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

Author

Zak Seidov, Jul 17 2003

Keywords

• nonn
• easy

Comments

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

Examples

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

Links

• 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).

Crossrefs

Row 2 of A329118.

Row sums of A235113.

Cf. A086346, A086348.

Cf. A028859.

Cf. A126473. - Johannes W. Meijer, Aug 01 2010

Programs

Maple

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

Mathematica

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

Formula

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

Extensions

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