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 1-3 of 3 results.

A008927 Number of increasing sequences of star chain type with maximal element n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 20, 36, 70, 130, 252, 475, 916, 1745, 3362, 6438, 12410, 23852, 46020, 88697, 171339, 330938, 640189, 1238751, 2399677, 4650819, 9021862, 17510819, 34013311, 66106491, 128568177, 250191797, 487168941, 949133722, 1850211247, 3608650388
Offset: 1

Views

Author

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

Keywords

Comments

a(n) counts the Brauer addition chains for n, which are equivalent to star chains. In a Brauer chain, each element after the first is the sum of any previous element with the immediately previous element. This sequence counts all Brauer chains for n, not just the minimal ones, which are given by A079301. - David W. Wilson, Apr 01 2006
In other words, a(n) = the number of increasing star addition chains ending in n.

Examples

			a(5)=3 because 1,2,3,4,5; 1,2,3,5; 1,2,4,5 are star-kind addition chains.
a(8)=20 because there are 21 increasing addition chains up to 8, but 1,2,4,5,8 is not a star chain.

References

  • M. Torelli, Increasing integer sequences and Goldbach's conjecture, preprint, 1996.
  • D. E. Knuth, The Art of Computer Programming; Addison-Wesley. Section 4.6.3.

Links

Crossrefs

Cf. A008928, A079301.

Formula

Conjecture: a(n) ~ 2^n/n. - Martin Fuller, Apr 29 2025

Extensions

More terms from David W. Wilson, Apr 01 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

Views

Author

Tatsuru Murai, Aug 08 2003

Keywords

Comments

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]

Examples

			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.

Links

Crossrefs

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

Formula

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

Extensions

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

Views

Author

Franklin T. Adams-Watters, Mar 22 2006

Keywords

Comments

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

Examples

			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,

Crossrefs

Row sums A008933, column sums A008928.
Showing 1-3 of 3 results.

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