A086347 -id:A086347 - cp's OEIS frontend

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

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

A180033 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 5x - 5x^2).

Original entry on oeis.org

1, 6, 35, 205, 1200, 7025, 41125, 240750, 1409375, 8250625, 48300000, 282753125, 1655265625, 9690093750, 56726796875, 332084453125, 1944056250000, 11380703515625, 66623798828125, 390022511718750, 2283231552734375

Offset: 0

Johannes W. Meijer, Aug 09 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in the corner and side squares (m = 1, 3, 7, 9; 2, 4, 6, 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a white queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen, see A180032.
The sequence above corresponds to 56 red queen vectors, i.e., A[5] vector, with decimal values between 47 and 488. The central squares lead for these vectors to A057088.
Inverse binomial transform of A004187 (without the leading 0).
Equals the INVERT transform of A086347 and the INVERTi transform of A180167. - Gary W. Adamson, Aug 14 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2025. See p. 14.
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 7.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Index entries for linear recurrences with constant coefficients, signature (5,5).

Cf. A057088, A180032.

Cf. A086347, A180167. - Gary W. Adamson, Aug 14 2010

I:=[1,6]; [n le 2 select I[n] else 5*Self(n-1)+5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011

with(LinearAlgebra): nmax:=20; m:=1; A[5]:= [0,0,0,1,0,1,1,1,1]: A:=Matrix([[0,1,1,1,1,0,1,0,1], [1,0,1,1,1,1,0,1,0], [1,1,0,0,1,1,1,0,1], [1,1,0,0,1,1,1,1,0], A[5], [0,1,1,1,1,0,0,1,1], [1,0,1,1,1,0,0,1,1], [0,1,0,1,1,1,1,0,1], [1,0,1,0,1,1,1,1,0]]): 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);

LinearRecurrence[{5,5},{1,6}, 30] (* Vincenzo Librandi, Nov 15 2011 *)

my(x='x+O('x^30)); Vec((1+x)/(1-5*x-5*x^2)) \\ G. C. Greubel, Apr 07 2019

((1+x)/(1-5*x-5*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 07 2019

G.f.: (1+x)/(1 - 5*x - 5*x^2).
a(n) = 5*a(n-1) + 5*a(n-2) with a(0) = 1 and a(1) = 6.
a(n) = ((7+5*A)*A^(-n-1) + (7+5*B)*B^(-n-1))/45 with A = (-5+3*sqrt(5))/10 and B = (-5-3*sqrt(5))/10.
Limit_{k->oo} a(n+k)/a(k) = 2*5^(n/2)/(L(2*n) - F(2*n)*sqrt(5)) with L(n) = A000032(n) and F(n) = A000045(n).
Limit_{k->oo} a(2*n+k)/a(k) = 2*A000351(n)/(A056854(n) - 3*A004187(n)*sqrt(5)) for n >= 1.
Limit_{k->oo} a(2*n-1+k)/a(k) = 2*A000351(n)/(3*A049685(n-1)*sqrt(5) - 5*A033890(n-1)) for n >= 1.
a(n) = A057088(n+1)/5. a(2*n) = 5^n*F(4*(n+1))/3, a(2*n+1) = 5^n*L(2*(2*n+3))/3. - Ehren Metcalfe, Apr 04 2019
E.g.f.: exp(5*x/2)*(15*cosh(3*sqrt(5)*x/2) + 7*sqrt(5)*sinh(3*sqrt(5)*x/2))/15. - Stefano Spezia, Mar 17 2025

A180165 Triangle read by rows, derived from an array of sequences generated from (1 + x)/ (1 - rx - rx^2).

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 8, 5, 1, 5, 15, 22, 8, 1, 6, 24, 57, 60, 13, 1, 7, 35, 116, 216, 164, 21, 1, 8, 48, 205, 560, 819, 448, 34, 1, 9, 63, 330, 1200, 2704, 3105, 1224, 55, 1, 10, 80, 497, 2268, 7025, 13056, 11772, 3344, 89, 1, 11, 99, 712, 3920, 15588, 41125, 63040, 44631, 9136, 144

Offset: 1

Gary W. Adamson, Aug 14 2010

Row sums = A180166: (1, 3, 7, 18, 51, 161, 560, 2163, ...).

