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A147980
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Given a set of positive integers A={1,2,...,n-1,n}, n>=2. Take subsets of A of the form {1,...,n} so only subsets containing numbers 1 and n are allowed. Then a(1)=1 and a(n) is the number of subsets where arithmetic mean of the subset is an integer.
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1
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1, 0, 2, 0, 4, 4, 8, 12, 28, 44, 84, 156, 288, 540, 1020, 1904, 3616, 6860, 13024, 24836, 47448, 90772, 174072, 334348, 643112, 1238928, 2389956, 4615916, 8925808, 17278680, 33482196, 64944060, 126083448, 244989096, 476416560, 927167752, 1805691728, 3519062820
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OFFSET
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1,3
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COMMENTS
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For n odd the value of the arithmetic mean for each possible subset equals (n+1)/2. For n even this value is n/2 or (n+2)/2. If looking after RootMeanSquare for the subset we obtain a sequence [1,0,0,0,0,0,2,...]. We see for example for n=7, A={1,2,3,4,5,6,7} and the only 2 subsets with an integer RootMeanSquare are {1,7}, {1,5,7}. Interestingly the value of RootMeanSquare is 5 for both subsets. So the sequence A140480 RMS numbers is a subsequence of it as a set of divisors of n is clearly a subset of n of the form {1,...,n}.
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LINKS
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EXAMPLE
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n=5, A={1,2,3,4,5}. Subsets of A starting with 1 and ending with 5 are : {1,5}, {1,2,5}, {1,3,5}, {1,4,5}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {1,2,3,4,5}. Arithmetic mean of the subset is an integer for subsets : {1,5}, {1,3,5}, {1,2,4,5}, {1,2,3,4,5}. Thus a(5) = 4. The value of the arithmetic mean is 3 for all 4 subsets.
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MAPLE
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b:= proc(i, s, c) option remember; `if` (i=1, `if` (irem (s, c)=0, 1, 0), b(i-1, s, c)+ b(i-1, s+i, c+1)) end: a:= n-> `if` (n=1, 1, b (n-1, n+1, 2)): seq (a(n), n=1..40); # Alois P. Heinz, May 06 2010
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MATHEMATICA
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b[i_, s_, c_] := b[i, s, c] = If[i==1, If[Mod[s, c]==0, 1, 0], b[i-1, s, c] + b[i-1, s+i, c+1]];
a[n_] := If[n==1, 1, b[n-1, n+1, 2]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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