A179597 -id:A179597 - cp's OEIS frontend

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

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

R. H. Hardin, Dec 27 2006

[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

Shanzhen Gao and Keh-Hsun Chen, Tackling Sequences From Prudent Self-Avoiding Walks, FCS'14, The 2014 International Conference on Foundations of Computer Science.
S. Gao and H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks, 2010.
Index entries for linear recurrences with constant coefficients, signature (4,3).

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.

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

LinearRecurrence[{4, 3}, {1, 5}, 24] (* Jean-François Alcover, Dec 10 2024 *)

a(n)=([0,1; 3,4]^n*[1;5])[1,1] \\ Charles R Greathouse IV, May 10 2016

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

Edited by Johannes W. Meijer, Aug 10 2010

A015530 Expansion of x/(1 - 4x - 3x^2).

Original entry on oeis.org

0, 1, 4, 19, 88, 409, 1900, 8827, 41008, 190513, 885076, 4111843, 19102600, 88745929, 412291516, 1915403851, 8898489952, 41340171361, 192056155300, 892245135283, 4145149007032, 19257331433977, 89464772757004

Offset: 0

Olivier Gérard

Let b(1)=1, b(k) = floor(b(k-1)) + 3/b(k-1); then for n>1, b(n) = a(n)/a(n-1). - Benoit Cloitre, Sep 09 2002
In general, x/(1 - a*x - b*x^2) has a(n) = Sum_{k=0..floor((n-1)/2)} C(n-k-1,k)*b^k*a^(n-2k-1). - Paul Barry, Apr 23 2005
Pisano period lengths: 1, 2, 1, 4, 24, 2, 21, 4, 3, 24, 40, 4, 84, 42, 24, 8, 288, 6, 18, 24, ... . - R. J. Mathar, Aug 10 2012
This is the Lucas sequence U(4,-3). - Bruno Berselli, Jan 09 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Lucyna Trojnar-Spelina and Iwona Włoch, On Generalized Pell and Pell-Lucas Numbers, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.
Wikipedia, Lucas sequence.
Index entries for linear recurrences with constant coefficients, signature (4,3).

Appears in A179596, A126473 and A179597. - Johannes W. Meijer, Aug 01 2010

Cf. A080042: Lucas sequence V(4,-3).

I:=[0, 1]; [n le 2 select I[n] else 4*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 19 2012

LinearRecurrence[{4,3},{0,1},30] (* Vincenzo Librandi, Jun 19 2012 *)

x='x+O('x^30); concat([0], Vec(x/(1-4*x-3*x^2))) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,4,-3) for n in range(0, 23)]# Zerinvary Lajos, Apr 23 2009

a(n) = 4*a(n-1) + 3*a(n-2).
a(n) = (A086901(n+2) - A086901(n+1))/6. - Ralf Stephan, Feb 01 2004
a(n) = Sum_{k=0..floor((n-1)/2)} C(n-k-1, k)*3^k*4^(n-2k-1). - Paul Barry, Apr 23 2005
a(n) = ((2+sqrt(7))^n - (2-sqrt(7))^n)/sqrt(28). Offset 1. a(3)=19. - Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009
From Johannes W. Meijer, Aug 01 2010: (Start)
Limit(a(n+k)/a(k), k=infinity) = A108851(n)+a(n)*sqrt(7).
Limit(A108851(n)/a(n), n=infinity) = sqrt(7). (End)
G.f.: x*G(0) where G(k)= 1 + (4*x+3*x^2)/(1 - (4*x+3*x^2)/(4*x + 3*x^2 + 1/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jul 28 2012
G.f.: G(0)*x/(2-4*x), where G(k)= 1 + 1/(1 - x*(7*k-4)/(x*(7*k+3) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 16 2013

A179596 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2x - 11x^2 - 6*x^3).

Original entry on oeis.org

1, 3, 17, 73, 351, 1607, 7513, 34809, 161903, 751783, 3493353, 16227737, 75393055, 350251335, 1627192697, 7559508409, 35119644495, 163157037671, 757987215241, 3521419711833, 16359641017343, 76002822156295, 353090213774361

Offset: 0

Johannes W. Meijer, Jul 28 2010; edited Jun 21 2013

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 center square the king goes crazy and turns into a red king.
On a 3 X 3 chessboard there are 2^9 = 512 ways to go crazy on the center square (off the center the piece behaves like a normal king). The red king is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the corner squares the 512 red kings lead to 47 different red king sequences, see the overview of the red king sequences.
The sequence above corresponds to four A[5] vectors with decimal [binary] values 367 [101 101 111], 463 [111 001 111], 487 [111 100 111] and 493 [111 101 101]. These vectors lead for the side squares to A126473 and for the central square to A179597.
This sequence belongs to a family of sequences with g.f. (1+x)/(1 - 2*x - (k+8)*x^2 - 2*k*x^3). Red king sequences that are members of this family are A083424 (k=0), A179604 (k=1), A179600 (k=2), A179596 (k=3; this sequence) and A086346 (k=4). Other members of this family are A015528 (k=5) and A179608 (k=-4).

