A015448 -id:A015448 - cp's OEIS frontend

A179604 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2x - 9x^2 - 2*x^3).

1, 3, 15, 59, 259, 1079, 4607, 19443, 82507, 349215, 1479879, 6267707, 26552755, 112474631, 476459471, 2018296131, 8549676763, 36216937647, 153417558423, 649886909195, 2752965719491, 11661748738583, 49399962770975

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 corner square (m = 1, 3, 7 or 9) 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 4 red king vectors, i.e., A[5] vectors, with decimal [binary] values 327 [1,0,1,0,0,0,1,1,1], 333 [1,0,1,0,0,1,1,0,1], 357 [1,0,1,1,0,0,1,0,1] and 453 [1,1,1,0,0,0,1,0,1]. These vectors lead for the side squares to A015448 and for the central square to A179605.

Index entries for linear recurrences with constant coefficients, signature (2,9,2).

Cf. A001076, A001077, A015448, A179605, A179596.

with(LinearAlgebra): nmax:=22; m:=1; 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]:= [1,0,1,1,0,0,1,0,1]: 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);

LinearRecurrence[{2,9,2},{1,3,15},30] (* or *) CoefficientList[ Series[ (x+1)/(-2 x^3-9 x^2-2 x+1),{x,0,30}],x] (* Harvey P. Dale, Mar 17 2012 *)

G.f.: ( -1-x ) / ( (2*x+1)*(x^2 + 4*x - 1) ).
a(n) = 2*a(n-1) + 9*a(n-2) + 2*a(n-3) with a(0)=1, a(1)=3 and a(2)=15.
a(n) = (20*(-1/2)^(-n) + (5+7*sqrt(5))*A^(-n-1) + (5-7*sqrt(5))*B^(-n-1))/110 with A = (-2+sqrt(5)) and B:= (-2-sqrt(5)).
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n+1)/(A001076(n)*sqrt(5) - A001077(n)).

A179605 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3x - 2x^2)/(1 - 2x - 9x^2 - 2*x^3).

Original entry on oeis.org

1, 5, 17, 81, 325, 1413, 5913, 25193, 106429, 451421, 1911089, 8097825, 34298293, 145299189, 615478665, 2607246617, 11044399597, 46784976077, 198184041761, 839521667409, 3556269662821, 15064602415845, 63814675131897

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 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 toes crazy and turns into a red king, see A179596.

The sequence above corresponds to 4 red king vectors, A[5] vectors, with decimal [binary] values 327 [1,0,1,0,0,0,1,1,1], 333 [1,0,1,0,0,1,1,0,1], 357 [1,0,1,1,0,0,1,0,1] and 453 [1,1,1,0,0,0,1,0,1]. These vectors lead for the corner squares to A179604 and for the side squares to A015448.

Index entries for linear recurrences with constant coefficients, signature (2,9,2).

Cf. A001076, A001077, A015448, A179596, A179597 (central square), A179604.

with(LinearAlgebra): nmax:=21; 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]:= [1,0,1,1,0,0,1,0,1]: 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 - 3*x + 2*x^2 ) / ( (2*x+1)*(x^2 + 4*x - 1) ).
a(n) = 2*a(n-1) + 9*a(n-2) + 2*a(n-3) with a(0)=1, a(1)=5 and a(2)=17.
a(n) = (-4/11)*(-1/2)^(-n) + ((17+41*A)*A^(-n-1) + (17+41*B)*B^(-n-1))/110 with A = (-2+sqrt(5)) and B =(-2-sqrt(5)).
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n+1)/(A001076(n)*sqrt(5) - A001077(n)).

A020712 Pisot sequences E(5,8), P(5,8).

Original entry on oeis.org

5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141

Offset: 0

David W. Wilson

Pisano period lengths: 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60,.. - R. J. Mathar, Aug 10 2012

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1,1).

Subsequence of A020701 and hence A020695, A000045. See A008776 for definitions of Pisot sequences.

Trisections: A015448, A014445, A033887.

