0,2

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 (maxal(AT)cs.ucsd.edu), 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 (FrankTAW(AT)Netscape.net), 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 (jblewis(AT)fas.harvard.edu), Nov 14 2006

Antidiagonal sums of square array A081277 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 04 2006

F. R. K. Chung and R. L. Graham, Primitive juggling sequences, preprint, 2006.

T. D. Noe, Table of n, a(n) for n=0..200

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. (2exp(3x)+exp(0))/3. - Paul Barry (pbarry(AT)wit.ie), Apr 20 2003

a(0)=1, a(n)=sum(k=0, n-1, a(k)+a(n-k-1) ) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 24 2003

Row sums of triangle A134318 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 19 2007

PLUSCONV := proc(a, b) local c, i, k, n; n := min( nops(a), nops(b) ); c := []; for i from 0 to n-1 do c := [ op(c), add((a[k+1]+b[i-k+1]), k=0..i)]; od; RETURN(c); end;

with(combstruct):ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), a):ZL1:=Prod(begin_blockP, Z, end_blockP):ZL2:=Prod(begin_blockLR, Z, Sequence(Prod(mu_length, Z), card>=1), end_blockLR): ZL3:=Prod(begin_blockRL, Sequence(Prod(mu_length, Z), card>=0), Z, end_blockRL):Q:=subs([a=Union(ZL1, ZL1), b=ZL1], ZL0), begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Epsilon, end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon, mu_length=Epsilon:temp15:=draw([S, {Q}, unlabelled], size=15):seq(count([S, {Q}, unlabelled], size=n)/2, n=1..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 08 2008

First differences of 3^n (A000244). Other self-convolved sequences: A000108, A007460 - A007464, A061922.

Apart from initial term, same as A008776.

Cf. A134318.

Adjacent sequences: A025189 A025190 A025191 this_sequence A025193 A025194 A025195

Sequence in context: A072850 A072852 A072853 this_sequence A008776 A134635 A114464

nonn,nice,eigen

Clark Kimberling (ck6(AT)evansville.edu)

Additional comments from Barry E. Williams, May 27 2000

a(22) corrected by T. D. Noe, Feb 08 2008