A081601 Numbers m such that 3 does not divide Sum_{k=0..m} binomial(2k,k) = A006134(m).

0, 3, 9, 12, 27, 30, 36, 39, 81, 84, 90, 93, 108, 111, 117, 120, 243, 246, 252, 255, 270, 273, 279, 282, 324, 327, 333, 336, 351, 354, 360, 363, 729, 732, 738, 741, 756, 759, 765, 768, 810, 813, 819, 822, 837, 840, 846, 849, 972, 975, 981, 984, 999, 1002, 1008, 1011

Offset: 1

Author

Benoit Cloitre, Apr 22 2003

Keywords

• easy
• nonn

Comments

Apparently a(n)/3 mod 2 = A010060(n-1), the Thue-Morse sequence.

a(n+1) is the smallest number with exactly n+1 partitions into distinct powers of 2 or of 3: A131996(a(n+1)) = n+1 and A131996(m) < n+1 for m < a(n+1). - Reinhard Zumkeller, Aug 06 2007

Examples

			For n=0, A006134(0) = 1, hence 0 is a term.
		

Links

• Ralf Stephan, Some divide-and-conquer sequences ...
• Ralf Stephan, Table of generating functions

Crossrefs

Cf. A005836, A006134, A006996, A010060, A083096, A131996.

Equals A089118(n-2) + 1, n > 1.

Programs

Mathematica

Select[Range[0, 1020], Mod[Sum[Binomial[2 k, k], {k, 0, #}], 3] != 0 &] (* Michael De Vlieger, Nov 28 2015 *)

PARI

for(n=0, 1e3, if(sum(k=0, n, binomial(2*k, k)) % 3 > 0, print1(n,", "))) \\ Altug Alkan, Nov 26 2015

Formula

Apparently a(n) = 3*A005836(n).

G.f.: (x/(1 - x))*Sum_{k>=0} 3^(k+1)*x^(2^k)/(1 + x^(2^k)) (conjecture). - Ilya Gutkovskiy, Jul 23 2017

Extensions

Zero prepended to the sequence and formulas modified accordingly by L. Edson Jeffery, Nov 25 2015