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Search: id:A056010
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| A056010 |
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Number of words of length n in a simple grammar. |
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+0
5
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| 1, 1, 3, 8, 23, 68, 207, 644, 2040, 6558, 21343, 70186, 232864, 778550, 2620459, 8872074, 30195288, 103246502, 354508628, 1221846856, 4225644866, 14659644348, 51002664023, 177909901566, 622093882290, 2180123564130, 7656055966092
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Essentially a duplicate of A025262.
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LINKS
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M. Somos, Number Walls in Combinatorics.
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FORMULA
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L = 1 + L(e+w) + LnLs - w.
a(n) = 2*a(n-1) + a(0)*a(n-2) + ... + a(n-2)*a(0) for n>1.
The Somos-4 sequence A006720(n+2) is the Hankel transform of a(n-1). See A001906 for definition of Hankel transform.
Let s(n)= A006769(n). Then 0= f( -s(n-1)* s(n+1)/ s(n)^2, -s(n)* s(n+2)/ s(n+1)^2 ) where f(u, v)= u+v -(1+u*v)^2 .
G.f. A(x) satisfies 0= f(x, A(x)) where f(u, v)= u+v -(1+u*v)^2 .
G.f.: (1 -2*x -sqrt( 1 -4*x +4*x^3) )/(2*x^2) .
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EXAMPLE
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L(0) = 1, L(1) = e, L(2) = ee + ew + ns, L(3) = eee + ewe + nse + eew + eww + nsw + nes + ens.
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PROGRAM
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(PARI) {a(n)= if(n<0, 0, polcoeff( (1 -2*x -sqrt( 1 -4*x +4*x^3 +x^3*O(x^n)) )/(2*x^2), n))}
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CROSSREFS
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A025262(n)=a(n-2), n>1. Cf. A006720.
Sequence in context: A038151 A057198 A025262 this_sequence A002712 A005960 A061557
Adjacent sequences: A056007 A056008 A056009 this_sequence A056011 A056012 A056013
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, Aug 01 2000.
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