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A055244
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Number of certain stackings of n+1 squares on a double staircase.
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7
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1, 1, 3, 6, 12, 23, 43, 79, 143, 256, 454, 799, 1397, 2429, 4203, 7242, 12432, 21271, 36287, 61739, 104791, 177476, 299978, 506111, 852457, 1433593, 2407443, 4037454, 6762708, 11314391, 18909139, 31569799, 52657247, 87751624
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n)= G_{n+1} of Turban reference eq.(3.9).
Equals A046854 * [1,2,3,...]. [From Gary W. Adamson, Dec 23 2008]
(1 + x + 3x^2 + 6x^3 + ...) = (1 + x + 2x^2 + 3x^3 + 5x^4 + 8x^5 + ...) * (1 + x^2 + 2x^3 + 3x^4 + 5x^5 + 8x^6 + ...). [Gary W. Adamson, Jul 27 2010]
Column 1 of A194540 [From R. H. Hardin, Aug 28 2011]
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REFERENCES
| L. Turban, Lattice animals on a staircase and Fibonacci numbers, J.Phys. A 33 (2000) 2587-2595.
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FORMULA
| G.f.: (1-x+x^3)/(1-x-x^2)^2 (from Turban reference eq.(3.3) with t=1).
a(n)=((n+5)*F(n+1)+(2*n-3)*F(n))/5 with F(n)=A000045(n) (Fibonacci numbers) (from Turban reference eq.(3.9)).
a(n) = A001629(n+1) + F(n-1). Example: a(5) = 23 = A001629(6) + F(4) = (20 + 3). Sequence starting (1, 3, 6, 12, 23,...) = A046854 * (1, 2, 3,...). - Gary W. Adamson, Jul 27 2007
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MAPLE
| (Maple) a := n -> (Matrix([[1, -1, 2, -4]]). Matrix(4, (i, j)-> if (i=j-1) then 1 elif j=1 then [2, 1, -2, -1][i] else 0 fi)^(n))[1, 1] ; seq (a(n), n=0..33); [From Alois P. Heinz, Aug 05 2008]
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CROSSREFS
| A000045, A055245.
Cf. A001629, A046854.
A046854 [From Gary W. Adamson, Dec 23 2008]
Sequence in context: A174201 A181844 A162506 * A089068 A018180 A079735
Adjacent sequences: A055241 A055242 A055243 * A055245 A055246 A055247
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), May 10 2000
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