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.

A383331 Number of pairs of nonnegative integers, not both equal to 0, with a shortest vectorial addition chain of length n.

Original entry on oeis.org

2, 3, 7, 16, 37, 91, 229, 585, 1528, 4034, 10862
Offset: 0

Views

Author

Pontus von Brömssen, Apr 26 2025

Keywords

Comments

See A383330 for details.

Crossrefs

Row 2 of A383333.
Cf. A003065, A383330, A383332.

A384480 Square array read by antidiagonals: T(n,k) is the length of a shortest addition-composition chain for n*x+k, starting with 1 and x; n, k >= 0.

Original entry on oeis.org

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

Views

Author

Pontus von Brömssen, Jun 02 2025

Keywords

Comments

An addition-composition chain for the affine function f is a finite sequence of affine functions, starting with 1, x and ending with f, in which each element except 1 and x equals g(x)+h(x) or g(h(x)) for two preceding, not necessarily distinct, elements g(x) and h(x) in the chain. The length of the chain is the number of elements in the chain, excluding 1 and x. Such chains exist only for functions of the form f(x) = n*x+k, where n and k are nonnegative integers, not both 0.
T(0,0) = 0 by convention.
Equivalently, the chains can be defined on pairs (s,t) of nonnegative integers (corresponding to the function f(x) = s*x+t) with the operations (s,t)+(u,v) = (s+t,u+v) (addition) and (s,t)o(u,v) = (s*u,s*v+t) (composition).

Examples

			Array begins:
  n\k| 0  1  2  3  4  5  6  7  8  9 10 11 12
  ---+--------------------------------------
   0 | 0  0  1  2  2  3  3  4  3  4  4  5  4
   1 | 0  1  2  3  3  4  4  5  4  5  5  5  5
   2 | 1  2  2  3  3  4  4  4  4  5  5  5  5
   3 | 2  3  3  3  4  4  4  5  5  5  5  5  5
   4 | 2  3  3  3  3  4  3  4  4  4  4  5  4
   5 | 3  4  4  4  4  4  4  4  5  5  5  5  5
   6 | 3  4  4  4  4  4  4  5  4  4  5  5  5
   7 | 4  5  5  5  5  5  5  5  5  5  5  6  6
   8 | 3  4  4  4  4  4  4  4  4  5  4  5  4
   9 | 3  4  5  4  4  5  5  5  4  5  5  5  4
  10 | 4  5  5  5  5  5  5  5  5  5  5  6  5
  11 | 4  5  6  5  5  5  6  6  6  5  5  6  6
  12 | 4  5  5  5  5  5  5  5  5  5  5  5  5
For (n,k) = (4,6), the unique shortest chain for 4*x+6 is (1, x,) x+1, 2*x+2, 4*x+6 of length T(4,6) = 3. The last term of the chain is the composition of 2*x+2 with itself.
For (n,k) = (6,4), a shortest chain for 6*x+4 is (1, x,) x+1, 2*x+2, 3*x+2, 6*x+4 of length T(6,4) = 4. This chain uses only additions.

Crossrefs

Cf. A383330 (addition only), A384481, A384482, A384483 (row 0).

Formula

T(n,k) <= T(n,k-1) + 1.
T(n,k) <= T(n-1,k) + 1.

A383332 Smallest positive weight of a pair of nonnegative integers with a shortest vectorial addition chain of length n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 20, 29, 44, 70, 104
Offset: 0

Views

Author

Pontus von Brömssen, Apr 26 2025

Keywords

Comments

See A383330 for details.
The weight of a pair is the sum of its elements.

Examples

			   n | a(n) | pairs (x,y) with x <= y, x+y = a(n), and shortest chain length n
  ---+------+-----------------------------------------------------------------
   0 |   1  | (0,1)
   1 |   2  | (0,2), (1,1)
   2 |   3  | (0,3), (1,2)
   3 |   4  | (1,3)
   4 |   6  | (1,5)
   5 |   8  | (1,7)
   6 |  12  | (1,11)
   7 |  20  | (1,19), (3,17)
   8 |  29  | (6,23)
   9 |  44  | (7,37)
  10 |  70  | (11,59)
  11 | 104  | (15,89)

Crossrefs

Row 2 of A383334.
Cf. A003064, A383330, A383331.
Showing 1-3 of 3 results.

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