A214843 -id:A214843 - cp's OEIS frontend

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

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

Alois P. Heinz, Mar 07 2013

nonn

			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.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Shalosh B. Ekhad, Everything About Formulas Representing Integers Using Additions and Multiplication for integers from 1 to 8000
Edinah K. Gnang, Maksym Radziwill, and Carlo Sanna, Counting arithmetic formulas, arXiv:1406.1704 [math.CO], 2014.
Edinah K. Gnang, Maksym Radziwill, and Carlo Sanna, Counting arithmetic formulas, European Journal of Combinatorics 47 (2015), pp. 40-53.
Edinah K. Ghang and Doron Zeilberger, Zeroless Arithmetic: Representing Integers ONLY using ONE, arXiv:1303.0885 [math.CO], 2013.
Wikipedia, Postfix notation
Index to sequences related to the complexity of n

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

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);

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 *)

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

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

A214836 Number of formula representations of n using addition, multiplication, exponentiation and the constant 1.

Original entry on oeis.org

1, 1, 2, 7, 18, 58, 180, 613, 2076, 7270, 25752, 92918, 338432, 1246092, 4624536, 17290646, 65047436, 246079536, 935484928, 3571960668, 13692523960, 52675401248, 203299385584, 786949008100, 3054440440486, 11884949139900, 46351113658232, 181153317512536

Offset: 1

Alois P. Heinz, Mar 07 2013

nonn

			a(1) = 1: 1.
a(2) = 1: 11+.
a(3) = 2: 111++, 11+1+.
a(4) = 7: 1111+++, 111+1++, 11+11++, 111++1+, 11+1+1+, 11+11+*, 11+11+^.
a(5) = 18: 11111++++, 1111+1+++, 111+11+++, 1111++1++, 111+1+1++, 111+11+*+, 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+, 11+11+^1+.
All formulas are given in postfix (reverse Polish) notation but other notations would give the same results.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Edinah K. Ghang and Doron Zeilberger, Zeroless Arithmetic: Representing Integers ONLY using ONE, arXiv:1303.0885 [math.CO], 2013.
Shalosh B. Ekhad, Everything About Formulas Representing Integers Using Additions, Multiplication and Exponentiation for integers from 1 to 8000
Wikipedia, Postfix notation
Index to sequences related to the complexity of n

Cf. A025280, A214833, A214843, A217250, A217253.

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})+
       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);

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] ~Complement~ {1, n}}] + Sum[a[n^(1/p)] * a[p], {p, Divisors[GCD @@ Table[i[[2]], {i, FactorInteger[n]}]] ~Complement~ {0, 1}}]]; Array[a, 40] (* Jean-François Alcover, Apr 11 2017, translated from Maple *)

A213924 Minimal lengths of formulas representing n only using addition, exponentiation and the constant 1.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 9, 9, 11, 13, 15, 17, 19, 21, 11, 13, 15, 17, 19, 21, 23, 25, 21, 13, 15, 11, 13, 15, 17, 19, 13, 15, 17, 19, 15, 17, 19, 21, 23, 23, 25, 23, 25, 25, 27, 29, 25, 17, 19, 21, 23, 25, 23, 25, 27, 27, 27, 25, 27, 29, 31, 27, 13, 15, 17, 19, 21, 23, 25, 27, 23, 23, 25, 27, 29, 31, 33

Offset: 1

Jonathan Vos Post, Mar 06 2013

nonn

			There are 502 different formulas for n=8. Two of them have shortest length 9: 11+111++^, 11+11+1+^. Thus a(8) = 9.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Shalosh B. Ekhad, Everything About Formulas Representing Integers Using Additions and Exponentiation for integers from 1 to 8000.
Edinah K. Ghang, Doron Zeilberger, Zeroless Arithmetic: Representing Integers ONLY using ONE, arXiv:1303.0885v1 [math.CO], 2013

Cf. A003037, A025280, A213923, A214843.

with(numtheory):
a:= proc(n) option remember; 1+ `if`(n=1, 0, min(
       seq(a(i)+a(n-i), i=1..n-1),
       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..100);  # Alois P. Heinz, Mar 12 2013

a[n_] := a[n] = 1 + If[n==1, 0, Min[Table[a[i]+a[n-i], {i, 1, n-1}], Table[ a[Floor[n^(1/p)]] + a[p], {p, Divisors[GCD @@ FactorInteger[n][[All, 2]]] ~Complement~ {0, 1}}]]]; Array[a, 100] (* Jean-François Alcover, Mar 22 2017, after Alois P. Heinz *)

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

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A214836 Number of formula representations of n using addition, multiplication, exponentiation and the constant 1.

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A213924 Minimal lengths of formulas representing n only using addition, exponentiation and the constant 1.

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