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A086347
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On a 3 X 3 board, number of n-move routes of chess king ending in a given side square.
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23
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1, 5, 24, 116, 560, 2704, 13056, 63040, 304384, 1469696, 7096320, 34264064, 165441536, 798822400, 3857055744, 18623512576, 89922273280, 434183143424, 2096421666816, 10122419240960, 48875363631104, 235991131488256, 1139465980477440, 5501828447862784
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OFFSET
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0,2
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COMMENTS
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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]
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
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LINKS
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FORMULA
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a(n) = (Sqrt[2]/32)((2+Sqrt[8])^(n+2)-(2-Sqrt[8])^(n+2))
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EXAMPLE
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a(3) = 116 = 5^3 - 9 (aaa, aab, aac, aad, aae, baa, caa, daa, eaa). [Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009]
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MAPLE
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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]:
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MATHEMATICA
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Table[(Sqrt[2]/32)((2+Sqrt[8])^(n+2)-(2-Sqrt[8])^(n+2)), {n, 0, 19}]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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