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

A288760 Number of distinct nonnegative 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, 6, 24, 300, 37761, 451572162
Offset: 0

Views

Author

Hugo Pfoertner, Jun 15 2017

Keywords

Comments

The conjectured value of a(6)=451572162 needs independent verification.
For an explanation of the difference from a straight-line program (SLP) see comment in A288759. A288850 provides the corresponding cardinalities of the sets that can be obtained by an n-step SLP.

Examples

			The sets of numbers >=0 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, 1/8, 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 }.

Links

Crossrefs

Cf. A214872, A216999, A288759, A288850.

Extensions

Wrong a(6) removed by Hugo Pfoertner, Jun 19 2017
a(6) from Markus Sigg, Jul 01 2017

A288849 Number of distinct 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, 7, 21, 83, 484, 4084, 49479
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) = { -1, 0, 1/2, 1, 2, 3, 4 },
S(3) = { -3, -2, -3/2, -1, -1/2, 0, 1/4, 1/3, 1/2, 2/3, 1, 3/2, 2, 5/2, 3, 4, 5, 6, 8, 9, 16 }.

Crossrefs

Cf. A214872, A288759, A288850.
A216999 provides the corresponding results if division is not used.

Extensions

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