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Charles Krauthammer, Did Chess Make Him Crazy?, Time, April 26, 2005.
Johannes W. Meijer, The red king sequences.
Index entries for linear recurrences with constant coefficients, signature (2,11,6).

Cf. A180140 (berserker sequences).

Cf. Red king sequences corner squares [decimal value A[5]]: A086346 [495], A015525 [239], A179596 [367], A179600 [335], A015524 [95], A083858 [31], A179604 [327], A015523 [27], A179610 [85], A083424 [325], A015521 [11], A007482 [2], A014335 [16].

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]:= [1,0,1,1,0,1,1,1,1]: 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[{2,11,6},{1,3,17},30] (* Harvey P. Dale, May 18 2011 *)

Vec((1+x)/(1-2*x-11*x^2-6*x^3)+O(x^99)) \\ Charles R Greathouse IV, Jul 16 2011

G.f.: (1+x)/(1 - 2*x - 11*x^2 - 6*x^3).
a(n) = 2*a(n-1) + 11*a(n-2) + 6*a(n-3) with a(0)=1, a(1)=3 and a(2)=17.
a(n) = (-1)^(-n)*2^(n+1)/9 + ((49+17*sqrt(7))*A^(-n) + (49-17*sqrt(7))*B^(-n))/126 with A = (-2+sqrt(7))/3 and B = (-2-sqrt(7))/3.
Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n+1)*A000244(n)/(A015530(n)*sqrt(7) - A108851(n)).

A179606 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 3x - 5x^2).

Original entry on oeis.org

1, 4, 17, 71, 298, 1249, 5237, 21956, 92053, 385939, 1618082, 6783941, 28442233, 119246404, 499950377, 2096083151, 8788001338, 36844419769, 154473265997, 647641896836, 2715292020493, 11384085545659, 47728716739442

Offset: 0

Johannes W. Meijer, Jul 28 2010

a(n) represents 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 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 side squares to A152187.
This sequence belongs to a family of sequences with g.f. (1 + (k-4)*x)/(1 - 3*x - k*x^2). Red king sequences that are members of this family are A007483 (k= 2), A015521 (k=4), A179606 (k=5; this sequence), A154964 (k=6), A179603 (k=7) and A179599 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A006190 (k=1), A133494 (k=0) and A168616 (k=-2).
Inverse binomial transform of A052918.
The sequence b(n+1) = 6*a(n), n >= 0 with b(0)=1, is a berserker sequence, see A180147. The b(n) sequence corresponds to 16 A[5] vectors with decimal values between 111 and 492. These vectors lead for the corner squares to sequence c(n+1)=4*A179606(n), n >= 0 with c(0)=1, and for the side squares to A180140. - Johannes W. Meijer, Aug 14 2010
Equals the INVERT transform of A063782: (1, 3, 10, 32, 104, ...). Example: a(3) = 71 = (1, 1, 4, 7) dot (32, 10, 3, 1) = (32 + 10 + 12 + 17). - Gary W. Adamson, Aug 14 2010

Indranil Ghosh, Table of n, a(n) for n = 0..1603
Index entries for linear recurrences with constant coefficients, signature (3,5).

Cf. A179597 (central square).

Cf. A072263, A072264, A152187, A197189.

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]:= [0,0,0,1,1,1,0,0,1]: 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);

CoefficientList[Series[(1+x)/(1-3*x-5*x^2), {x, 0, 22}],x] (* or *) LinearRecurrence[{3,5,0},{1,4},23] (* Indranil Ghosh, Mar 05 2017 *)

print(Vec((1 + x)/(1- 3*x - 5*x^2) + O(x^23))); \\ Indranil Ghosh, Mar 05 2017

G.f.: (1+x)/(1 - 3*x - 5*x^2).
a(n) = A015523(n) + A015523(n+1).
a(n) = 3*a(n-1) + 5*a(n-2) with a(0) = 1 and a(1) = 4.
a(n) = ((29 + 7*sqrt(29))*A^(-n-1) + (29-7*sqrt(29))*B^(-n-1))/290 with A = (-3+sqrt(29))/10 and B = (-3-sqrt(29))/10
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n+1)*A000351(n)*A130196(n)/(A015523(n)*sqrt(29) - A072263(n)) for n >= 1.

