A104429 -id:A104429 - cp's OEIS frontend

A360334 Array read by antidiagonals downwards: A(n,m) = number of set partitions of [3n] into 3-element subsets {i, i+k, i+2k} with 1 <= k <= m.

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 2, 5, 7, 8, 1, 1, 2, 5, 12, 13, 13, 1, 1, 2, 5, 15, 25, 24, 21, 1, 1, 2, 5, 15, 35, 56, 44, 34, 1, 1, 2, 5, 15, 46, 84, 126, 81, 55, 1, 1, 2, 5, 15, 55, 129, 211, 281, 149, 89, 1, 1, 2, 5, 15, 55, 185, 346, 537, 625, 274, 144, 1

Offset: 1

Peter Dolland, Feb 03 2023

			Square array begins:
  1,  1,   1,   1,    1,    1,    1,     1,     1, ...
  1,  2,   2,   2,    2,    2,    2,     2,     2, ...
  1,  3,   4,   5,    5,    5,    5,     5,     5, ...
  1,  5,   7,  12,   15,   15,   15,    15,    15, ...
  1,  8,  13,  25,   35,   46,   55,    55,    55, ...
  1, 13,  24,  56,   84,  129,  185,   232,   232, ...
  1, 21,  44, 126,  211,  346,  567,   831,  1040, ...
  1, 34,  81, 281,  537,  973, 1781,  2920,  4242, ...
  1, 55, 149, 625, 1352, 2732, 5643, 10213, 16110, ...
  ...

Main diagonal is A334250.
Columns 1..3 are A000012, A000045(n+1), A000073(n+2).
Cf. A104429, A104443, A360333, A360335.

A(n,m) = A104429(n) = A104443(n,3) for m >= floor((3n - 1) / 2).

A104434 Number of ways to split 1, 2, 3, ..., 8n into n arithmetic progressions each with 8 terms.

Original entry on oeis.org

1, 1, 2, 4, 10, 20, 56, 116, 321, 739, 1881, 4200, 12776, 28528, 74020, 179197, 492839, 1146192

Offset: 0

Jonas Wallgren, Mar 17 2005

Cf. A104429-A104433, A104435-A104443.

a(0), a(10)-a(17) from Alois P. Heinz, Nov 18 2020

A104436 Number of ways to split 1, 2, 3, ..., 3n into 3 arithmetic progressions each with n terms.

Original entry on oeis.org

1, 15, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Jonas Wallgren, Mar 17 2005

nonn

Cf. A104429-A104435, A104437-A104443.

A202954 Number of partitions of [1,...,3n] into triples satisfying x+y=4z.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 6, 0, 5, 0, 0, 349, 0, 443, 0, 0, 110757, 0, 1254452, 0, 0, 152965479, 0

Offset: 0

N. J. A. Sloane, Dec 26 2011

a(n)=0 when n is not congruent to 0 or 3 mod 5. - Edward Moody, Jan 17 2021

R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.] See Table II.6.

Cf. A108235, A104429.

a(14)-a(17) from Alois P. Heinz, Dec 28 2011
a(18) from Alois P. Heinz, Jan 04 2012
a(19)-a(24) from Edward Moody, Jan 17 2021

A284737 A condensed version of A108235.

Original entry on oeis.org

1, 1, 8, 21, 3040, 20505, 10567748, 103372655

Offset: 1

N. J. A. Sloane, Apr 09 2017

Lists only the terms A108235(n) for n == 0 or 3 mod 12.

See A108235, the main entry for this sequence, for more information.

R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]
Ana Rechtman, Août 2018, 3e défi, Images des Mathématiques, CNRS, 2018 (in French).

Cf. A108235.

A284738 A condensed version of A202951.

Original entry on oeis.org

1, 1, 6, 10, 700

Offset: 1

N. J. A. Sloane, Apr 09 2017

Lists only the terms A202951(n) for n == 0 or 3 mod 12.

See A202951, the main entry for this sequence, for more information.

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

Cf. A202951.

A284739 A condensed version of A202952.

Original entry on oeis.org

0, 0, 2, 11, 2300

Offset: 1

N. J. A. Sloane, Apr 09 2017

Lists only the terms A202952(n) for n == 0 or 3 mod 12.

See A202952, the main entry for this sequence, for more information.

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

Cf. A202952.

A284740 A condensed version of A202954.

Original entry on oeis.org

1, 0, 0, 6, 5, 349, 443, 110757

Offset: 1

N. J. A. Sloane, Apr 09 2017

Lists only the terms A202954(n) for n == 0 or 3 mod 12.

See A202954, the main entry for this sequence, for more information.

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

Cf. A202954.

A330285 The maximum number of arithmetic progressions in a sequence of length n.

Original entry on oeis.org

0, 0, 1, 3, 7, 12, 20, 29, 41, 55, 72, 90, 113, 137, 164, 194, 228, 263, 303, 344, 390, 439, 491, 544, 604, 666, 731, 799, 872, 946, 1027, 1109, 1196, 1286, 1379, 1475, 1579, 1684, 1792, 1903, 2021, 2140, 2266, 2393, 2525, 2662, 2802, 2943, 3093, 3245, 3402, 3562, 3727

Offset: 1

Joseph Wheat, Dec 21 2019

nonn

The partial arithmetic density D_n(A) up to n is merely the number of arithmetic progressions, A(s(n)), divided by the total number of nonempty subsets of {s(1), s(2), ..., s(n)}, i.e., A(s(n))/(2^n - 1). As n approaches infinity, D_n(A) converges to zero. Furthermore, the infinite sum of the partial densities for any sequence always converges to the total density D(A). Every infinite arithmetic progression has the same total density, Sum_{n >= 1} a(n)/(2^n - 1) = alpha ~ 1.25568880818612911696845537; sequences with a finite number of progressions have D(A) < alpha; and sequences without any arithmetic progressions have D(A) = 0.

Encyclopedia of Mathematics, Density of a sequence
Eric Weisstein's World of Mathematics, Arithmetic Progression

Cf. A049988, A130518, A104429, A065825, A092482.

Partial sums of A002541.

a(n) = sum(i=1, n, sum(j=1, i, floor((i - 1)/(j + 1))))

a(n) = Sum_{i=1..n} Sum_{j=1..i} floor((i - 1)/(j + 1)).

cp's OEIS Frontend

A360334 Array read by antidiagonals downwards: A(n,m) = number of set partitions of [3n] into 3-element subsets {i, i+k, i+2k} with 1 <= k <= m.

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A104434 Number of ways to split 1, 2, 3, ..., 8n into n arithmetic progressions each with 8 terms.

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A104436 Number of ways to split 1, 2, 3, ..., 3n into 3 arithmetic progressions each with n terms.

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A202954 Number of partitions of [1,...,3n] into triples satisfying x+y=4z.

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Extensions

A284737 A condensed version of A108235.

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A284738 A condensed version of A202951.

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A284739 A condensed version of A202952.

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A284740 A condensed version of A202954.

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A330285 The maximum number of arithmetic progressions in a sequence of length n.

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