Rows of the array, with other offsets: (row 1 = A000045 starting with offset 2: (1, 2, 3, 5, 8, 13, ...); and for rows > 1, the entries: A028859, A125145, A086347, and A180033 start with offset 0; with the offset in the present array = 1.

			First few rows of the triangle:
  1;
  1, 2;
  1, 3, 3;
  1, 4, 8, 5;
  1, 5, 15, 22, 8;
  1, 6, 24, 57, 60, 13;
  1, 7, 35, 116, 216, 164, 21;
  1, 8, 48, 205, 560, 819, 448, 34;
  1, 9, 63, 330, 1200, 2704, 3105, 1224, 55;
  1, 10, 80, 497, 2268, 7025, 13056, 11772, 3344, 89;
  1, 11, 99, 712, 3920, 15588, 41125, 63040, 44631, 9136, 144;
  1, 12, 120, 981, 6336, 30919, 107136, 240750, 304384, 169209, 24960, 233;
  ...
As an array A(r,k) by upwards antidiagonals:
        k=1  k=2  k=3   k=4    k=5
  r=1:   1,   2,    3,    5,     8, ...
  r=2:   1,   3,    8,   22,    60, ...
  r=3:   1,   4,   15,   57,   216, ...
  r=4:   1,   5,   24,  116,   560, ...
  r=5:   1,   6,   35,  205,  1200, ...
Row r=5 = A180033 = (1, 6, 35, 205,...) and is generated from (1+x)/(1-5*x-5*x^2); is the INVERT transform of row r=4; and the array term A(5,4) = 205 = 5*35 + 5*6.
Terms A(2,4) and A(2,5) = [22,60] = [0,1; 2,2]^3 * [1,3].

Robert P. P. McKone, Antidiagonals n = 0..199, flattened

Cf. A180166, A000045, A028859, A125145, A086347, A180033.

Cf. A340156.

A180165[a_] := Reverse[Table[Table[CoefficientList[Series[(1 + x)/(1 - r*x - r*x^2), {x, 0, a - 2}], x], {r, 1, a + 1}][[k, n - k]], {n, 1, a}, {k, 1, n - 1}], 2] // Flatten;
A180165[12] (* Robert P. P. McKone, Jan 19 2021 *)

Triangle read by rows, generated from an array of sequences generated from (1 + x)/(1 - r*x - r*x^2); r > 0.
Alternatively, given the array with offset 1, the sequence r-th sequence is generated from a(k) = r*a(k-1) + r*(k-2); a(1) = 1, a(2) = r+1.
With a matrix method, the array can be generated from a 2 X 2 matrix of the form [0,1; r,r] = M, such that M^n * [1,r+1] = [r,n+1; r,n+2].
Also, for r > 1, the (r+1)-th row of the array is the INVERT transform of the r-th row.

a(35) corrected by Robert P. P. McKone, Dec 31 2020

A340156 Square array read by upward antidiagonals: T(n, k) is the number of n-ary strings of length k containing 00.

Original entry on oeis.org

1, 1, 3, 1, 5, 8, 1, 7, 21, 19, 1, 9, 40, 79, 43, 1, 11, 65, 205, 281, 94, 1, 13, 96, 421, 991, 963, 201, 1, 15, 133, 751, 2569, 4612, 3217, 423, 1, 17, 176, 1219, 5531, 15085, 20905, 10547, 880, 1, 19, 225, 1849, 10513, 39186, 86241, 92935, 34089, 1815

Offset: 2

Robert P. P. McKone, Dec 29 2020

			For n = 3 and k = 4, there are 21 strings: {0000, 0001, 0002, 0010, 0011, 0012, 0020, 0021, 0022, 0100, 0200, 1000, 1001, 1002, 1100, 1200, 2000, 2001, 2002, 2100, 2200}.
Square table T(n,k):
     k=2:  k=3:  k=4:   k=5:    k=6:     k=7:
n=2:   1     3     8     19      43       94
n=3:   1     5    21     79     281      963
n=4:   1     7    40    205     991     4612
n=5:   1     9    65    421    2569    15085
n=6:   1    11    96    751    5531    39186
n=7:   1    13   133   1219   10513    87199
n=8:   1    15   176   1849   18271   173608
n=9:   1    17   225   2665   29681   317817

