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A086347 On a 3 X 3 board, number of n-move routes of chess king ending in a given side square. 19
1, 5, 24, 116, 560, 2704, 13056, 63040, 304384, 1469696, 7096320, 34264064, 165441536, 798822400, 3857055744, 18623512576, 89922273280, 434183143424, 2096421666816, 10122419240960 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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}.

Row sums of A235113.

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..1459

Joerg Arndt, Matters Computational (The Fxtbook)

Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

M. 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.

Sleephound, KingMovesForPrimes.

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

FORMULA

a(n) = (Sqrt[2]/32)((2+Sqrt[8])^(n+2)-(2-Sqrt[8])^(n+2))

G.f.: (1+x)/(1-4*x-4*x^2). a(n) = A057087(n) + A057087(n-1). - Ralf Stephan, Feb 01 2004

a(n) = 4*a(n-1) + 4*a(n-2). - Tanya Khovanova, Jan 11 2007

Limit(a(n+k)/a(k),k=infinity) = A084128(n) + 2*A057087(n-1)*sqrt(2). - Johannes W. Meijer, Aug 01 2010

EXAMPLE

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

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

MATHEMATICA

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

CROSSREFS

Cf. A086346, A086348.

Cf. A028859.

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

Sequence in context: A242509 A057969 A004254 * A200739 A026707 A235115

Adjacent sequences:  A086344 A086345 A086346 * A086348 A086349 A086350

KEYWORD

nonn,easy

AUTHOR

Zak Seidov, Jul 17 2003

EXTENSIONS

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

STATUS

approved

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Last modified July 20 16:53 EDT 2017. Contains 289628 sequences.