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Number of new colors that can be mixed with n units of yellow, blue, red.
7

%I #23 Apr 11 2024 10:33:47

%S 1,3,3,7,9,18,15,33,30,45,42,75,54,102,81,108,108,168,117,207,156,210,

%T 195,297,204,330,270,351,306,462,300,525,408,510,456,612,450,738,567,

%U 708,600,900,594,987,750,900,825,1173,792,1239,930,1200

%N Number of new colors that can be mixed with n units of yellow, blue, red.

%H T. D. Noe, <a href="/A048240/b048240.txt">Table of n, a(n) for n = 0..1000</a>

%H Vadym Kurylenko, <a href="https://arxiv.org/abs/2404.03975">Thin Simplices via Modular Arithmetic</a>, arXiv:2404.03975 [math.CO], 2024. See p. 33.

%F a(n) = number of triples (i, j, k) with i+j+k = n and gcd(i, j, k) = 1.

%F a(n) = Sum_{d|n} mu(n/d)*(d+1)*(d+2)/2. G.f.: Sum_{k>0} mu(k)/(1-x^k)^3. - _Vladeta Jovovic_, Dec 22 2002

%p A048240 := proc(n) local ans, i, j, k; ans := 0; for i from n by -1 to 0 do for j from n by -1 to 0 do k := n - i - j; if 0 <= k and k <= n and gcd(gcd(i, j), k) = 1 then ans := ans + 1; fi; od; od; RETURN(ans); end;

%t a[n_] := Sum[ MoebiusMu[n/d]*(d+1)*(d+2)/2, {d, Divisors[n]}]; a[0] = 1; Table[a[n], {n, 0, 51}] (* _Jean-François Alcover_, Jun 14 2012, after _Vladeta Jovovic_ *)

%Y Cf. A048134, A048241.

%Y A032125(n) = a(2^n).

%Y Cf. A000217, A008683, A000741, A023023.

%K nonn,easy

%O 0,2

%A _Jurjen N.E. Bos_, _N. J. A. Sloane_, Robin Trew (trew(AT)hcs.harvard.edu)