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.

A216999 Number of integers obtainable from 1 in n steps using addition, multiplication, and subtraction.

Original entry on oeis.org

1, 3, 6, 13, 38, 153, 867, 6930, 75986, 1109442, 20693262, 477815647
Offset: 0

Views

Author

Stan Wagon, Sep 22 2012

Keywords

Comments

A straight-line program is a sequence that starts at 1 and has each entry obtained from two preceding entries by addition, multiplication, or subtraction. S(n) is the set of integers obtainable at any point in a straight-line program using n steps. Thus S(0) = {1}, S(1) = {0,1,2}, S(2) = {-1,0,1,2,3,4}; the sequence here is the cardinality of S(n).

Links

Crossrefs

Cf. A003065, A141414, A173419, A288849, A288850.

Programs

  • Mathematica
    extend[p_] :=  Module[{q = Tuples[p, {2}], new},
  new = Flatten[Table[{Total[t], Subtract @@ t, Times @@ t}, {t, q}]];
  Union[ Sort /@  DeleteCases[ Table[If[! MemberQ[p, n], Append[p, n]], {n, new}], Null]]] ;
P[0] = {{1}};
P[n_] := P[n] = DeleteDuplicates[Flatten[extend /@ P[n - 1], 1]];
S[n_] := DeleteDuplicates[Flatten[P[n]]];
Length /@ S /@ Range[6]

Extensions

a(9)-a(11) (Michael Collier verified independently the 1109442, 20693262 values) by Gil Dogon, Sep 27 2013

A288850 Number of distinct nonnegative rational numbers that can be obtained in n steps by a straight-line program (SLP) starting at 1 using addition, subtraction, multiplication and division.

Original entry on oeis.org

1, 3, 6, 16, 58, 321, 2585, 30198
Offset: 0

Views

Author

Hugo Pfoertner, Jun 18 2017

Keywords

Examples

			The sets of numbers obtainable at the n-th step are:
S(0) = { 1 },
S(1) = { 0, 1, 2 },
S(2) = { 0, 1/2, 1, 2, 3, 4 },
S(3) = { 0, 1/4, 1/3, 1/2, 2/3, 1, 3/2, 2, 5/2, 3, 4, 5, 6, 8, 9, 16 }.

Links

Crossrefs

Cf. A216999, A288760, A288849.

Extensions

a(7) from Alois P. Heinz, Jun 18 2017

A288759 Number of distinct rational numbers that can be obtained in n steps by applying addition, subtraction, multiplication and division to any two potentially identical numbers from the complete set of numbers created in n-1 steps, starting with the set {1}.

Original entry on oeis.org

1, 3, 8, 38, 555, 74423, 902663448
Offset: 0

Views

Author

Hugo Pfoertner, Jun 15 2017

Keywords

Comments

This is different from a straight-line program (SLP), which can only use numbers created in the path to its own result at level n-1. A288849 provides the cardinalities of the sets that can be created by the related SLPs.

Examples

			The sets of numbers obtainable at the n-th step are:
S(0) = { 1 },
S(1) = { 0, 1, 2 },
S(2) = { -2, -1, 0, 1/2, 1, 2, 3, 4 },
S(3) = { -8, -6, -5, -4, -7/2, -3, -5/2, -2, -3/2, -1, -2/3, -1/2, -1/3, -1/4, 0, 1/8, 1/6, 1/4, 1/3, 1/2, 2/3, 3/4, 1, 4/3, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5, 6, 7, 8, 9, 12, 16 }.

Crossrefs

Cf. A214872, A216999, A288760, A288849.

Extensions

a(6) from Hugo Pfoertner and Markus Sigg, Aug 06 2017
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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