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A164283 Number of ways to write n as the root-mean-square (RMS) of a set of distinct positive integers. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Table of n, a(n) for n=1..32.

Eric Weisstein's World of Math, Root-Mean-Square

EXAMPLE

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).

MAPLE

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);

MATHEMATICA

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-Fran├žois Alcover, Dec 30 2013, translated from Maple *)

PROG

(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

-- Reinhard Zumkeller, Feb 13 2013

CROSSREFS

Cf. A163974, A066572, A066571, A072701.

Cf. A000196, A211868.

Sequence in context: A130586 A147146 A146066 * A178963 A033315 A200612

Adjacent sequences:  A164280 A164281 A164282 * A164284 A164285 A164286

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Aug 12 2009

STATUS

approved

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Last modified August 1 13:00 EDT 2021. Contains 346391 sequences. (Running on oeis4.)