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 1-4 of 4 results.

A214833 Number of formula representations of n using addition, multiplication and the constant 1.

Original entry on oeis.org

1, 1, 2, 6, 16, 52, 160, 536, 1796, 6216, 21752, 77504, 278720, 1013184, 3712128, 13701204, 50880808, 190003808, 712975648, 2687114976, 10167088608, 38605365712, 147060726688, 561853414896, 2152382687488, 8265949250848, 31817041756880, 122728993889056
Offset: 1

Views

Author

Alois P. Heinz, Mar 07 2013

Keywords

Examples

			a(1) = 1: 1.
a(2) = 1: 11+.
a(3) = 2: 111++, 11+1+.
a(4) = 6: 1111+++, 111+1++, 11+11++, 111++1+, 11+1+1+, 11+11+*.
a(5) = 16: 11111++++, 1111+1+++, 111+11+++, 1111++1++, 111+1+1++, 111+11+*+, 11+111+++, 11+11+1++, 111++11++, 11+1+11++, 1111+++1+, 111+1++1+, 11+11++1+, 111++1+1+, 11+1+1+1+, 11+11+*1+.
All formulas are given in postfix (reverse Polish) notation but other notations would give the same results.

Links

Crossrefs

Cf. A213923 (minimal length of formula), A005408(n-1) (maximal length of formula), A214835 (total sum of lengths), A214836, A214843, A242970, A242955.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=1, 1,
       add(a(i)*a(n-i), i=1..n-1)+
       add(a(d)*a(n/d), d=divisors(n) minus {1, n}))
    end:
seq(a(n), n=1..40);
  • Mathematica
    a[n_] := a[n] = If[n == 1, 1, Sum[a[i]*a[n-i], {i, 1, n-1}] + Sum[a[d]*a[n/d], {d, Divisors[n][[2 ;; -2]]}]]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)
  • PARI
    A214833_vec=[1]; alias(A,A214833_vec); A214833(n)={n>#A&&A=concat(A,vector(n-#A));if(A[n],A[n],A[n]=sum(i=1,n-1,A214833(i)*A214833(n-i))+sumdiv(n,d,if(d>1&&dA214833(d)*A214833(n/d))))} \\ M. F. Hasler, May 04 2017

Formula

a(n) = Sum_{i=1..n-1} a(i)*a(n-i) + Sum_{d|n, 11, a(1)=1.
a(n) ~ c * d^n / n^(3/2), where d = 4.076561785276... = A242970, c = 0.145691854699979... = A242955. - Vaclav Kotesovec, Sep 12 2014

A214843 Number of formula representations of n using addition, exponentiation and the constant 1.

Original entry on oeis.org

1, 1, 2, 6, 16, 48, 152, 502, 1694, 5832, 20420, 72472, 260096, 942304, 3441584, 12658128, 46842920, 174289108, 651610504, 2446686568, 9222628592, 34886505168, 132387975040, 503857644160, 1922782984688, 7355686851696, 28203617340756, 108368274550664
Offset: 1

Views

Author

Alois P. Heinz, Mar 08 2013

Keywords

Examples

			a(1) = 1: 1.
a(2) = 1: 11+.
a(3) = 2: 111++, 11+1+.
a(4) = 6: 1111+++, 111+1++, 11+11++, 111++1+, 11+1+1+, 11+11+^.
a(5) = 16: 11111++++, 1111+1+++, 111+11+++, 1111++1++, 111+1+1++, 111+11+^+, 11+111+++, 11+11+1++, 111++11++, 11+1+11++, 1111+++1+, 111+1++1+, 11+11++1+, 111++1+1+, 11+1+1+1+, 11+11+^1+.
All formulas are given in postfix (reverse Polish) notation but other notations would give the same results.

Links

Crossrefs

Cf. A213924, A214833, A214836.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=1, 1,
       add(a(i)*a(n-i), i=1..n-1)+
       add(a(root(n, p))*a(p), p=divisors(igcd(seq(i[2],
           i=ifactors(n)[2]))) minus {0, 1}))
    end:
seq(a(n), n=1..40);
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[a[i]*a[n-i], {i, 1, n-1}] + Sum[a[n^(1/p)] * a[p], {p, Divisors[GCD @@ FactorInteger[n][[All, 2]]] ~Complement~ {0, 1} } ];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *)

A217250 Minimal length of formulas representing n only using addition, multiplication, exponentiation and the constant 1.

