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.

A383330 Triangle read by rows: T(n,k) is the length of a shortest vectorial addition chain for (n,k), 0 <= k <= n.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 2, 3, 3, 3, 2, 3, 3, 4, 3, 3, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 5, 4, 4, 5, 5, 5, 5, 5, 5, 5, 3, 4, 4, 5, 4, 5, 5, 6, 4, 4, 5, 5, 5, 5, 5, 5, 6, 5, 5, 4, 5, 5, 5, 5, 5, 5, 6, 5, 6, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 5, 5, 5, 5, 6, 5, 6, 5, 6, 6, 7, 5
Offset: 0

Views

Author

Pontus von Brömssen, Apr 26 2025

Keywords

Comments

Starting with (1,0) and (0,1), each pair of the chain must be equal to the sum of two preceding pairs. The length of the chain is defined to be the number of pairs in the chain, excluding (1,0) and (0,1).
Also, T(n,k) is the least number of multiplications needed to obtain x^n*y^k, starting with x and y.
T(0,0) = 0 by convention.

Examples

			Triangle begins:
  n\k| 0  1  2  3  4  5  6  7  8  9 10
  ---+--------------------------------
   0 | 0
   1 | 0  1
   2 | 1  2  2
   3 | 2  3  3  3
   4 | 2  3  3  4  3
   5 | 3  4  4  4  4  4
   6 | 3  4  4  4  4  5  4
   7 | 4  5  5  5  5  5  5  5
   8 | 3  4  4  5  4  5  5  6  4
   9 | 4  5  5  5  5  5  5  6  5  5
  10 | 4  5  5  5  5  5  5  6  5  6  5
A shortest addition chain for (11,7) is [(1,0), (0,1),] (1,1), (2,1), (4,2), (5,3), (10,6), (11,7) of length T(11,7) = 6.

Links

Crossrefs

Cf. A003313 (column k=0, excluding T(0,0)), A265690 (column k=1 and main diagonal; apparently also column k=2), A383331, A383332.

A265691 Irregular triangle read by rows: row n gives lexicographically first addition chain of minimal length for n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 1, 2, 3, 5, 1, 2, 3, 6, 1, 2, 3, 4, 7, 1, 2, 4, 8, 1, 2, 3, 6, 9, 1, 2, 3, 5, 10, 1, 2, 3, 4, 7, 11, 1, 2, 3, 6, 12, 1, 2, 3, 5, 8, 13, 1, 2, 3, 4, 7, 14, 1, 2, 3, 5, 10, 15, 1, 2, 4, 8, 16, 1, 2, 4, 8, 9, 17, 1, 2, 3, 6, 9, 18, 1, 2, 3, 4, 8, 11, 19, 1, 2, 3, 5, 10, 20
Offset: 1

Views

Author

Franklin T. Adams-Watters, Dec 13 2015

Keywords

Comments

The addition chains in this sequence for 1, 2, 3, 7 start respectively with 1, 1,2, 1,2,3, and 1,2,3,4. Are there minimal addition chains starting 1,2,3,4,5? 1,2,...,n for larger n?
Needs a b-file.

Examples

			The table starts:
1
1 2
1 2 3
1 2 4
1 2 3 5
1 2 3 6
1 2 3 4 7
1 2 4 8
...

Crossrefs

Cf. A265690 (row lengths), A265692 (largest).

A265692 Irregular triangle read by rows: lexicographically last addition chain of minimal length for n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 1, 2, 4, 5, 1, 2, 4, 6, 1, 2, 4, 6, 7, 1, 2, 4, 8, 1, 2, 4, 8, 9, 1, 2, 4, 8, 10, 1, 2, 4, 8, 10, 11, 1, 2, 4, 8, 12, 1, 2, 4, 8, 12, 13, 1, 2, 4, 8, 12, 14, 1, 2, 4, 5, 10, 15, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 17, 1, 2, 4, 8, 16, 18, 1, 2, 4, 8, 16, 18, 19, 1, 2, 4, 8, 16, 20
Offset: 1

Views

Author

Franklin T. Adams-Watters, Dec 13 2015

Keywords

Comments

While addition chains are normally entered with terms in increasing order, the usual definition requires only that each term follows a pair of terms of which it is the sum. This sequence will eventually contain rows that are not monotonic; in particular, any non-Brauer number will have such a chain. What is the first number whose chain in this sequence is not monotonic?
47 is the first number with a non-monotonic lexicographically last addition chain of minimal length: [1, 2, 4, 8, 16, 5, 21, 42, 47]. - Alois P. Heinz, Dec 22 2015
Needs a b-file.

Examples

			The table starts:
1
1 2
1 2 3
1 2 4
1 2 4 5
1 2 4 6
1 2 4 6 7
1 2 4 8
...

Links

Crossrefs

Cf. A265690 (row lengths), A265691 (smallest).
Showing 1-3 of 3 results.

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