CoefficientList[Series[(-3 z - 5)/(z^2 + z - 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{1,1},{5,8},40] (* Harvey P. Dale, Dec 28 2013 *)

a(n)=fibonacci(n+5) \\ Charles R Greathouse IV, Jan 17 2012

a(n) = Fib(n+5). a(n) = a(n-1) + a(n-2).
O.g.f.: (5+3x)/(1-x-x^2). a(n)=A020701(n+1). - R. J. Mathar, May 28 2008
a(n) = (2^(-1-n)*((1-sqrt(5))^n*(-11+5*sqrt(5))+(1+sqrt(5))^n*(11+5*sqrt(5))))/sqrt(5). - Colin Barker, Jun 05 2016

A153764 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 3, 1, 1, 0, 1, 3, 3, 4, 1, 1, 0, 1, 3, 6, 4, 5, 1, 1, 0, 1, 4, 6, 10, 5, 6, 1, 1, 0, 1, 4, 10, 10, 15, 6, 7, 1, 1, 0, 1, 5, 10, 20, 15, 21, 7, 8, 1, 1, 0, 1, 5, 15, 20, 35, 21, 28, 8, 9, 1, 1, 0, 1, 6, 15, 35, 35, 56, 28, 36, 9, 10, 1, 1, 0

Offset: 0

Philippe Deléham, Jan 01 2009

A130595*A153342 as infinite lower triangular matrices. Reflected version of A103631. Another version of A046854. Row sums are Fibonacci numbers (A000045).

A055830*A130595 as infinite lower triangular matrices.

			Triangle begins:
  1;
  1, 0;
  1, 1, 0;
  1, 1, 1, 0;
  1, 2, 1, 1, 0;
  1, 2, 3, 1, 1, 0;
  1, 3, 3, 4, 1, 1, 0;
  ...

G. C. Greubel, Table of n, a(n) for the first 45 rows

Cf. A046854, A103631.

/* As triangle */ [[Binomial(Floor((n+k-1)/2),k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 28 2016

Table[Binomial[Floor[(n + k - 1)/2], k], {n, 0, 45}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 27 2016 *)

T(n,k) = binomial(floor((n+k-1)/2),k).
Sum_{k=0..n} T(n,k)*x^k = A122335(n-1), A039834(n-2), A000012(n), A000045(n+1), A001333(n), A003688(n), A015448(n), A015449(n), A015451(n), A015453(n), A015454(n), A015455(n), A015456(n), A015457(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively. - Philippe Deléham, Dec 17 2011
Sum_{k=0..n} T(n,k)*x^(n-k) = A152163(n), A000007(n), A000045(n+1), A026597(n), A122994(n+1), A158608(n), A122995(n+1), A158797(n), A122996(n+1), A158798(n), A158609(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Dec 17 2011
G.f.: (1+(1-y)*x)/(1-y*x-x^2). - Philippe Deléham, Dec 17 2011
T(n,k) = T(n-1,k-1) + T(n-2,k), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 09 2013

A287811 Number of septenary sequences of length n such that no two consecutive terms have distance 5.

Original entry on oeis.org

1, 7, 45, 291, 1881, 12159, 78597, 508059, 3284145, 21229047, 137226717, 887047443, 5733964809, 37064931183, 239591481525, 1548743682699, 10011236540769, 64713650292711, 418315611378573, 2704034619149571, 17479154549033145, 112987031151647583

Offset: 0

David Nacin, Jun 01 2017

			For n=2 the a(2) = 49-4 = 45 sequences contain every combination except these four: 05, 50, 16, 61.

Index entries for linear recurrences with constant coefficients, signature (6, 3).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473.

Cf. A287804-A287819.

Mathematica
```
LinearRecurrence[{6, 3}, {1,7}, 40]
```

def a(n):
 if n in [0, 1]:
  return [1, 7][n]
 return 6*a(n-1)-3*a(n-2)

a(n) = 6*a(n-1) + 3*a(n-2), a(0)=1, a(1)=7.
G.f.: (1 + x)/(1 - 6*x - 3*x^2).
a(n) = A090018(n-1)+A090018(n). - R. J. Mathar, Oct 20 2019

A287838 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 8.

