%I A025192
%S A025192 1,2,6,18,54,162,486,1458,4374,13122,39366,118098,354294,1062882,3188646,
%T A025192 9565938,28697814,86093442,258280326,774840978,2324522934,6973568802,
%U A025192 20920706406,62762119218,188286357654,564859072962,1694577218886
%N A025192 2*3^(n-1), n >= 1.
%C A025192 Warning: there is a considerable overlap between this entry and the essentially 
               identical A008776.
%C A025192 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
%C A025192 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.
%C A025192 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 
               (maxale(AT)gmail.com), Feb 26 2003
%C A025192 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
%C A025192 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
%C A025192 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
%C A025192 Antidiagonal sums of square array A081277 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 04 2006
%C A025192 Equals row sums of triangle A160760 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 25 2009]
%D A025192 F. R. K. Chung and R. L. Graham, Primitive juggling sequences, preprint, 
               2006.
%H A025192 T. D. Noe, <a href="b025192.txt">Table of n, a(n) for n=0..200</a>
%H A025192 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A025192 a(n) = phi[3^n] = A000010[A000244(n)]. - Labos E. (labos(AT)ana.sote.hu), 
               Apr 14 2003
%F A025192 G.f.: (1-x)/(1-3x)
%F A025192 E.g.f. (2exp(3x)+exp(0))/3. - Paul Barry (pbarry(AT)wit.ie), Apr 20 2003
%F A025192 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
%F A025192 Row sums of triangle A134318 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Oct 19 2007
%F A025192 a(n)=A002326((3^n-1)/2) - Vladimir Shevelev (shevelev(AT)bgu.ac.il), 
               May 26 2008
%p A025192 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;
%p A025192 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
%p A025192 with(finance):seq(ceil(futurevalue(2,2,n)), n=-1..25);# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 25 2009]
%p A025192 with(finance):seq(ceil(futurevalue(6,2,n)), n=-2..24);# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 25 2009]
%Y A025192 First differences of 3^n (A000244). Other self-convolved sequences: A000108, 
               A007460 - A007464, A061922.
%Y A025192 Apart from initial term, same as A008776.
%Y A025192 Cf. A134318.
%Y A025192 A160760 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 25 2009]
%Y A025192 Sequence in context: A160175 A072852 A072853 this_sequence A008776 A134635 
               A114464
%Y A025192 Adjacent sequences: A025189 A025190 A025191 this_sequence A025193 A025194 
               A025195
%K A025192 nonn,nice,eigen
%O A025192 0,2
%A A025192 Clark Kimberling (ck6(AT)evansville.edu)
%E A025192 Additional comments from Barry E. Williams, May 27 2000
%E A025192 a(22) corrected by T. D. Noe, Feb 08 2008