A104429 -id:A104429 - cp's OEIS frontend

A104433 Number of ways to split 1, 2, 3, ..., 7n into n arithmetic progressions each with 7 terms.

Original entry on oeis.org

1, 1, 2, 4, 10, 20, 56, 117, 323, 745, 1896, 4242, 12883, 29108, 75725, 183366, 504215, 1176776

Offset: 0

Jonas Wallgren, Mar 17 2005

Cf. A104429-A104432, A104434-A104443.

a(0), a(11)-a(17) from Alois P. Heinz, Nov 18 2020

A104435 Number of ways to split 1, 2, 3, ..., 2n into 2 arithmetic progressions each with n terms.

Original entry on oeis.org

1, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Jonas Wallgren, Mar 17 2005

The common difference in an arithmetic progression must be a positive integer. - David A. Corneth, Apr 14 2024

			From _R. J. Mathar_, Apr 14 2024: (Start)
a(2)=3 offers 3 ways of splitting (1,2,3,4): {(1,2),(3,4)}, {(1,3),(2,4)}, {(1,4),(2,3)}.
a(n)=2 for n>=3 because there are at least the two ways of splitting (1,2,..,2n) into the even and odd numbers. (End)

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A104429, A104430, A104431, A104432, A104433, A104434, A104436, A104437, A104438, A104439, A104440, A104441, A104442, A104443.

a(n) = if(n <= 2, [1,3][n], 2) \\ David A. Corneth, Apr 14 2024

a(1) = 1, a(2) = 3, a(n) = 2 for n >= 3. Proof of the latter: if the common difference in an arithmetic progression, starting with a number at least 1, is at least 3 then the largest term in that arithmetic progression is at least 1 + 3*(n-1) = 3*n - 2. But 3*n - 2 > 2*n for n > 2. - David A. Corneth, Apr 14 2024

G.f.: x*(1 + 2*x - x^2)/(1 - x). - Stefano Spezia, Apr 14 2024

More terms from David A. Corneth, Apr 14 2024

A104437 Number of ways to split 1, 2, 3, ..., 4n into 4 arithmetic progressions each with n terms.

Original entry on oeis.org

1, 105, 15, 11, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

Jonas Wallgren, Mar 17 2005

nonn

Cf. A104429-A104436, A104438-A104443.

A104438 Number of ways to split 1, 2, 3, ..., 5n into 5 arithmetic progressions each with n terms.

Original entry on oeis.org

1, 945, 55, 23, 21, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20

Offset: 1

Jonas Wallgren, Mar 17 2005

nonn

Cf. A104429-A104437, A104439-A104443.

A104439 Number of ways to split 1, 2, 3, ..., 6n into 6 arithmetic progressions each with n terms.

Original entry on oeis.org

1, 10395, 232, 68, 59, 57, 56, 56, 56, 56, 56, 56, 56, 56, 56

Offset: 1

Jonas Wallgren, Mar 17 2005

nonn

Cf. A104429-A104438, A104440-A104443.

A104440 Number of ways to split 1, 2, 3, ..., 7n into 7 arithmetic progressions each with n terms.

Original entry on oeis.org

1, 135135, 1161, 161, 125, 119, 117, 116, 116, 116, 116, 116, 116, 116, 116, 116

Offset: 1

Jonas Wallgren, Mar 17 2005

nonn

Cf. A104429-A104439, A104441-A104443.

A104441 Number of ways to split 1, 2, 3, ..., 8n into 8 arithmetic progressions each with n terms.

Original entry on oeis.org

1, 2027025, 6643, 488, 349, 329, 323, 321, 320, 320, 320, 320, 320, 320, 320, 320

Offset: 1

Jonas Wallgren, Mar 17 2005

nonn

Cf. A104429-A104440, A104442-A104443.

A202951 Number of Nickerson-type partitions of [1,...,3n] into triples satisfying x+y=z.

Original entry on oeis.org

1, 1, 0, 0, 6, 10, 0, 0, 700

Offset: 0

N. J. A. Sloane, Dec 26 2011

Perhaps an incorrect version of A004075? Sequence values are from p. 51 of Nowakowski. - Martin Fuller, Jul 06 2025

R. S. Nickerson, Problem E1845, Amer. Math. Monthly, p. 81, 73 (1966).
R. S. Nickerson and D. C. B. Marsh, E1845: A variant of Langford's Problem, American Math. Monthly, 1967, 74, 591-595.
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

Cf. A108235, A202952.

A202952 A108235(n)-A202951(n).

Original entry on oeis.org

0, 0, 0, 0, 2, 11, 0, 0, 2300

Offset: 0

N. J. A. Sloane, Dec 26 2011

R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

Cf. A108235, A202951.

A334250 Number of set partitions of [3n] into 3-element subsets {i, i+k, i+2k} with 1<=k<=n.

Original entry on oeis.org

1, 1, 2, 4, 12, 35, 129, 567, 2920, 16110, 103467, 717608, 5748214, 47937957, 441139750, 4319093093, 45963368076, 510202534002, 6150655137844, 76789781005325, 1028853084775725, 14294680087131380

Offset: 0

Alois P. Heinz, Apr 20 2020

Differs from A331621 first at n=7.

			a(2) = 2: 123|456, 135|246.
a(3) = 4: 123|456|789, 123|468|579, 135|246|789, 147|258|369.

Wikipedia, Partition of a set

Cf. A014307 (the same for 2-element subsets), A025035, A059108, A104429 (where k is not restricted), A285527, A331621, A337520.

Main diagonal of A360334.

b:= proc(s, t) option remember; `if`(s={}, 1, (m-> add(
     `if`({m-j, m-2*j} minus s={}, b(s minus {m, m-j, m-2*j},
            t), 0), j=1..min(t, iquo(m-1, 2))))(max(s)))
    end:
a:= proc(n) option remember; forget(b): b({$1..3*n}, n) end:
seq(a(n), n=0..12);

b[s_List, t_] := b[s, t] = If[s == {}, 1, Function[m, Sum[If[{m - j, m - 2j} ~Complement~ s == {}, b[s ~Complement~ {m, m - j, m - 2j}, t], 0], {j, 1, Min[t, Quotient[m - 1, 2]]}]][Max[s]]];
a[n_] := a[n] = b[Range[3n], n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 12}] (* Jean-François Alcover, May 10 2020, after Maple *)

a(n) <= A104429(n) <= A025035(n).

a(17)-a(21) from Martin Fuller, Jul 19 2025

cp's OEIS Frontend

A104433 Number of ways to split 1, 2, 3, ..., 7n into n arithmetic progressions each with 7 terms.

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A104435 Number of ways to split 1, 2, 3, ..., 2n into 2 arithmetic progressions each with n terms.

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A104437 Number of ways to split 1, 2, 3, ..., 4n into 4 arithmetic progressions each with n terms.

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A104438 Number of ways to split 1, 2, 3, ..., 5n into 5 arithmetic progressions each with n terms.

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A104439 Number of ways to split 1, 2, 3, ..., 6n into 6 arithmetic progressions each with n terms.

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A104440 Number of ways to split 1, 2, 3, ..., 7n into 7 arithmetic progressions each with n terms.

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A104441 Number of ways to split 1, 2, 3, ..., 8n into 8 arithmetic progressions each with n terms.

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A202951 Number of Nickerson-type partitions of [1,...,3n] into triples satisfying x+y=z.

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A202952 A108235(n)-A202951(n).

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A334250 Number of set partitions of [3n] into 3-element subsets {i, i+k, i+2k} with 1<=k<=n.

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Programs

Maple

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Formula

Extensions