Original entry on oeis.org

1, 11, 115, 1205, 12625, 132275, 1385875, 14520125, 152130625, 1593906875, 16699721875, 174966753125, 1833166140625, 19206495171875, 201230782421875, 2108340300078125, 22089556912890625, 231437270629296875, 2424820490857421875, 25405391261720703125

Offset: 0

David Nacin, Jun 07 2017

In general, the number of sequences on {0,1,...,10} such that no two consecutive terms have distance 6+k for k in {0,1,2,3,4} has generating function (-1 - x)/(-1 + 10*x + (2*k+1)*x^2).

Colin Barker, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (10,5).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287839.

LinearRecurrence[{10, 5}, {1, 11, 115}, 20]

Vec((1 + x) / (1 - 10*x - 5*x^2) + O(x^40)) \\ Colin Barker, Nov 25 2017

def a(n):
 if n in [0,1,2]:
  return [1, 11, 115][n]
 return 10*a(n-1) + 5*a(n-2)

For n > 2, a(n) = 10*a(n-1) + 5*a(n-2), a(0)=1, a(1)=11, a(2)=115.
G.f.: (-1 - x)/(-1 + 10*x + 5*x^2).
a(n) = (((5-sqrt(30))^n*(-6+sqrt(30)) + (5+sqrt(30))^n*(6+sqrt(30)))) / (2*sqrt(30)). - Colin Barker, Nov 25 2017

A122186 First row sum of the 4 X 4 matrix M^n, where M={{10, 9, 7, 4}, {9, 8, 6, 3}, {7, 6, 4, 2}, {4, 3, 2, 1}}.

Original entry on oeis.org

1, 30, 707, 16886, 403104, 9623140, 229729153, 5484227157, 130922641160, 3125460977225, 74612811302754, 1781200165693270, 42521840081752984, 1015105948653689061, 24233196047277585233, 578508865448619225434

Offset: 0

Gary W. Adamson and Roger L. Bagula, Oct 17 2006

These matrices resemble Hankel matrices.

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Index entries for linear recurrences with constant coefficients, signature (23,21,-4,-1).

Cf. A015448.

with(linalg): M[1]:=matrix(4,4,[10,9,7,4,9,8,6,3,7,6,4,2,4,3,2,1]): for n from 2 to 15 do M[n]:=multiply(M[1],M[n-1]) od: 1,seq(M[n][1,1]+M[n][1,2]+M[n][1,3]+M[n][1,4],n=1..15);
a[0]:=1: a[1]:=30: a[2]:=707: a[3]:=16886: for n from 4 to 15 do a[n]:=23*a[n-1]+21*a[n-2]-4*a[n-3]-a[n-4] od: seq(a[n],n=0..15);

M = {{10, 9, 7, 4}, {9, 8, 6, 3}, {7, 6, 4, 2}, {4, 3, 2, 1}}; v[1] = {1, 1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 50}]

a(n) = 23a(n-1)+21a(n-2)-4a(n-3)-a(n-4) for n>=4; a(0)=1, a(1)=30, a(2)=707, a(3)=16886 (follows from the minimal polynomial x^4-23x^3-21x^2+4x+1 of the matrix M).

G.f.: -(x^3+4*x^2-7*x-1) / ((x+1)*(x^3+3*x^2-24*x+1)). [Colin Barker, Dec 07 2012]

Edited by N. J. A. Sloane, Nov 07 2006

A122187 First row sum of the matrix M^n, where M is the 3 X 3 matrix [[6, 5, 3], [5, 4, 2], [3, 2, 1]] (n>=0).

Original entry on oeis.org

1, 14, 157, 1782, 20216, 229347, 2601899, 29518061, 334876920, 3799116465, 43100270734, 488964567014, 5547212203625, 62932092237197, 713952898856653, 8099663044168346, 91889172989041221, 1042465602157270162

Offset: 1

Gary W. Adamson and Roger L. Bagula, Oct 17 2006

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Index entries for linear recurrences with constant coefficients, signature (11,4,-1).

