A048134 -id:A048134 - cp's OEIS frontend

A048240 Number of new colors that can be mixed with n units of yellow, blue, red.

1, 3, 3, 7, 9, 18, 15, 33, 30, 45, 42, 75, 54, 102, 81, 108, 108, 168, 117, 207, 156, 210, 195, 297, 204, 330, 270, 351, 306, 462, 300, 525, 408, 510, 456, 612, 450, 738, 567, 708, 600, 900, 594, 987, 750, 900, 825, 1173, 792, 1239, 930, 1200

Offset: 0

Jurjen N.E. Bos, N. J. A. Sloane, Robin Trew (trew(AT)hcs.harvard.edu)

T. D. Noe, Table of n, a(n) for n = 0..1000
Vadym Kurylenko, Thin Simplices via Modular Arithmetic, arXiv:2404.03975 [math.CO], 2024. See p. 33.

Cf. A048134, A048241.
A032125(n) = a(2^n).
Cf. A000217, A008683, A000741, A023023.

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;

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 *)

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

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

A048241 Number of colors that can be mixed with n >= 0 units of yellow, blue, red.

Original entry on oeis.org

1, 4, 7, 14, 23, 41, 56, 89, 119, 164, 206, 281, 335, 437, 518, 626, 734, 902, 1019, 1226, 1382, 1592, 1787, 2084, 2288, 2618, 2888, 3239, 3545, 4007, 4307, 4832, 5240, 5750, 6206, 6818, 7268, 8006, 8573, 9281, 9881, 10781, 11375, 12362

Offset: 0

Jurjen N.E. Bos, N. J. A. Sloane, Robin Trew (trew(AT)hcs.harvard.edu)

			a(2)=7: white, primary and secondary colors (null, Y^1, B^1, R^1, Y^1*B^1, Y^1*R^1, B^1*R^1).

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A048134, A048240.

Accumulate[ Table[ Sum[ MoebiusMu[n/d]*(d+1)*(d+2)/2, {d, Divisors[n]}], {n, 0, 43}]] + 1 (* Jean-François Alcover, Oct 16 2013, after T. D. Noe *)

a(n) = number of triples (i, j, k) with 0 <= i+j+k <= n and gcd(i, j, k) = 1.

a(n) = A048134(n)+1. - T. D. Noe, Jan 16 2007

A295849 Number of nonnegative solutions to gcd(x,y,z) = 1 and x^2 + y^2 + z^2 <= n.

Original entry on oeis.org

0, 3, 6, 7, 7, 13, 16, 16, 16, 19, 25, 28, 28, 34, 40, 40, 40, 49, 52, 55, 55, 61, 64, 64, 64, 70, 82, 85, 85, 97, 103, 103, 103, 109, 118, 124, 124, 130, 139, 139, 139, 154, 160, 163, 163, 169, 175, 175, 175, 181, 193, 199, 199, 211, 220, 220, 220, 226, 232, 241

Offset: 0

Seiichi Manyama, Nov 29 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000606, A048134, A295820, A295848.

N:= 100:
V:= Vector(N):
for x from 0 to floor(sqrt(N/3)) do
  for y from x to floor(sqrt((N-x^2)/2)) do
    for z from y to floor(sqrt(N-x^2-y^2)) do
      if igcd(x,y,z) = 1 then
        r:= x^2 + y^2 + z^2;
        m:= nops({x,y,z});
        if m=3 then V[r]:= V[r]+6
        elif m=2 then V[r]:= V[r]+3
        else V[r]:= V[r]+1
        fi
      fi
od od od:
0,op(ListTools:-PartialSums(convert(V,list))); # Robert Israel, Nov 30 2017

a[n_] := Sum[Boole[GCD[i, j, k] == 1], {i, 0, Sqrt[n]}, {j, 0, Sqrt[n - i^2]}, {k, 0, Sqrt[n - i^2 - j^2]}];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jul 07 2018, after Andrew Howroyd *)

a(n) = {sum(i=0, sqrtint(n), sum(j=0, sqrtint(n-i^2), sum(k=0, sqrtint(n-i^2-j^2), gcd([i, j, k]) == 1)))} \\ Andrew Howroyd, Dec 12 2017

a(n) = a(n-1) + A295848(n) for n > 0.

A048600 Array a(n,k) = number of colors that can be produced by n units of paint from k primary colors, read by descending antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 5, 6, 4, 1, 7, 13, 10, 5, 1, 11, 22, 26, 15, 6, 1, 13, 40, 51, 45, 21, 7, 1, 19, 55, 103, 100, 71, 28, 8, 1, 23, 88, 161, 221, 176, 105, 36, 9, 1, 29, 118, 277, 386, 422, 287, 148, 45, 10

Offset: 1

Jurjen N.E. Bos

			Table array begins:
  1  1  1   1   1
  2  3  5   7  11
  3  6 13  22  40
  4 10 26  51 103
  5 15 45 100 221
  ...
a(3,2) = 6 because you can take each color once, or mix two colors.

Cf. A005728 (row 2), A048134 (row 3). Cf. A048240, A048241.

max = 10; col[k_] := Accumulate[ Table[ Sum[ MoebiusMu[n/d]*Product[d+j, {j, 1, k}]/k!, {d, Divisors[n]}], {n, 1, max}]]; t = Table[col[k], {k, 0, max-1}] // Transpose; Flatten[ Table[ t[[n-k+1, k]], {n, 1, max}, {k, 1, n}]] (* Jean-François Alcover, Dec 26 2012 *)

All partitions of size n: if GCD is not 1, skip; else: fill the partition with zeros to get k numbers; count occurrences of each number (e.g.: 2 2 1 0 0 0 becomes 2 1 3); compute multinomial of k over these digits (e.g. 2 1 3 becomes 6!/(2!*1!*3!) = 60); sum.

Name edited by Michel Marcus, Aug 11 2024

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A048240 Number of new colors that can be mixed with n units of yellow, blue, red.

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A048241 Number of colors that can be mixed with n >= 0 units of yellow, blue, red.

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A295849 Number of nonnegative solutions to gcd(x,y,z) = 1 and x^2 + y^2 + z^2 <= n.

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A048600 Array a(n,k) = number of colors that can be produced by n units of paint from k primary colors, read by descending antidiagonals.

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