cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A100450 Number of ordered triples (i,j,k) with |i| + |j| + |k| <= n and gcd(i,j,k) <= 1.

Original entry on oeis.org

1, 7, 19, 51, 99, 195, 291, 483, 675, 963, 1251, 1731, 2115, 2787, 3363, 4131, 4899, 6051, 6915, 8355, 9507, 11043, 12483, 14595, 16131, 18531, 20547, 23139, 25443, 28803, 31107, 34947, 38019, 41859, 45315, 49923, 53379, 58851, 63171, 68547
Offset: 0

Views

Author

N. J. A. Sloane, Nov 21 2004

Keywords

Comments

Note that gcd(0,m) = m for any m.
I would also like to get the sequences of the numbers of distinct sums i+j+k (also distinct products i*j*k) over all ordered triples (i,j,k) with |i| + |j| + |k| <= n; also over all ordered triples (i,j,k) with |i| + |j| + |k| <= n and gcd(i,j,k) <= 1.
Also the sequences of the numbers of distinct sums i+j+k (also distinct products i*j*k) over all ordered triples (i,j,k) with i >= 0, j >= 0, k >= 0 and i + j + k = n; also over all ordered triples (i,j,k) with i >= 0, j >= 0, k >= 0, i + j + k = n and gcd(i,j,k) <= 1.
Also the number of ordered triples (i,j,k) with i >= 0, j >= 0, k >= 0, i + j + k = n and gcd(i,j,k) <= 1.
From Robert Price, Mar 05 2013: (Start)
The sequences that address the previous comments are:
Distinct sums i+j+k with or without the GCD qualifier results in a(n)=2n+1 (A005408).
Distinct products i*j*k without the GCD qualifier is given by A213207.
Distinct products i*j*k with the GCD qualifier is given by A213208.
With the restriction i,j,k >= 0 ...
Distinct sums or products equal to n is trivial and always equals one (A000012).
Distinct sums <= n results in a(n)=n (A001477).
Distinct products <= n without the GCD qualifier is given by A213213.
Distinct products <= n with the GCD qualifier is given by A213212.
Ordered triples with sum = n without the GCD qualifier is A000217(n+1).
Ordered triples with sum = n with the GCD qualifier is A048240.
Ordered triples with sum <= n without the GCD qualifier is A000292.
Ordered triples with sum <= n with the GCD qualifier is A048241. (End)
This sequence (A100450) without the GCD qualifier results in A001845. - Robert Price, Jun 04 2013

Links

Crossrefs

Cf. A000124, A000292, A018805, A027430, A048240, A048241, A100448, A100449, A213207, A213208, A213212, A213213.

Programs

  • Maple
    f:=proc(n) local i,j,k,t1,t2,t3; t1:=0; for i from -n to n do for j from -n to n do t2:=gcd(i,j); for k from -n to n do if abs(i) + abs(j) + abs(k) <= n then t3:=gcd(t2,k); if t3 <= 1 then t1:=t1+1; fi; fi; od: od: od: t1; end;
  • Mathematica
    f[n_] := Length[ Union[ Flatten[ Table[ If[ Abs[i] + Abs[j] + Abs[k] <= n && GCD[i, j, k] <= 1, {i, j, k}, {0, 0, 0}], {i, -n, n}, {j, -n, n}, {k, -n, n}], 2]]]; Table[ f[n], {n, 0, 40}] (* Robert G. Wilson v, Dec 14 2004 *)

Formula

G.f.: (3 + Sum_{k>=1} (moebius(k)*((1+x^k)/(1-x^k))^3))/(1-x). - Vladeta Jovovic, Nov 22 2004. [Sketch of proof: Let b(n) = number of ordered triples (i, j, k) with |i| + |j| + |k| = n and gcd(i, j, k) <= 1. Then a(n) = A100450(n) = partial sums of b(n) and Sum_{d divides n} b(d) = 4*n^2+2 = A005899(n) with g.f. ((1+x)/(1-x))^3.]

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

Original entry on oeis.org

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

Views

Author

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

Keywords

Links

Crossrefs

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

Programs

  • Maple
    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;
  • Mathematica
    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 *)

Formula

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

A048134 Number of colors that can be mixed with up to n units of yellow, blue, red.

Original entry on oeis.org

0, 3, 6, 13, 22, 40, 55, 88, 118, 163, 205, 280, 334, 436, 517, 625, 733, 901, 1018, 1225, 1381, 1591, 1786, 2083, 2287, 2617, 2887, 3238, 3544, 4006, 4306, 4831, 5239, 5749, 6205, 6817, 7267, 8005, 8572, 9280, 9880, 10780, 11374, 12361
Offset: 0

Views

Author

Jurjen N.E. Bos

Keywords

Examples

			a(2)=6: primary and secondary colors (Y^1, B^1, R^1, Y^1*B^1, Y^1*R^1, B^1*R^1).

Links

Crossrefs

Two colors gives A005728.
Cf. A048240, A048241.

Programs

Formula

a(n) = number of triples (i, j, k) with 1 <= i+j+k <= n and gcd(i, j, k) = 1.
Cumulative sums of A048240(k) for k>0.

Extensions

More terms from Robin Trew (trew(AT)hcs.harvard.edu).

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

Views

Author

Jurjen N.E. Bos

Keywords

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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.

Extensions

Name edited by Michel Marcus, Aug 11 2024
Showing 1-4 of 4 results.

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