1,1

Can be stated as the number of animals starting from a single pair if any pair of animals can produce a single offspring (as in the game Minecraft, if the player allows offspring to fully grow before breeding again). - Denis Moskowitz, Dec 05 2012

Maximum number of partial products that can be added in a Wallace tree multiplier with n-1 full adder stages. - Chinmaya Dash, Aug 19 2020

From Jianing Song, Nov 01 2025: (Start)

In general, let {f_n}_{n>=0} be the sequence defined by f_{n+1} = alpha*f_n + e_n, where alpha > 1, a <= e_n <= b, then f_n = (f_0 + e_0/alpha + ... + e_{n-1}/alpha^n)*alpha^n = c*alpha^n - (e_n/alpha + e_{n+1}/alpha^2 + ...), where c = f_0 + Sum_{n>=0} e_n/alpha^{n+1}. We conclude that c*alpha^n - b/(alpha - 1) <= f_n <= c*alpha^n - a/(alpha - 1).

Here alpha = 3/2, a = -1/2, b = 0, and c = K(3) = A083286, so we conclude that K(3)*(3/2)^n < a(n+1) = f_n < K(3)*(3/2)^n + 1, since we have neither e_n = -1/2 for all sufficiently large n nor e_n = 0 for all sufficiently large n. (End)

Iain Fox, Table of n, a(n) for n = 1..1000 (first 500 terms from Harry J. Smith)

Artūras Dubickas, On integer sequences generated by linear maps, Glasg. Math. J. 51(2) (2009), 243-252.

Leopold Flatto, Jeffrey C. Lagarias, and Andrew D. Pollington, On the range of fractional parts {ξ(p/q)}^n, Acta Arithmetica (1995), Volume: 70, Issue: 2, page 125-147.

Don Knuth, Ambidextrous Numbers, Preprint, September 2022.

M. van de Vel, Determination of msd(L^n), J. Algebraic Combin. 9(2) (1999), 161-171. See Table 5. - N. J. A. Sloane, Mar 26 2012

Mark van Wijk, The Quest for the Best Thread-Safe Java List, Univ. of Twente (Netherlands 2022).

Wikipedia, Wallace tree.

a(n) = A061419(n) + 1 = ceiling(K*(3/2)^n) where K = 1.08151366859...

The constant K is (2/3)*K(3) (see A083286). - Ralf Stephan, May 29 2003

a(6) = floor(9*3/2) = 13.

NestList[Quotient[3*#, 2] &, 2, 50] (* Paolo Xausa, Mar 09 2026 *)

(Magma) [ n eq 1 select 2 else Floor(Self(n-1)*(3/2)): n in [1..39] ]; // Klaus Brockhaus, Nov 14 2008

(PARI) { a=4/3; for (n=1, 500, a=a*3\2; write("b061418.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 22 2009

(PARI) first(n) = my(v=vector(n)); v[1]=2; for(i=2, n, v[i]=v[i-1]*3\2); v \\ Iain Fox, Jul 15 2022

(Python)

from itertools import islice

def A061418_gen(): # generator of terms

a = 2

while True:

yield a

a += a>>1

A061418_list = list(islice(A061418_gen(), 70)) # Chai Wah Wu, Sep 20 2022

Cf. A002379, A003312, A034082, A061419, A083286.

First differences are in A073941.

Sequence in context: A375922 A238434 A384728 * A355909 A136423 A215245

Adjacent sequences: A061415 A061416 A061417 * A061419 A061420 A061421

nonn,easy

Henry Bottomley, May 02 2001

approved