Robert P. P. McKone, Antidiagonals n = 2..100, flattened

Cf. A008466 (row 2), A186244 (row 3), A000567 (column 4).
Cf. A180165 (not containing 00), A340242 (containing 000).
Cf. A000045, A028859, A125145, A086347, A180033, A180167, A322054.
Cf. A005563, A033445.

m[r_] := Normal[With[{p = 1/n}, SparseArray[{Band[{1, 2}] -> p, {i_, 1} /; i <= r -> 1 - p, {r + 1, r + 1} -> 1}]]];
T[n_, k_, r_] := MatrixPower[m[r], k][[1, r + 1]]*n^k;
Reverse[Table[T[n, k - n + 2, 2], {k, 2, 11}, {n, 2, k}], 2] // Flatten (* Robert P. P. McKone, Jan 26 2021 *)

T(n, k) = n^k - A180165(n+1,k-1), where A180165 in the number of strings not containing 00.

m(2) = [1 - 1/n, 1/n, 0; 1 - 1/n, 0, 1/n; 0, 0, 1], is the probability/transition matrix for two consecutive "0" -> "containing 00".

A218988 Power floor sequence of 2+sqrt(8).

Original entry on oeis.org

4, 19, 91, 439, 2119, 10231, 49399, 238519, 1151671, 5560759, 26849719, 129641911, 625966519, 3022433719, 14593600951, 70464138679, 340230958519, 1642780388791, 7932045389239, 38299303112119, 184925394005431, 892898788470199, 4311296729902519

Offset: 0

Clark Kimberling, Nov 11 2012

See A214992 for a discussion of power floor sequence and the power floor function, p1(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(8), and the limit p1(r) = 3.8983688904482395322594950087206...

See A218989 for the power floor function, p4. For comparison with p1, limit(p4(r)/p1(r) = 4/3.

			a(0) = [r] = 4, where r = 2+sqrt(8).
a(1) = [4*r] = 19; a(2) = [19*r] = 91.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (5,0,-4).

Cf. A214992, A057087, A086347, A218989.

x = 2 + Sqrt[8]; z = 30; (* z = # terms in sequences *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
t1 = Table[p1[n], {n, 0, z}]  (* this sequence *)
t2 = Table[p2[n], {n, 0, z}]  (* A057087 *)
t3 = Table[p3[n], {n, 0, z}]  (* A086347 *)
t4 = Table[p4[n], {n, 0, z}]  (* A218989 *)

Vec((4 - x - 4*x^2) / ((1 - x)*(1 - 4*x - 4*x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017

a(n) = floor(x*a(n-1)), where x=2+sqrt(8), a(0) = floor(x).
a(n) = 5*a(n-1) - 4*a(n-3).
G.f.: (4 - x - 4*x^2) / ((1 - x)*(1 - 4*x - 4*x^2)). [Corrected by Colin Barker, Nov 13 2017]
a(n) = (1/28)*(4 + (54-39*sqrt(2))*(2-2*sqrt(2))^n + (2*(1+sqrt(2)))^n*(54+39*sqrt(2))). - Colin Barker, Nov 13 2017
From Philippe Deléham, Mar 18 2024: (Start)
a(n) = 4*a(n-1) + 4*a(n-2) - 1.
a(n-1) = Sum_{k = 0..n} A370174(n,k)*3^k. (End)

A218989 Power ceiling sequence of 2+sqrt(8).

Original entry on oeis.org

5, 25, 121, 585, 2825, 13641, 65865, 318025, 1535561, 7414345, 35799625, 172855881, 834622025, 4029911625, 19458134601, 93952184905, 453641278025, 2190373851721, 10576060518985, 51065737482825, 246567192007241, 1190531717960265, 5748395639870025

Offset: 0

Clark Kimberling, Nov 11 2012

See A214992 for a discussion of power ceiling sequence and the power ceiling function, p4(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(8), and the limit p4(r) = (18 + 13*sqrt(2))/2 = 5.1978251872643193763459933449608678602008191971286...

See A218988 for the power floor function, p1(x); for comparison of p1 and p4, we have limit(p4(r)/p1(r) = 4 - sqrt(7).

			a(0) = ceiling(r) = 5, where r = 2+sqrt(8);
a(1) = ceiling(5*r) = 25; a(2) = ceiling(25*r) = 121.

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (5,0,-4).

Cf. A214992, A057087, A086347, A218988.

