A242955
Decimal expansion of the constant c = lim f(n)*n^(3/2)/rho^n where f(n) = A214833(n) is the number of arithmetic formulas for n, and rho = A242970.
Original entry on oeis.org
1, 4, 5, 6, 9, 1, 8, 5, 4, 6, 9, 9, 9, 7, 9, 2, 9, 4, 5, 6, 0, 4, 2, 4, 8, 3, 6, 0, 5, 7, 9
Offset: 0
c = 0.1456918546999792945604248360579...
- Edinah K. Gnang, Maksym Radziwill, Carlo Sanna, Counting arithmetic formulas, arXiv:1406.1704 [math.CO], (6-June-2014).
- Edinah K. Gnang, Maksym Radziwill, Carlo Sanna, Counting arithmetic formulas, European Journal of Combinatorics 47 (2015), pp. 40-53.
The constant rho is given in
A242970.
Offset corrected, definition improved, more terms from
M. F. Hasler, May 04 2017
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
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
-
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
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