A086348 On a 3 X 3 board, number of n-move routes of chess king ending in the central square.

Original entry on oeis.org

1, 8, 32, 168, 784, 3840, 18432, 89216, 430336, 2078720, 10035200, 48457728, 233967616, 1129709568, 5454692352, 26337640448, 127169265664, 614027755520, 2964787822592, 14315262836736

Offset: 0

Zak Seidov, Jul 17 2003

From Johannes W. Meijer, Aug 01 2010: (Start)
The a(n) represent the number of n-move paths of a chess king on a 3 X 3 board that end or start in the central square m (m = 5).
Inverse binomial transform of A090390 (without the first leading 1).
(End)
From R. J. Mathar, Oct 12 2010: (Start)
The row n=3 of an array T(n,k) counting king walks on an n X n board starting on a square on the diagonal next to a corner:
1,8,32,168,784,3840,18432,89216,430336,2078720,10035200,48457728,233967616,
1,8,47,275,1610,9425,55175,323000,1890875,11069375,64801250,379353125,
1,8,47,318,2013,13140,84555,547722,3537081,22874400,147831399,955690326,
1,8,47,318,2134,14539,99267,679189,4650100,31848677,218164072,1494530576,
1,8,47,318,2134,14880,103920,733712,5187856,36796224,261164848,1855327584,
1,8,47,318,2134,14880,104885,748845,5382180,38880243,281743740,2045995632,
1,8,47,318,2134,14880,104885,751590,5430735,39556080,289541500,2127935700,
1,8,47,318,2134,14880,104885,751590,5438580,39710495,291852880,2156410817,
1,8,47,318,2134,14880,104885,751590,5438580,39733008,292340803,2164218694,
1,8,47,318,2134,14880,104885,751590,5438580,39733008,292405638,2165752797, (End)

Mike Oakes, KingMovesForPrimes.
Zak Seidov, KingMovesForPrimes.
Zak Seidov et al., New puzzle? King moves for primes, digest of 28 messages in primenumbers group, Jul 13 - Jul 23, 2003. [Cached copy]
Sleephound, KingMovesForPrimes.
Index entries for linear recurrences with constant coefficients, signature (2, 12, 8).

Cf. A086346, A086347.

Cf. A179597.

with(LinearAlgebra): nmax:=19; m:=5; A[5]:= [1,1,1,1,0,1,1,1,1]: 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

Table[(1/16)(4(-2)^(n+1)+(2+Sqrt[8])^(n+2)+(2-Sqrt[8])^(n+2)), {n, 0, 19}]

a(n) = (1/16)(4(-2)^(n+1) + (2+sqrt(8))^(n+2) + (2-sqrt(8))^(n+2)).
From Johannes W. Meijer, Aug 01 2010: (Start)
G.f.: ( 1+6*x+4*x^2 ) / ( (2*x+1)*(-4*x^2-4*x+1) ).
a(n) = 2*a(n-1) + 12*a(n-2) + 8*a(n-3) with a(0)=1, a(1)=8 and a(2)=32.
Lim_{k->infinity} a(n+k)/a(k) = A084128(n) + 2*A057087(n-1)*sqrt(2). (End)
2*a(n) = 3*A057087(n) + 2*A057087(n-1) - (-2)^n. - R. J. Mathar, May 21 2019

Offset changed and edited by Johannes W. Meijer, Jul 15 2010

A108851 a(n) = 4a(n-1) + 3a(n-2), a(0) = 1, a(1) = 2.

Original entry on oeis.org

1, 2, 11, 50, 233, 1082, 5027, 23354, 108497, 504050, 2341691, 10878914, 50540729, 234799658, 1090820819, 5067682250, 23543191457, 109375812578, 508132824683, 2360658736466, 10967033419913, 50950109889050

Offset: 0

Philippe Deléham, Jul 11 2005

Binomial transform of A083098, second binomial transform of (1, 0, 7, 0, 49, 0, 243, 0, ...).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,3).

