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A055842
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Expansion of (1-x)^2/(1-5*x).
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7
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1, 3, 16, 80, 400, 2000, 10000, 50000, 250000, 1250000, 6250000, 31250000, 156250000, 781250000, 3906250000, 19531250000, 97656250000, 488281250000, 2441406250000, 12207031250000, 61035156250000, 305175781250000, 1525878906250000, 7629394531250000
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OFFSET
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0,2
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COMMENTS
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For n>=2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4,5} we have f(x_1)<>y_1 and f(x_2)<> y_2. - Milan Janjic, Apr 19 2007
a(n) is the number of generalized compositions of n when there are 4 *i-1 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010
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REFERENCES
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A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
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LINKS
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FORMULA
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a(n) = 16*5^(n-2), a(0)=1, a(1)=3.
a(n) = 5*a(n-1) + (-1)^n*binomial(2,2-n).
G.f.: (1-x)^2/(1-5*x).
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MAPLE
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seq( `if`(n<2, 2*n+1, 16*5^(n-2)), n=0..30); # G. C. Greubel, Jan 21 2020
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MATHEMATICA
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PROG
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(PARI) Vec((1-x)^2/(1-5*x) + O(x^30)) \\ Altug Alkan, Mar 13 2016
(Magma) [1, 3] cat [16*5^(n-2): n in [2..30]]; // G. C. Greubel, Jan 21 2020
(Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1-x)^2/(1-5*x))); // Marius A. Burtea, Jan 21 2020
(Sage) [1, 3]+[16*5^(n-2) for n in (2..30)] # G. C. Greubel, Jan 21 2020
(GAP) Concatenation([1, 3], List([2..30], n-> 16*5^(n-2) )); # G. C. Greubel, Jan 21 2020
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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