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A164283
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Number of ways to write n as the root-mean-square (RMS) of a set of distinct positive integers.
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4
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1, 1, 1, 1, 3, 9, 19, 79, 225, 693, 1901, 5597, 17641, 57503, 195431, 647139, 2182987, 7344451, 25057681, 85742999, 295284367, 1028155825, 3596134963, 12659796475, 44696280143, 158226554179, 562623263251, 2006471222195, 7182910999719, 25795458946677, 92875047372825, 335362896810137
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OFFSET
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1,5
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LINKS
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EXAMPLE
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a(6) = 9, because 6 is the RMS of 9 sets of distinct positive integers: 6 = RMS(6) = RMS(1,3,5,8,9) = RMS(3,4,5,7,9) = RMS(1,2,4,5,7,11) = RMS(1,3,5,6,8,9) = RMS(3,4,5,6,7,9) = RMS(1,2,3,5,7,8,10) = RMS(1,2,4,5,6,7,11) = RMS(1,2,3,5,6,7,8,10).
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MAPLE
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sns:= proc(i) option remember; `if`(i=1, 1, sns(i-1) +i^2) end: b:= proc(n, i, t) if n<0 or i<t then 0 elif n=0 then `if`(t=0, 1, 0) elif i=1 then `if`(n=1 and t=1, 1, 0) else b(n, i, t):= b(n, i-1, t) +b(n-i^2, i-1, t-1) fi end: a:= proc(n) option remember; local s, k; s:= 1; for k from 2 while sns(k)<=k*n^2 do s:= s +b(k*n^2, floor(sqrt(k*n^2 -sns(k-1))), k) od; s end: seq(a(n), n=1..15);
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MATHEMATICA
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sns[i_] := sns[i] = If[i == 1, 1, sns[i-1] + i^2] ; b[n_, i_, t_] := Which[n < 0 || i < t, 0, n == 0, If[t == 0, 1, 0], i == 1, If[n == 1 && t == 1, 1, 0], True, b[n, i, t] = b[n, i-1, t] + b[n - i^2, i-1, t-1]]; a[n_] := a[n] = Module[{s = 1, k}, For[k = 2, sns[k] <= k*n^2, k++, s = s + b[k*n^2, Floor[Sqrt[k*n^2 - sns[k-1]]], k]]; s]; Table[Print[an = a[n]]; an, {n, 1, 29}] (* Jean-François Alcover, Dec 30 2013, translated from Maple *)
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PROG
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(Haskell)
a164283 n = f [1..] 1 nn 0 where
f (k:ks) l nl xx
| yy > nl = 0
| yy < nl = f ks (l + 1) (nl + nn) yy + f ks l nl xx
| otherwise = if w == n then 1 else 0
where w = if r == 0 then a000196 m else 0
(m, r) = divMod yy l
yy = xx + k * k
nn = n ^ 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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STATUS
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approved
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