A126501 -id:A126501 - cp's OEIS frontend

A254664 Numbers of words on alphabet {0,1,...,8} with no subwords ii, where i is from {0,1,...,5}.

1, 9, 75, 627, 5241, 43809, 366195, 3060987, 25586481, 213874809, 1787757915, 14943687747, 124912775721, 1044133269009, 8727804479235, 72954835640907, 609822098564961, 5097441295442409, 42608996659234155, 356164297160200467

Offset: 0

Milan Janjic, Feb 04 2015

a(n) is the number of nonary sequences of length n such that no two consecutive terms have distance 6. - David Nacin, May 31 2017

Index entries for linear recurrences with constant coefficients, signature (8,3).

Cf. A015574, A055099, A126473, A126501, A126528, A254598, A254602.

[n le 1 select 9^n else 8*Self(n)+3*Self(n-1): n in [0..20]];

RecurrenceTable[{a[0] == 1, a[1] == 9, a[n] == 8 a[n - 1] + 3 a[n - 2]}, a[n], {n, 0, 20}]
LinearRecurrence[{8,3},{1,9},20] (* Harvey P. Dale, Feb 16 2024 *)

G.f.: (1 + x)/(1 - 8*x -3*x^2).
a(n) = 8*a(n-1) + 3*a(n-2) with n>1, a(0) = 1, a(1) = 9.
a(n) = ((1+t)*(4-t)^(n+1)+(-1+t)*(4+t)^(n+1))/(6*t), where t=sqrt(19). [Bruno Berselli, Feb 04 2015]

A179598 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2x)/(1 - 3x - 8*x^2).

Original entry on oeis.org

1, 5, 23, 109, 511, 2405, 11303, 53149, 249871, 1174805, 5523383, 25968589, 122092831, 574027205, 2698824263, 12688690429, 59656665391, 280479519605, 1318691881943, 6199911802669, 29149270463551, 137047105812005

Offset: 0

Johannes W. Meijer, Jul 28 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 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 central square to A179599.
Inverse binomial transform of A126501.

Index entries for linear recurrences with constant coefficients, signature (3, 8).

Cf. A126473 (side squares).

with(LinearAlgebra): nmax:=21; m:=2; 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);

G.f.: (1+2*x)/(1 - 3*x - 8*x^2).
a(n) = 3*a(n-1) + 8*a(n-2) with a(0) = 1 and a(1) = 5.
a(n) = ((41+5*sqrt(41))*A^(-n-1) + (41-5*sqrt(41))*B^(-n-1))/328 with A = (-3+sqrt(41))/16 and B = (-3-sqrt(41))/16.

A235118 Number of independent vertex subsets of the graph obtained by attaching two pendant edges to each vertex of the ladder graph L_n (L_n is the 2 X n grid graph; see A235117).

Original entry on oeis.org

1, 24, 544, 12416, 283136, 6457344, 147267584, 3358621696, 76597559296, 1746902974464, 39840303284224, 908607856050176, 20721936531193856, 472589633411088384, 10777996606218174464, 245805668662673145856, 5605905156051426082816, 127849665915439602991104

Offset: 0

Emeric Deutsch, Jan 14 2014

Row sums of A235117.

Colin Barker, Table of n, a(n) for n = 0..700
E. Mandrescu, Unimodality of some independence polynomials via their palindromicity, Australasian J. of Combinatorics, 53, 2012, 77-82.
D. Stevanovic, Graphs with palindromic independence polynomial, Graph Theory Notes of New York, 34, 1998, 31-36.
Index entries for linear recurrences with constant coefficients, signature (20,64).

Cf. A235117.

G := (1+4*x)/(1-20*x-64*x^2): Gser := series(G, x = 0, 22): seq(coeff(Gser, x, j), j = 0 .. 20);

Vec((1 + 4*x) / (1 - 20*x - 64*x^2) + O(x^30)) \\ Colin Barker, Jul 31 2017

a(0)=1, a(1)=24, a(n) = 20*a(n-1) + 64*a(n-2) for n>=2.
G.f.: (1 + 4*x)/(1 - 20*x - 64*x^2).
a(n) = (((-7+sqrt(41))*(-2*(-5+sqrt(41)))^n + (2*(5+sqrt(41)))^n*(7+sqrt(41))) / (2*sqrt(41))). - Colin Barker, Jul 31 2017
a(n) = 4^n*A126501(n). - R. J. Mathar, Jul 26 2022

A254599 Numbers of words on alphabet {0,1,...,9} with no subwords ii, for i from {0,1}.

Original entry on oeis.org

1, 10, 98, 962, 9442, 92674, 909602, 8927810, 87627106, 860066434, 8441614754, 82855064258, 813228496354, 7981896981250, 78342900802082, 768941283068738, 7547214754035298, 74076463050867586, 727065885490090658, 7136204673817756610, 70042369148280534754

Offset: 0

Milan Janjic, Feb 02 2015

a(n) is the number of sequences over {0,1,...,9} of length n such that no two consecutive terms have distance 9. - David Nacin, May 31 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,8).

