OFFSET
1,1
COMMENTS
When read as a triangle in which the n-th row has 2^n terms, every row is the last half of the next one. All the terms are powers of 2. First column = 2*A000079.
a(n) seems to be the denominator of Euler(2*n+1,1) but I have no proof of this.
a(n) is also gcd[C(2n,1), C(2n,3), ..., C(2n,2n-1)]. - Franz Vrabec, Oct 22 2012
a(n) is also the ratio r(2n) = s2(2n)/s1(2n) where s1(2n) is the sum of the odd unitary divisors of 2n and s2(2n) is the sum of the even unitary divisors of 2n. - Michel Lagneau, Dec 19 2013
a(n) or a(n)/2 = A006519(n) is known as the Steinhaus sequence in probability theory, proposed as a sequence of asymptotically fair premiums for the St. Petersburg game. - Peter Kern, Aug 28 2015
After the all-1's sequence this is the next sequence in lexicographical order such that the gap between a(n) and the next occurrence of a(n) is given by a(n). - Scott R. Shannon, Oct 16 2019
First 2^(k-1) - 1 terms are also the areas of the successive rectangles and squares of width 2 that are adjacent to any of the four sides of the toothpick structure of A139250 after 2^k stages, with k >= 2. For example: if k = 5 the areas after 32 stages are [2, 4, 2, 8, 2, 4, 2, 16, 2, 4, 2, 8, 2, 4, 2] respectively, the same as the first 15 terms of this sequence. - Omar E. Pol, Dec 29 2020
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16383
Sandor Csörgö and Gordon Simons, On Steinhaus' resolution of the St. Petersburg paradox, Probab. Math. Statist. 14 (1993), 157--172. MR1321758 (96b:60017). - Peter Kern, Aug 28 2015
Roger B. Eggleton, Aviezri S. Fraenkel, and R. Jamie Simpson, Beatty sequences and Langford sequences, Graph theory and combinatorics (Marseille-Luminy, 1990). Discrete Math. 111 (1993), no. 1-3, 165--178. MR1210094 (94a:11018). See Example 2.6. - N. J. A. Sloane, Mar 18 2012
Hugo Steinhaus, The so-called Petersburg paradox, Colloq. Math. 2 (1949), 56--58. MR0039937 (12,619e).
FORMULA
a(n) = -(-n XOR n). XOR the bitwise operation on the two's complement representation for negative integers. - Peter Luschny, Jun 01 2011
a(n) = A038712(n)+1. - Franz Vrabec, Mar 03 2012
a(n) = 2^A001511(n). - Franz Vrabec, Oct 22 2012
a(n) = 2*A006519(n). - Antti Karttunen, Nov 06 2018
MAPLE
a := proc(n) local k: k:=1: while frac(n/2^k) = 0 do k := k+1 end do: k := k-1: a(n) := 2^(k+1) end: seq(a(n), n=1..63); # Johannes W. Meijer, Nov 04 2012
seq(2^(1 + padic[ordp](n, 2)), n = 1..63); # Peter Luschny, Nov 27 2020
MATHEMATICA
Table[-BitXor[-i, i], {i, 200}] (* Peter Luschny, Jun 01 2011 *)
a[n_] := 2^(IntegerExponent[n, 2] + 1); Array[a, 100] (* Jean-François Alcover, May 09 2017 *)
PROG
(PARI) A171977(n) = 2^(1+valuation(n, 2)); \\ Antti Karttunen, Nov 06 2018
(Python)
def A171977(n): return (n&-n)<<1 # Chai Wah Wu, Jul 13 2022
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Paul Curtz, Nov 19 2010
EXTENSIONS
I edited this sequence, based on an email message from the author. - N. J. A. Sloane, Nov 20 2010
Definition simplified by N. J. A. Sloane, Mar 18 2012
STATUS
approved