Cf. A080042. - Zerinvary Lajos, May 14 2009
Cf. A015530, A083098.
Appears in A179596, A179597 and A126473. - Johannes W. Meijer, Aug 01 2010

[Floor(((2 + Sqrt(7))^n + (2 - Sqrt(7))^n) / 2): n in [0..30]]; // Vincenzo Librandi, Jul 18 2011

LinearRecurrence[{4,3},{1,2},30] (* Harvey P. Dale, Jan 02 2022 *)

a(n)=round(((2+sqrt(7))^n+(2-sqrt(7))^n)/2) \\ Charles R Greathouse IV, Dec 06 2011

[lucas_number2(n,4,-3)/2 for n in range(0, 22)] # Zerinvary Lajos, May 14 2009

a(n) = ((2 + sqrt(7))^n + (2 - sqrt(7))^n) / 2.
G.f.: (1 - 2*x) / (1 - 4*x - 3*x^2).
E.g.f.: exp(2*x)*cosh(sqrt(7)*x).
a(n+1)/a(n) converges to 2 + sqrt(7) = 4.645751311064...
Limit_{k->oo} a(n+k)/a(k) = A108851(n) + A015530(n)*sqrt(7); also lim_{n->oo} A108851(n)/A015530(n) = sqrt(7). - Johannes W. Meijer, Aug 01 2010
a(n) = Sum_{k=0..n} A201730(n,k)*6^k. - Philippe Deléham, Dec 06 2011
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(7*k-4)/(x*(7*k+3) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013
a(n) = (2 + sqrt(7))^n - A015530(n)*sqrt(7). - Robert FERREOL, Aug 04 2025

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

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 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.

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

Cf. A179597 (central square).

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);

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.

A179603 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 3x - 7*x^2).

Original entry on oeis.org

1, 6, 25, 117, 526, 2397, 10873, 49398, 224305, 1018701, 4626238, 21009621, 95412529, 433304934, 1967802505, 8936542053, 40584243694, 184308525453, 837015282217, 3801205524822, 17262723549985, 78396609323709

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 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 95, 119, 125, 215, 221, 245, 287, 311, 317, 347, 350, 371, 374, 377, 380, 407, 413, 437, 467, 470, 473, 476, 497 and 500. These vectors lead for the corner squares to A015524 and for the side squares to A179602.

Index entries for linear recurrences with constant coefficients, signature (3, 7).

Cf. A179597 (central square).

with(LinearAlgebra): nmax:=23; 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]:= [1,1,1,0,1,0,0,1,1]: 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)/(1 - 3*x - 7*x^2).
a(n) = 3*a(n-1) + 7*a(n-2) with a(0) = 1 and a(1) = 6.
a(n) = ((9+5*A)*A^(-n-1) + (9+5*B)*B^(-n-1))/37 with A = (-3+sqrt(37))/14 and B = (-3-sqrt(37))/14.

A179601 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1+4x)/(1 - 2x - 10x^2 - 4x^3).

Original entry on oeis.org

1, 6, 22, 108, 460, 2088, 9208, 41136, 182704, 813600, 3618784, 16104384, 71651008, 318820992, 1418569600, 6311953152, 28084886272, 124963582464, 556023840256, 2474023050240, 11008138832896, 48980603529216, 217938687588352

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

The sequence above corresponds to 6 red king vectors, i.e., A[5] vectors, with decimal values 335, 359, 365, 455, 461 and 485. These vectors lead for the corner squares to A179600 and for the side squares to A123347.

Index entries for linear recurrences with constant coefficients, signature (2,10,4).

Cf. A041006, A041007, A123347, A179596, A179597 (central square), A179600.

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]:= [1,1,1,0,0,0,1,1,1]: 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.: ( -1-4*x ) / ( (2*x+1)*(2*x^2 + 4*x - 1) ).
a(n) = 2*a(n-1) + 10*a(n-2) + 4*a(n-3) with a(0)=1, a(1)=6 and a(2)=22.
a(n) = (-2/5)*(-1/2)^(-n) + ((2+3*A)*A^(-n-1) + (2+3*B)*B^(-n-1))/10 with A = (-1+sqrt(6)/2) and B = (-1-sqrt(6)/2).
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n+1)*A016116(n+1)/(A041007(n-1)*sqrt(6) - A041006(n-1)) for n => 1.

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)).

cp's OEIS Frontend

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

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A015530 Expansion of x/(1 - 4*x - 3*x^2).

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A179596 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2*x - 11*x^2 - 6*x^3).

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A179606 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 3*x - 5*x^2).

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A086348 On a 3 X 3 board, number of n-move routes of chess king ending in the central square.

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A108851 a(n) = 4*a(n-1) + 3*a(n-2), a(0) = 1, a(1) = 2.

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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).

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A015530 Expansion of x/(1 - 4x - 3x^2).

A179596 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2x - 11x^2 - 6*x^3).

A179606 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 3x - 5x^2).

A108851 a(n) = 4a(n-1) + 3a(n-2), a(0) = 1, a(1) = 2.

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

A179603 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 3x - 7*x^2).

A179601 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1+4x)/(1 - 2x - 10x^2 - 4x^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).