
  0,  0, -3, -1, 3, 1;
...

Cf. A000007, A155020, A155116, A155117, A155119, A155127, A155130, A155132, A155144, A155157.

Cf. A084938, A130595, A155112.

from functools import cache
@cache
def T(n, k):
    if k > n: return 0
    if n == 0: return 1
    if k == 0: return -1 if n == 1 or n == 2 else 0
    return T(n-1, k-1) + T(n-2, k-1)
for n in range(9):
    print([T(n, k) for k in range(n+1)])  # Peter Luschny, Feb 28 2024

T(n,k) = T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = -1, T(n,0) = 0 for n>2, T(n,k) = 0 if k>n.
T(n,k) = Sum_{j = k..n} A155112(n,j)*A130595(j,k).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A155020(n), A155116(n), A155117(n), A155119(n), A155127(n), A155130(n), A155132(n), A155144(n), A155157(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.

cp's OEIS Frontend

A123620 Expansion of (1 + x + x^2) / (1 - 3x - 3x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A180142 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - x^2)/(1 - 3x - 3x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A370173 Riordan array (1-x-x^2, x*(1+x)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

Formula

cp's OEIS Frontend

A123620 Expansion of (1 + x + x^2) / (1 - 3*x - 3*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A180142 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - x^2)/(1 - 3*x - 3*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A370173 Riordan array (1-x-x^2, x*(1+x)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

Formula

A123620 Expansion of (1 + x + x^2) / (1 - 3x - 3x^2).

A180142 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - x^2)/(1 - 3x - 3x^2).