Mathematica
```
(See A218988.)
```

Vec((5 - 4*x^2) / ((1 - x)*(1 - 4*x - 4*x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017

a(n) = ceiling(x*a(n-1)), where x=2+sqrt(8), a(0) = ceiling(x).
a(n) = 5*a(n-1) - 4*a(n-3).
G.f.: (5 - 4*x^2) / ((1 - x)*(1 - 4*x - 4*x^2)). Corrected by Colin Barker, Nov 13 2017
a(n) = (1/7)*(-1 + (18-13*sqrt(2))*(2-2*sqrt(2))^n + (2*(1+sqrt(2)))^n*(18+13*sqrt(2))). - Colin Barker, Nov 13 2017

A193168 Number of simple paths from (1, 1) to (n, 3) on an n X 3 grid with king moves allowed.

Original entry on oeis.org

1, 24, 235, 2922, 38169, 494596, 6375379, 82191766, 1059980385, 13670322763, 176299392634, 2273637717194, 29321902354841, 378149186319554, 4876791443282017, 62893416210079645, 811103334429880838, 10460373436120693109, 134901938876902858230, 1739759409373842739031

Offset: 1

Matías Benzo, Jul 17 2011

Simple paths are self-avoiding walks.
Note: Number of paths from (1, 1) to (n, 2) on a n X 2 grid matches the sequence A086347.
Example: a(2) = 24 paths from (1, 1) to (2, 3)
[(1, 1), (2, 1), (1, 2), (1, 3), (2, 3)]
[(1, 1), (2, 2), (1, 3), (2, 3)]
[(1, 1), (2, 2), (2, 1), (1, 2), (2, 3)]
[(1, 1), (1, 2), (2, 1), (2, 2), (1, 3), (2, 3)]
[(1, 1), (2, 2), (2, 1), (1, 2), (1, 3), (2, 3)]
[(1, 1), (1, 2), (2, 2), (1, 3), (2, 3)]
[(1, 1), (1, 2), (1, 3), (2, 3)]
[(1, 1), (2, 2), (1, 3), (1, 2), (2, 3)]
[(1, 1), (2, 2), (2, 3)]
[(1, 1), (1, 2), (2, 3)]
[(1, 1), (2, 2), (1, 2), (1, 3), (2, 3)]
[(1, 1), (2, 1), (2, 2), (1, 3), (2, 3)]
[(1, 1), (2, 1), (2, 2), (1, 2), (2, 3)]
[(1, 1), (2, 1), (1, 2), (1, 3), (2, 2), (2, 3)]
[(1, 1), (1, 2), (2, 1), (2, 2), (2, 3)]
[(1, 1), (2, 1), (2, 2), (1, 3), (1, 2), (2, 3)]
[(1, 1), (2, 1), (1, 2), (2, 3)]
[(1, 1), (1, 2), (2, 2), (2, 3)]
[(1, 1), (1, 2), (1, 3), (2, 2), (2, 3)]
[(1, 1), (2, 1), (2, 2), (1, 2), (1, 3), (2, 3)]
[(1, 1), (2, 2), (1, 2), (2, 3)]
[(1, 1), (2, 1), (2, 2), (2, 3)]
[(1, 1), (2, 1), (1, 2), (2, 2), (2, 3)]
[(1, 1), (2, 1), (1, 2), (2, 2), (1, 3), (2, 3)]

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row 3 of A329118.

Cf. A086347, A140518.

Conjectures from Andrew Howroyd, Nov 05 2019: (Start)
a(n) = 12*a(n-1) + 6*a(n-2) + 50*a(n-3) + 300*a(n-4) - 194*a(n-5) - 833*a(n-6) + 352*a(n-7) + 661*a(n-8) - 84*a(n-9) - 219*a(n-10) + 72*a(n-11) + 54*a(n-12) for n > 12.
G.f.: x*(1 + 12*x - 59*x^2 - 92*x^3 + 195*x^4 + 280*x^5 + 102*x^6 - 178*x^7 - 67*x^8 + 109*x^9 - 75*x^10 - 54*x^11)/((1 + x + 9*x^2 + 24*x^3 + 9*x^4)*(1 - 13*x - 2*x^2 + 45*x^3 - 24*x^4 - 22*x^5 + 9*x^6 + 8*x^7 - 6*x^8)).
(End)

Offset changed and more terms from Andrew Howroyd, Nov 05 2019

A351530 The number of quinary strings of length n containing 00.

