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A277873
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Number of ways of writing n as a sum of powers of 5, each power being used at most five times.
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6
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1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 2, 2, 2, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 2, 2, 2, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 2, 2, 2, 3, 1, 1, 1, 1, 2
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OFFSET
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0,6
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COMMENTS
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Also known as the hyper 5-ary partition sequence, often denoted h_5(n).
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LINKS
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FORMULA
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G.f.: Product_{j >= 0} (1-x^(6*5^j))/(1-x^(5^j)).
G.f.: Product_{j >= 0} Sum_{k=0..5} x^(k*5^j).
a(0)=1; for k>0, a(5*k) = a(k)+a(k-1) and a(5*k+r) = a(k) with r=1,2,3,4.
G.f. A(x) satisfies: A(x) = (1 + x + x^2 + x^3 + x^4 + x^5) * A(x^5). - Ilya Gutkovskiy, Jul 09 2019
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EXAMPLE
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a(140) = 4 because 140 = 125+5+5+5 = 125+5+5+1+1+1+1+1 = 25+25+25+25+25+5+5+5 = 25+25+25+25+25+5+5+1+1+1+1+1.
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
add(b(n-j*5^i, i-1), j=0..min(5, n/5^i))))
end:
a:= n-> b(n, ilog[5](n)):
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MATHEMATICA
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n:=250; r:=3; (* To get up to n-th term, need r such that 5^r < n < 5^(r+1) *) h5 := CoefficientList[ Series[ Product[ (1 - q^(6*5^i))/(1 - q^(5^i)) , {i, 0, r}], {q, 0, n} ], q]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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