Cf. A015448.

a[0]:=1: a[1]:=14: a[2]:=157: for n from 3 to 20 do a[n]:=11*a[n-1]+4*a[n-2]-a[n-3] od: seq(a[n],n=0..20);

M = {{6, 5, 3}, {5, 4, 2}, {3, 2, 1}}; v[1] = {1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 50}]
LinearRecurrence[{11,4,-1},{1,14,157},30] (* Harvey P. Dale, Oct 18 2022 *)

a(n) = 11a(n-1)+4a(n-2)-a(n-3), a(0)=1, a(1)=14, a(2)=157 (derived from the minimal polynomial of the matrix M).

G.f.: -x*(x^2-3*x-1) / (x^3-4*x^2-11*x+1). [Colin Barker, Dec 07 2012]

Edited by N. J. A. Sloane, Oct 29 2006

A284732 Square array read by antidiagonals downwards: T(n,k) = number of linear extensions of the North-East rectangular partial order NE_{n,k} that avoid the pattern 2143.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 4, 1, 1, 14, 21, 8, 1, 1, 42, 121, 89, 16, 1, 1, 132, 728, 1094, 377, 32, 1

Offset: 1

N. J. A. Sloane, Apr 07 2017

			The square array begins:
  1,  1,   1,    1,      1, ...
  1,  2,   5,   14,     42, ...
  1,  4,  21,  121,    728, ...
  1,  8,  89, 1094,  14041, ...
  1, 16, 377, 9841, 266110, ...
  ...
As a triangular array:
  1,
  1,   1,
  1,   2,   1,
  1,   5,   4,    1,
  1,  14,  21,    8,   1,
  1,  42, 121,   89,  16,  1,
  1, 132, 728, 1094, 377, 32, 1,
  ...

David Anderson, E. S. Egge, M. Riehl, L. Ryan, R. Steinke, Y. Vaughan, Pattern Avoiding Linear Extensions of Rectangular Posets, arXiv preprint arXiv:1605.06825 [math.CO], 2016.

Cf. A281731.

For early rows and columns see A000108, A000079 and (apparently) A274969, A015448.

A287805 Number of quinary sequences of length n such that no two consecutive terms have distance 2.

Original entry on oeis.org

1, 5, 19, 73, 281, 1083, 4175, 16097, 62065, 239307, 922711, 3557761, 13717913, 52893147, 203943935, 786361409, 3032030689, 11690820555, 45077144455, 173807214241, 670161078089, 2583988659867, 9963272432111, 38416111919777, 148123788152017, 571131629935179

Offset: 0

David Nacin, Jun 01 2017

			For n=2 the a(2)=19=25-6 sequences contain every combination except these six: 02,20,13,31,24,42.

Index entries for linear recurrences with constant coefficients, signature (4, 1, -6).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819.

LinearRecurrence[{4, 1, -6}, {1, 5, 19, 73}, 40]

def a(n):
 if n in [0,1,2,3]:
  return [1,5,19,73][n]
 return 4*a(n-1)+a(n-2)-6*a(n-3)

For n>0, a(n) = 4*a(n-1) + a(n-2) - 6*a(n-3), a(1)=5, a(2)=19, a(3)=73.

G.f.: (1 + x - 2*x^2 - 2*x^3)/(1 - 4*x - x^2 + 6*x^3).

cp's OEIS Frontend

A179604 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2*x - 9*x^2 - 2*x^3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A179605 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3*x - 2*x^2)/(1 - 2*x - 9*x^2 - 2*x^3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A020712 Pisot sequences E(5,8), P(5,8).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A153764 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A287811 Number of septenary sequences of length n such that no two consecutive terms have distance 5.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A287838 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A122186 First row sum of the 4 X 4 matrix M^n, where M={{10, 9, 7, 4}, {9, 8, 6, 3}, {7, 6, 4, 2}, {4, 3, 2, 1}}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A179604 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2x - 9x^2 - 2*x^3).

A179605 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3x - 2x^2)/(1 - 2x - 9x^2 - 2*x^3).