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COMMENTS
| Warning: there is a considerable overlap between this entry and the essentially identical A008776.
Shifts one place left when plus-convolved (PLUSCONV) with itself. a(n) = 2*Sum_{i=0..n-1} a(i) - Antti Karttunen May 15 2001
Let M = { 0, 1, ..., 2^n-1 } be the set of all n-bit numbers. Consider two operations on this set: ``sum modulo 2^n'' (+) and ``bitwise exclusive or'' (XOR). The results of these operations are correlated.
To give a numerical measure, consider the equations over M: u = x + y, v = x XOR y and ask for how many pairs (u,v) is there a solution? The answer is exactly a(n)=2*3^(n-1) for n>=1. The fraction a(n)/4^n of such pairs vanishes as n goes to infinity. - Max Alekseyev, Feb 26 2003
Number of (s(0), s(1), ..., s(2n+2)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n+2, s(0) = 3, s(2n+2) = 3. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 10 2004
Number of compositions of n into parts of two kinds. For a string of n objects, before the first, choose first kind or second kind; before each subsequent object, choose continue, first kind, or second kind. For example, compositions of 3 are 3; 2,1; 1,2; and 1,1,1. Using parts of two kinds, these produce respectively 2, 4, 4 and 8 compositions, 2+4+4+8 = 18. - Frank Adams-Watters, Aug 18 2006
Number of permutations of {1, 2, ..., n+1} such that no term is more than 2 larger than its predecessor. For example, a(3) = 18 because all permutations of {1, 2, 3, 4} are valid except 1423, 1432, 2143, 3142, 2314, 3214, in which 1 is followed by 4. Proof: removing (n + 1) gives a still-valid sequence. For n>=2, can insert (n + 1) either at the beginning or immediately following n or immediately following (n - 1), but nowhere else. Thus the number of such permutations triples when we increase the sequence length by 1. - Joel Lewis, Nov 14 2006
Antidiagonal sums of square array A081277 . - Philippe DELEHAM, Dec 04 2006
Equals row sums of triangle A160760 [From Gary W. Adamson, May 25 2009]
Let M = a triangle with (1, 2, 4, 8,...) as the left border and all other columns = (0, 1, 2, 4, 8,...). A025192 = lim_{n->inf} M^n, the left-shifted vector considered as a sequence. [Gary W. Adamson, Jul 27 2010]
Number of nonisomorphic graded posets with 0 and 1 and uniform hasse graph of rank n with no 3-element antichain. (Uniform used in the sense of http://arxiv.org/pdf/1010.6295.pdf.) - David Nacin, Feb 13 2012
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FORMULA
| a(n) = phi[3^n] = A000010[A000244(n)]. - Labos E. (labos(AT)ana.sote.hu), Apr 14 2003
G.f.: (1-x)/(1-3x).
E.g.f.: (2*exp(3*x)+exp(0))/3. - Paul Barry, Apr 20 2003
a(0) = 1, a(n) = sum(k=0, n-1, a(k)+a(n-k-1) ) - Benoit Cloitre, Jun 24 2003
Row sums of triangle A134318 - Gary W. Adamson, Oct 19 2007
a(n) = A002326((3^n-1)/2) - Vladimir Shevelev, May 26 2008
a(1) = 2, a(n) = 3*a(n-1). [From Vincenzo Librandi, Jan 01 2011]
a(n) = lcm(a(n-1), sum(k=1..n-1, a(k))) for n >= 3. - David W. Wilson, Sep 27 2011
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