A008933 -id:A008933 - cp's OEIS frontend

A010787 Duplicate of A008933.

1, 1, 2, 6, 25, 135, 913, 7499, 73191, 833597, 10917343, 162402263

Offset: 1

dead

A008928 Number of increasing sequences of addition chain type with maximal element n.

1, 1, 1, 2, 3, 6, 10, 21, 38, 77, 144, 293, 563, 1131, 2205, 4434, 8711, 17466, 34506, 69169, 137247, 274677, 546081, 1093217, 2177556, 4356756, 8688370, 17381926, 34691608, 69394626, 138578144, 277197191, 553794526, 1107654097, 2213527055, 4427345544, 8849519773

Offset: 1

Mauro Torelli (torelli(AT)hermes.mc.dsi.unimi.it)

nonn

This sequence counts all addition chains for n. - David W. Wilson, Apr 01 2006

In other words, a(n) = the number of increasing addition chains ending in n. - Don Reble, Apr 09 2006

Martin Fuller, Table of n, a(n) for n = 1..64
Don Reble, Python program
Mauro Torelli, Increasing integer sequences and Goldbach's conjecture, RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, 40:2 (2006), pp. 107-121.

Cf. A008927, A079301, A008933.

More terms from David W. Wilson, Apr 01 2006

More terms from Don Reble, Apr 09 2006

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

A116511 Table T(n,k) = number of strictly increasing addition chains of length n whose final value is k.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 2, 2, 0, 1, 0, 0, 0, 0, 1, 3, 5, 5, 3, 4, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 4, 9, 14, 17, 15, 17, 10, 14, 4, 10, 2, 7, 0, 6, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 5, 14, 28, 45, 60, 67, 78, 66, 81, 51, 73, 33, 65, 29, 40, 4, 47, 14, 24, 5, 23, 0, 12

Offset: 1

Franklin T. Adams-Watters, Mar 22 2006

Row n has 2^(n-1) entries (starting with n-1 zeros).

			Table starts:
1,
0,1,
0,0,1,1,
0,0,0,1,2,2,0,1,
0,0,0,0,1,3,5,5,3,4,0,3,0,0,0,1,

Row sums A008933, column sums A008928.

cp's OEIS Frontend

A010787 Duplicate of A008933.

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A008928 Number of increasing sequences of addition chain type with maximal element n.

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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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A116511 Table T(n,k) = number of strictly increasing addition chains of length n whose final value is k.

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