Original entry on oeis.org

0, 0, 1, 9, 65, 421, 2569, 15085, 86241, 483429, 2669305, 14564061, 78699089, 421880725, 2246459881, 11894065549, 62665617345, 328756309701, 1718275598809, 8951067087165, 46492068009521, 240846026714869, 1244719810538185, 6419100507215341

Offset: 0

R. J. Mathar, Feb 13 2022

Michael De Vlieger, Table of n, a(n) for n = 0..1430
Index entries for linear recurrences with constant coefficients, signature (9,-16,-20).

Cf. A008466 (2-ary), A186244 (3-ary), A351529 (4-ary), A086347 (not containing 00).

CoefficientList[Series[x^2/((5*x - 1)*(4*x^2 + 4*x - 1)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jun 22 2022 *)
LinearRecurrence[{9,-16,-20},{0,0,1},30] (* Harvey P. Dale, Mar 26 2024 *)

G.f.: x^2 / ( (5*x-1)*(4*x^2+4*x-1) ).
a(n) = 5^n - A086347(n).
a(n) = 9*a(n-1) - 16*a(n-2) - 20*a(n-3). - Wesley Ivan Hurt, Jun 22 2022

A086349 On a 3 X 3 board, the number of n-move paths for a chess king.

Original entry on oeis.org

1, 9, 40, 200, 952, 4624, 22272, 107648, 519552, 2509056, 12113920, 58492928, 282425344, 1363677184, 6584401920, 31792332800, 153506906112, 741197021184, 3578815578112, 17280050659328, 83435464425472

Offset: 0

Zak Seidov, Jul 17 2003

			a(1)=9 because there are 9 cells in the 3 X 3 board;
a(2)=40 because from each of 4 corner cells, king can move to 3 cells, this gives 4*3=12 moves, from each of 4 side cells, king can move to 5 cells, this gives 4*5=20 moves and from the central cell, king can move to 8 cells, this gives 8 moves and the total is 12+20+8=40.

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.

Cf. A086346, A086347, A086348.

a(n) = 4*A086346(n) + 4*A086347(n) + A086349(n).

a(n) = 2*a(n-1)+12*a(n-2)+8*a(n-3) for n>3. G.f.: (1+x)*(1+6*x+4*x^2)/((1+2*x)*(1-4*x-4*x^2)). - Colin Barker, Apr 12 2012

Edited by Johannes W. Meijer, Jul 15 2010

A164589 a(n) = ((4 + 3sqrt(2))(1 + 2sqrt(2))^n + (4 - 3sqrt(2))(1 - 2sqrt(2))^n)/8.

Original entry on oeis.org

1, 4, 15, 58, 221, 848, 3243, 12422, 47545, 182044, 696903, 2668114, 10214549, 39105896, 149713635, 573168542, 2194332529, 8400844852, 32162017407, 123129948778, 471394019405, 1804697680256, 6909153496347, 26451190754486, 101266455983401, 387691247248204

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

Binomial transform of A096886. Inverse binomial transform of A086347.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (2,7).

Cf. A086347, A096886.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((4+3*r)*(1+2*r)^n+(4-3*r)*(1-2*r)^n)/8: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 24 2009

CoefficientList[Series[(1+2x)/(1-2x-7x^2),{x,0,30}],x] (* or *) LinearRecurrence[{2,7},{1,4},30] (* Harvey P. Dale, Jun 22 2011 *)

Vec((1+2*x)/(1-2*x-7*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jul 16 2011

a(n) = 2*a(n-1) + 7*a(n-2) for n > 1; a(0) = 1, a(1) = 4.
G.f.: (1 + 2*x)/(1 - 2*x - 7*x^2).
E.g.f.: (1/4)*exp(x)*(4*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 12 2017

Edited and extended beyond a(5) by Klaus Brockhaus and R. J. Mathar, Aug 24 2009

cp's OEIS Frontend

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A193168 Number of simple paths from (1, 1) to (n, 3) on an n X 3 grid with king moves allowed.

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A180033 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 5x - 5x^2).

A180165 Triangle read by rows, derived from an array of sequences generated from (1 + x)/ (1 - rx - rx^2).

A164589 a(n) = ((4 + 3sqrt(2))(1 + 2sqrt(2))^n + (4 - 3sqrt(2))(1 - 2sqrt(2))^n)/8.