OFFSET
1,2
COMMENTS
It appears that this sequence is the (L)-sieve transform of {3,6,9,12,...,3n,...} = A008585. (See A152009 for the definition of the (L)-sieve transform.) - John W. Layman, Jan 06 2009
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 = 0, b = 1/2, and c = K(3) = A083286, so we conclude that K(3)*(3/2)^n - 1 < a(n+1) = f_n < K(3)*(3/2)^n, since we have neither e_n = 0 for all sufficiently large n nor e_n = 1/2 for all sufficiently large n. (End)
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.30.1, p. 196.
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..500
Zakir Deniz, Topology of acyclic complexes of tournaments and coloring, Applicable Algebra in Engineering, Communication, March 2015, Volume 26, Issue 1-2, pp. 213-226.
Artūras Dubickas, On integer sequences generated by linear maps, Glasg. Math. J. 51(2) (2009), 243-252.
Don Knuth, Ambidextrous Numbers, Preprint, September 2022.
Andrew M. Odlyzko and Herbert S. Wilf, Functional iteration and the Josephus problem, Glasgow Math. J. 33(2) (1991), 235-240.
Arya Saranathan, Quasigeodesic languages are not context-free in some non-hyperbolic groups, arXiv:2601.12520 [math.GR], 2026. See p. 11.
Eric Weisstein's World of Mathematics, Power Ceilings.
FORMULA
a(n) = A061418(n) - 1 = floor(K*(3/2)^n) where K = 1.08151366859...
The constant K is (2/3)*K(3) (see A083286). - Ralf Stephan, May 29 2003
a(1) = 1, a(n) = A070885(n)/3. - Benoit Cloitre, Aug 18 2002
a(n) = ceiling((a(n-1) + a(n-2))*9/10) - Franklin T. Adams-Watters, May 01 2006
EXAMPLE
a(6) = ceiling(8*3/2) = 12.
MAPLE
a:=proc(n) option remember: if n=1 then 1 else ceil(procname(n-1)*3/2) fi; end; seq(a(n), n=1..40); # Muniru A Asiru, Jun 07 2018
MATHEMATICA
a=1; a=Table[a=Ceiling[a*3/2], {n, 0, 4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *)
NestList[Ceiling[3#/2]&, 1, 39] (* Stefano Spezia, Dec 08 2024 *)
PROG
(Magma) [ n eq 1 select 1 else Ceiling(Self(n-1)*3/2): n in [1..40] ]; // Klaus Brockhaus, Nov 14 2008
(PARI) { a=2/3; for (n=1, 500, write("b061419.txt", n, " ", a=ceil(a*3/2)) ) } \\ Harry J. Smith, Jul 22 2009
(Python)
from itertools import islice
def A061419_gen(): # generator of terms
a = 2
while True:
yield a-1
a += a>>1
CROSSREFS
KEYWORD
nonn
AUTHOR
Henry Bottomley, May 02 2001
STATUS
approved
