A211868 - OEIS

%I #14 Oct 26 2024 22:59:40

%S 1,0,1,0,3,0,3,0,9,1,19,2,59,13,161,50,413,123,1201,352,3463,689,

%T 10921,1585,35365,5409,110773,20950,359725,82702,1192801,320873,

%U 3998397,1096384,13584075,3417934,45973713,10657777,157515581,33447019,543663919,111463220

%N Number of ways to write n as the root-mean-square (RMS) of a set of distinct odd integers.

%H Eric Weisstein's World of Math, <a href="http://mathworld.wolfram.com/Root-Mean-Square.html">Root-Mean-Square</a>

%e a(5) = 3: 5 = RMS(5) = RMS(1,7) = RMS(1,5,7);

%e a(7) = 3: 7 = RMS(7) = RMS(1,5,11) = RMS(1,5,7,11);

%e a(9) = 9: 9 = RMS(9) = RMS(5,7,13) = RMS(5,7,9,13) = RMS(3,5,11,13) = RMS(3,5,9,11,13) = RMS(1,3,7,11,15) = RMS(1,3,7,9,11,15) = RMS(1,3,5,17) = RMS(1,3,5,9,17);

%e a(10) = 1: 10 = RMS(1,3,5,7,9,11,15,17);

%e a(11) = 19: 11 = RMS(11) = RMS(3,9,13,15) = RMS(3,9,11,13,15) = RMS(5,7,17) = RMS(5,7,11,17) = RMS(1,5,13,17) = RMS(1,5,11,13,17) = RMS(1,3,9,15,17) = RMS(1,3,9,11,15,17) = RMS(3,5,7,9,13,15,17) = RMS(3,5,7,9,11,13,15,17) = RMS(1,5,7,13,19) = RMS(1,5,7,11,13,19) = RMS(1,3,7,9,15,19) = RMS(1,3,7,9,11,15,19) = RMS(3,5,7,9,21) = RMS(3,5,7,9,11,21) = RMS(1,3,5,9,13,21) = RMS(1,3,5,9,11,13,21);

%e a(12) = 2: 12 = RMS(1,5,7,9,11,15,17,19) = RMS(1,3,5,7,9,13,17,23).

%o (Haskell)

%o a211868 n = f a005408_list 1 nn 0 where

%o    f (o:os) l nl xx

%o      | yy > nl   = 0

%o      | yy < nl   = f os (l + 1) (nl + nn) yy + f os l nl xx

%o      | otherwise = if w == n then 1 else 0

%o      where w = if r == 0 then a000196 m else 0

%o            (m, r) = divMod yy l

%o            yy = xx + o * o

%o    nn = n ^ 2

%Y Cf. A000196, A005408, A163974, A164283.

%K nonn

%O 1,5

%A _Reinhard Zumkeller_, Feb 13 2013

%E a(37)-a(40) from _Alois P. Heinz_, Feb 25 2013

%E a(41)-a(42) from _Alois P. Heinz_, May 03 2015