Cf. A015584, A055099, A126473, A126501, A126528.

[n le 1 select 10^n else 9*Self(n)+8*Self(n-1): n in [0..20]]; // Bruno Berselli, Feb 02 2015

RecurrenceTable[{a[0] == 1, a[1] == 10, a[n] == 9 a[n - 1] + 8 a[n - 2]}, a[n], {n, 0, 20}] (* Bruno Berselli, Feb 02 2015 *)

Vec((1 + x)/(1 - 9*x - 8*x^2) + O(x^30)) \\ Colin Barker, Jan 22 2017

a(n) = 9*a(n-1) + 8*a(n-2) with n>1, a(0) = 1, a(1) = 10.
G.f.: (1 + x)/(1 - 9*x - 8*x^2). - Bruno Berselli, Feb 02 2015
a(n) = (2^(-1-n)*((9-r)^n*(-11+r) + (9+r)^n*(11+r))) / r, where r=sqrt(113). - Colin Barker, Jan 22 2017

A254659 Numbers of words on alphabet {0,1,...,8} with no subwords ii, where i is from {0,1,2,3}.

Original entry on oeis.org

1, 9, 77, 661, 5673, 48689, 417877, 3586461, 30781073, 264180889, 2267352477, 19459724261, 167014556473, 1433415073089, 12302393367077, 105586222302061, 906201745251873, 7777545073525289, 66751369314461677, 572898679883319861, 4916946285638867273

Offset: 0

Milan Janjic, Feb 04 2015

a(n) is the number of nonary sequences of length n such that no two consecutive terms have distance 7. - David Nacin, May 31 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,5).

Cf. A015575, A055099, A126473, A126501, A126528, A254598, A254602.

[n le 1 select 9^n else 8*Self(n)+5*Self(n-1): n in [0..20]];

RecurrenceTable[{a[0] == 1, a[1] == 9, a[n] == 8 a[n - 1] + 5 a[n - 2]}, a[n], {n, 0, 20}]

Vec((1 + x)/(1 - 8*x -5*x^2) + O(x^30)) \\ Colin Barker, Jan 22 2017

G.f.: (1 + x)/(1 - 8*x - 5*x^2).
a(n) = 8*a(n-1) + 5*a(n-2) with n>1, a(0) = 1, a(1) = 9.
a(n) = ((4-r)^n*(-5+r) + (4+r)^n*(5+r)) / (2*r), where r=sqrt(21). - Colin Barker, Jan 22 2017

A254662 Numbers of words on alphabet {0,1,...,7} with no subwords ii, where i is from {0,1,...,4}.

Original entry on oeis.org

1, 8, 59, 437, 3236, 23963, 177449, 1314032, 9730571, 72056093, 533584364, 3951258827, 29259564881, 216670730648, 1604473809179, 11881328856197, 87982723420916, 651523050515003, 4824609523867769, 35726835818619392, 264561679301939051, 1959112262569431533

Offset: 0

Milan Janjic, Feb 04 2015

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,3).

Cf. A015559, A055099, A126473, A126501, A126528, A254598, A254602.

[n le 1 select 8^n else 7*Self(n)+3*Self(n-1): n in [0..20]];

RecurrenceTable[{a[0] == 1, a[1] == 8, a[n] == 7 a[n - 1] + 3 a[n - 2]}, a[n], {n, 0, 20}]

Vec((1+x)/(1-7*x-3*x^2) + O(x^30)) \\ Colin Barker, Sep 08 2016

G.f.: (1 + x)/(1 - 7*x -3*x^2).
a(n) = 7*a(n-1) + 3*a(n-2) with n>1, a(0) = 1, a(1) = 8.
a(n) = (2^(-1-n) * ((7-sqrt(61))^n * (-9+sqrt(61)) + (7+sqrt(61))^n * (9+sqrt(61)))) / sqrt(61). - Colin Barker, Sep 08 2016

cp's OEIS Frontend

A254664 Numbers of words on alphabet {0,1,...,8} with no subwords ii, where i is from {0,1,...,5}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A179598 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2*x)/(1 - 3*x - 8*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A235118 Number of independent vertex subsets of the graph obtained by attaching two pendant edges to each vertex of the ladder graph L_n (L_n is the 2 X n grid graph; see A235117).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Formula

A254599 Numbers of words on alphabet {0,1,...,9} with no subwords ii, for i from {0,1}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A254659 Numbers of words on alphabet {0,1,...,8} with no subwords ii, where i is from {0,1,2,3}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A254662 Numbers of words on alphabet {0,1,...,7} with no subwords ii, where i is from {0,1,...,4}.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A179598 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2x)/(1 - 3x - 8*x^2).