Original entry on oeis.org

1, 3, 5, 7, 9, 9, 11, 9, 9, 11, 13, 13, 15, 15, 15, 11, 13, 13, 15, 15, 17, 17, 19, 15, 13, 15, 11, 13, 15, 17, 19, 13, 15, 17, 19, 13, 15, 17, 19, 19, 21, 21, 23, 21, 19, 21, 23, 17, 15, 17, 19, 19, 21, 15, 17, 17, 19, 19, 21, 21, 23, 23, 21, 13, 15, 17, 19
Offset: 1

Views

Author

Alois P. Heinz, Mar 16 2013

Keywords

Examples

			a(6) = 9: there are 58 formulas representing 6 only using addition, multiplication, exponentiation and the constant 1. The formulas with minimal length 9 are: 11+111++*, 11+11+1+*, 111++11+*, 11+1+11+*.
a(8) = 9: 11+111++^, 11+11+1+^.
a(9) = 9: 111++11+^, 11+1+11+^.
a(10) = 11: 1111++11+^+, 111+1+11+^+, 111++11+^1+, 11+1+11+^1+.
All formulas are given in postfix (reverse Polish) notation but other notations would give the same results.

Links

Crossrefs

Cf. A025280, A213923, A213924, A214836, A217253.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; 1+ `if`(n=1, 0, min(
      seq(a(i)+a(n-i), i=1..n/2),
      seq(a(d)+a(n/d), d=divisors(n) minus {1, n}),
      seq(a(root(n, p))+a(p), p=divisors(igcd(seq(i[2],
          i=ifactors(n)[2]))) minus {0, 1})))
    end:
seq(a(n), n=1..120);
  • Mathematica
    a[n_] := a[n] = 1 + If[n==1, 0, Min[Table[a[i] + a[n-i], {i, 1, n/2}] ~Join~ Table[a[d] + a[n/d], {d, Divisors[n] ~Complement~ {1, n}}] ~Join~ Table[a[Floor[n^(1/p)]] + a[p], {p, Divisors[GCD @@ FactorInteger[n][[ All, 2]]] ~Complement~ {0, 1}}]]];
Array[a, 120] (* Jean-François Alcover, Mar 22 2017, translated from Maple *)

Formula

a(n) = 2*A025280(n)-1.

A217253 Number of minimal length formulas representing n only using addition, multiplication, exponentiation and the constant 1.

Original entry on oeis.org

1, 1, 2, 7, 18, 4, 8, 2, 2, 4, 12, 36, 72, 16, 72, 14, 28, 4, 8, 8, 48, 24, 48, 8, 18, 36, 4, 8, 24, 96, 328, 18, 36, 164, 472, 4, 8, 24, 80, 144, 288, 224, 560, 216, 72, 144, 432, 56, 8, 52, 232, 72, 144, 8, 16, 16, 32, 48, 96, 256, 512, 656, 32, 20, 40, 120
Offset: 1

Views

Author

Alois P. Heinz, Mar 16 2013

Keywords

Examples

			a(6) = 4: there are 58 formulas representing 6 only using addition, multiplication, exponentiation and the constant 1. The 4 formulas with minimal length 9 are: 11+111++*, 11+11+1+*, 111++11+*, 11+1+11+*.
a(8) = 2: 11+111++^, 11+11+1+^.
a(9) = 2: 111++11+^, 11+1+11+^.
a(10) = 4: 1111++11+^+, 111+1+11+^+, 111++11+^1+, 11+1+11+^1+.
All formulas are given in postfix (reverse Polish) notation but other notations would give the same results.

Links

Crossrefs

Cf. A214836, A217250.

Programs

  • Maple
    with(numtheory):
b:= proc(n) option remember; local d, i, l, m, p, t;
      if n=1 then [1, 1] else l, m:= infinity, 0;
        for i to n-1 do  t:=  1+b(i)[1]+b(n-i)[1];
          if t=l then    m:= m +b(i)[2]*b(n-i)[2]
        elif t b(n)[2]:
seq(a(n), n=1..100);
  • Mathematica
    b[1] = {1, 1}; b[n_] := b[n] = Module[{d, i, l, m, p, t}, {l, m} = { Infinity, 0}; For[i=1, i <= n-1, i++, t = 1 + b[i][[1]] + b[n - i][[1]]; Which[t==l, m = m + b[i][[2]]*b[n-i][[2]], tJean-François Alcover, Mar 22 2017, translated from Maple *)
Showing 1-4 of 4 results.

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