A096886 -id:A096886 - cp's OEIS frontend

A086347 On a 3 X 3 board, number of n-move routes of chess king ending in a given side square.

1, 5, 24, 116, 560, 2704, 13056, 63040, 304384, 1469696, 7096320, 34264064, 165441536, 798822400, 3857055744, 18623512576, 89922273280, 434183143424, 2096421666816, 10122419240960, 48875363631104, 235991131488256, 1139465980477440, 5501828447862784

Offset: 0

Zak Seidov, Jul 17 2003

Number of aa-avoiding words of length n on alphabet {a,b,c,d,e}. - Tanya Khovanova, Jan 11 2007
Binomial transform of A164589 and second binomial transform of A096886. [Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009]
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 a given side square m (m = 2, 4, 6, 8).
Inverse binomial transform of A001109 (without the leading 0).
(End)
Number of independent vertex subsets of the graph obtained by attaching two pendant edges to each vertex of the path graph P_n (see A235116). Example: a(1)=5; indeed, P_1 is the one-vertex graph and after attaching two pendant vertices we obtain the path graph ABC; the independent vertex subsets are: empty, {A}, {B}, {C}, and {A,C}.
Number of simple paths from corner to diagonally opposite corner on a 2 X n grid with king moves allowed. - Andrew Howroyd, Nov 06 2019
Number of 4-compositions of n+1 restricted to parts 1 and 2 (and allowed zeros); see Hopkins & Ouvry reference. - Brian Hopkins, Aug 16 2020

			a(3) = 116 = 5^3 - 9 (aaa, aab, aac, aad, aae, baa, caa, daa, eaa). [Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009]

Indranil Ghosh, Table of n, a(n) for n = 0..1459
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.
Joerg Arndt, Matters Computational (The Fxtbook)
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.
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova, Recursive Sequences
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 (4,4).

Row 2 of A329118.
Row sums of A235113.
Cf. A086346, A086348.
Cf. A028859.
Cf. A126473. - Johannes W. Meijer, Aug 01 2010

with(LinearAlgebra): nmax:=19; m:=2; 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
# second Maple program:
a:= n-> (<<0|1>, <4|4>>^n. <<1, 5>>)[1,1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 12 2022

Table[(Sqrt[2]/32)((2+Sqrt[8])^(n+2)-(2-Sqrt[8])^(n+2)), {n, 0, 19}]

a(n) = (sqrt(2)/32)*((2+sqrt(8))^(n+2)-(2-sqrt(8))^(n+2)).
From Ralf Stephan, Feb 01 2004: (Start)
G.f.: (1+x)/(1-4*x-4*x^2).
a(n) = A057087(n) + A057087(n-1). (End)
a(n) = 4*a(n-1) + 4*a(n-2). - Tanya Khovanova, Jan 11 2007
Limit_{k->oo} a(n+k)/a(k) = A084128(n) + 2*A057087(n-1)*sqrt(2). - Johannes W. Meijer, Aug 01 2010
E.g.f.: exp(2*x)*(4*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x))/4. - Stefano Spezia, Mar 17 2025

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

A164591 a(n) = ((4 + sqrt(18))(4 + sqrt(8))^n + (4 - sqrt(18))(4 - sqrt(8))^n)/8 .

Original entry on oeis.org

1, 7, 48, 328, 2240, 15296, 104448, 713216, 4870144, 33255424, 227082240, 1550614528, 10588258304, 72301150208, 493703135232, 3371215880192, 23020101959680, 157191088635904, 1073367893409792, 7329414438191104, 50048372358250496

Offset: 0

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

Binomial transform of A001109 without initial 0. Fourth binomial transform of A096886. Inverse binomial transform of A164592.

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 (8,-8).

Cf. A001109, A096886, A164592.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((4+3*r)*(4+2*r)^n+(4-3*r)*(4-2*r)^n)/8: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 24 2009

LinearRecurrence[{8,-8}, {1,7}, 50] (* G. C. Greubel, Aug 12 2017 *)

Vec((1-x)/(1-8*x+8*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jul 16 2011

a(n) = 8*a(n-1) - 8*a(n-2) for n > 1; a(0) = 1, a(1) = 7.
G.f.: (1-x)/(1-8*x+8*x^2).
E.g.f.: (1/4)*exp(4*x)*(4*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 12 2017

Extended by Klaus Brockhaus and R. J. Mathar Aug 24 2009

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

A164592 a(n) = 10a(n-1) - 17a(n-2) for n > 1; a(0) = 1, a(1) = 8.

Original entry on oeis.org

1, 8, 63, 494, 3869, 30292, 237147, 1856506, 14533561, 113775008, 890679543, 6972620294, 54584650709, 427311962092, 3345180558867, 26187502233106, 205006952830321, 1604881990340408, 12563701705288623, 98354023217099294, 769957303181086349, 6027554637120175492

Offset: 0

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

Binomial transform of A164591. Fifth binomial transform of A096886.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..144 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (10,-17).

Cf. A164591, A096886.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((4+3*r)*(5+2*r)^n+(4-3*r)*(5-2*r)^n)/8: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 24 2009

CoefficientList[Series[(1 - 2*z)/(17*z^2 - 10*z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 12 2011 *)
LinearRecurrence[{10,-17},{1,8},30] (* Harvey P. Dale, Oct 14 2012 *)

my(x='x+O('x^30)); Vec((1-2*x)/(1-10*x+17*x^2)) \\ G. C. Greubel, Aug 12 2017

a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 8.
a(n) = ((4 + sqrt(18))*(5 + sqrt(8))^n + (4 - sqrt(18))*(5 - sqrt(8))^n)/8.
G.f.: (1-2*x)/(1-10*x+17*x^2).
E.g.f.: (1/4)*exp(5*x)*(4*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 12 2017

Extended by Klaus Brockhaus and R. J. Mathar Aug 24 2009

cp's OEIS Frontend

A086347 On a 3 X 3 board, number of n-move routes of chess king ending in a given side square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A164591 a(n) = ((4 + sqrt(18))*(4 + sqrt(8))^n + (4 - sqrt(18))*(4 - sqrt(8))^n)/8 .

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A164592 a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A164591 a(n) = ((4 + sqrt(18))(4 + sqrt(8))^n + (4 - sqrt(18))(4 - sqrt(8))^n)/8 .

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

A164592 a(n) = 10a(n-1) - 17a(n-2) for n > 1; a(0) = 1, a(1) = 8.