Tatsuru Murai - cp's OEIS frontend

A254243 Number of ways to partition the multiset consisting of 3 copies each of 1, 2, ..., n into n sets of size 3.

1, 1, 2, 10, 93, 1417, 32152, 1016489, 42737945, 2307295021, 155607773014, 12823004639504, 1267907392540573, 148160916629902965, 20199662575448858212, 3177820001990224608763, 571395567211112572679633, 116448309072281063992943561, 26700057600529091443246943530

Offset: 0

Tatsuru Murai, Jan 27 2015

nonn

			a(1) = 1: 111.
a(2) = 2: 111|222 and 112|122.
a(3) = 10: 111|222|333, 111|223|233, 112|122|333, 112|123|233, 112|133|223, 113|122|233, 113|123|223, 113|133|222, 122|123|133, and 123|123|123.

Andrew Howroyd, Table of n, a(n) for n = 0..100
P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41. See page 40 (but there is a typo).
StackExchange, Number of Partitioning a deck with m cards in n types into n-element sets, January 2015.

Cf. A002135 (2 instead of 3), A254233 (n copies each of 1, 2, and 3).

Column k=3 of A257463.

Name and example edited by Danny Rorabaugh, Apr 22 2015
a(6)-a(10) from Alois P. Heinz, Apr 22 2015
Terms a(11) and beyond from Andrew Howroyd, Apr 18 2020

A254233 Number of ways to partition the multiset consisting of n copies each of 1, 2, and 3 into n sets of size 3.

Original entry on oeis.org

1, 1, 4, 10, 25, 49, 103, 184, 331, 554, 911, 1424, 2204, 3278, 4817, 6896, 9746, 13487, 18480, 24882, 33192, 43683, 56994, 73512, 94131, 119340, 150300, 187732, 233065, 287248, 352153, 428944, 519949, 626737, 752095, 897994, 1067924, 1264241, 1491155, 1751672

Offset: 0

Tatsuru Murai, Jan 27 2015

			For n = 2, the set {1,1,2,2,3,3} can be partitioned into two sets in four ways: {{112},{233}}, {{113},{223}}, {{122},{133}}, and {{123},{123}}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A002135, A254243.

Column k=3 of A257462.

G.f.: (x^12-x^11+x^10+3*x^9+5*x^8+x^7+4*x^6+x^5+5*x^4+3*x^3+x^2-x+1) / ((x^2+1)*(x^2-x+1)*(x^2+x+1)^3*(x+1)^4*(x-1)^8). - Alois P. Heinz, Apr 21 2015

Fixed definition and examples by Kellen Myers, Apr 21 2015

a(14)-a(39) from Alois P. Heinz, Apr 21 2015

A086833 Minimum number of different addends occurring in any shortest addition chain of Brauer type for a given n, or 0 if n has no shortest addition chain of Brauer type.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 4, 3, 4, 5, 4, 4, 4, 3, 4, 4, 4, 4, 5, 5, 5, 4, 4, 4, 4, 4, 5, 5, 4, 6, 5, 4, 6, 4, 5, 5, 5, 5, 5, 5, 4, 4, 5, 4, 5, 5, 5, 5, 5, 4, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 7, 5, 5, 6, 4, 6, 7, 5, 6, 7, 5, 6, 6, 5, 5, 7, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5

Offset: 1

Tatsuru Murai, Aug 08 2003

nonn

n = 12509 is the first n for which a(n) = 0 because it is the smallest number that has no shortest addition chain of Brauer type. - Hugo Pfoertner, Jun 10 2006 [Edited by Pontus von Brömssen, Apr 25 2025]

			a(23)=5 because 23=1+1+2+1+4+9+5 is the shortest addition chain for 23.
For n=9 there are A079301(9)=3 different shortest addition chains, all of Brauer type:
[1 2 3 6 9] -> 9=1+1+1+3+3 -> 2 different addends {1,3}
[1 2 4 5 9] -> 9=1+1+2+1+4 -> 3 different addends {1,2,4}
[1 2 4 8 9] -> 9=1+1+2+4+1 -> 3 different addends {1,2,4}
The minimum number of different addends is 2, therefore a(9)=2.

Giovanni Resta, Tables of Shortest Addition Chains, computed by David W. Wilson.
Index to sequences related to the complexity of n

Cf. A003064, A003065, A003313, A005766, A008057, A008928, A008933, A079300, A079300, A079301, A349044.

a(n) = 0 if and only if n is in A349044. - Pontus von Brömssen, Apr 25 2025

Edited by Hugo Pfoertner, Jun 10 2006

Escape clause added by Pontus von Brömssen, Apr 25 2025

cp's OEIS Frontend

User: Tatsuru Murai

A254243 Number of ways to partition the multiset consisting of 3 copies each of 1, 2, ..., n into n sets of size 3.

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A254233 Number of ways to partition the multiset consisting of n copies each of 1, 2, and 3 into n sets of size 3.

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A086833 Minimum number of different addends occurring in any shortest addition chain of Brauer type for a given n, or 0 if n has no shortest addition chain of Brauer type.

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Extensions