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

Author

Alois P. Heinz, Mar 07 2013

Keywords

• nonn

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

• 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

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