A242970 Decimal expansion of the constant rho = lim f(n)^(1/n), where f(n) = A214833(n) is the number of arithmetic formulas for n (cf. comments).

4, 0, 7, 6, 5, 6, 1, 7, 8, 5, 2, 7, 6, 0, 4, 6, 1, 9, 8, 6, 0, 4, 0, 2, 2, 8, 5, 2, 8, 1, 5, 0, 2, 0, 2, 6

Offset: 1

Author

Jonathan Vos Post, Jun 09 2014

Keywords

• nonn
• more
• cons

Comments

This is the constant ρ, given on page 2 of E. K. Gnang and others. From the abstract: "An arithmetic formula is an expression involving only the constant 1, and the binary operations of addition and multiplication, with multiplication by 1 not allowed. We obtain an asymptotic formula for the number of arithmetic formulas evaluating to n as n goes to infinity, solving a conjecture of E. K. Gnang and D. Zeilberger."

More precisely: Let f(n) = A214833(n) be the number of arithmetic formulas for n. Then there exist constants c > 0 and ρ > 4 such that f(n) ~ c*ρ^n/n^(3/2) as n -> oo where ρ = 4.076561785276046... given in this sequence, and c = 0.145691854699979... given in A242955. - M. F. Hasler, May 04 2017

Examples

			ρ = 4.07656178527604619860402285281502026...
		

Links

• 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.

Crossrefs

The constant c = lim f(n)*n^(3/2)/rho^n is given in A242955.

Cf. A214833.

Formula

f(n) = A214833(n) ~ c*ρ^n/n^(3/2) = A242955*A242970^n/n^(3/2) as n -> oo, thus rho = A242970 = lim f(n)^(1/n) = lim f(n+1)/f(n) = lim (1+1/n)^(3/2)*f(n+1)/f(n), the latter expression being the most accurate/rapidly converging of the three. The values for n = 999, however, yield only 6 correct decimals (4.076559...). - M. F. Hasler, May 04 2017

Extensions

Edited by M. F. Hasler, May 03 2017