%I A102699
%S A102699 1,2,6,16,42,104,252,592,1370,3112,6996,15536,34244,74832,162616,351136,
%T A102699 754938,1615208,3443940,7314928,15493676,32714992,68918856,144815456,
%U A102699 303703972,635554064,1327816392,2769049312,5766417480,11989472672,24897569648
%N A102699 Number of strings of length n, using as symbols numbers from the set 
               {1, 2, ..., n}, in which consecutive symbols differ by exactly 1.
%C A102699 Equally, number of different n-digit numbers, using only the digits 1 
               through n, where consecutive digits differ by 1. It is assumed that 
               there are n different digits available even when n > 9.
%D A102699 Sr. Arworn, An algorithm for the number of endomorphisms on paths, Disc. 
               Math., 309 (2009), 94-103 (see p. 95).
%H A102699 T. D. Noe, <a href="b102699.txt">Table of n, a(n) for n=1..300</a>
%H A102699 Joseph Myers, <a href="http://www.srcf.ucam.org/~jsm28/publications/2008/
               bmo1-2009-q1.pdf">BMO 2008-2009 Round 1 Problem 1-Generalisation</
               a>
%F A102699 It appears that the limit of a(n)/a(n-1) is decreasing towards 2. - Ben 
               Thurston (benthurston27(AT)yahoo.com), Oct 04 2006
%F A102699 a(n) = (n+1)2^(n-1) - 4(n-1)binomial(n-2,(n-2)/2) for n even, a(n) = 
               (n+1)2^(n-1) - (2n-1)binomial(n-1,(n-1)/2) for n odd. [From Joseph 
               Myers (jsm(AT)polyomino.org.uk), Dec 23 2008]
%e A102699 For example, a(4)=16: the 16 strings are 1212, 1232, 1234, 2121, 2123, 
               2321, 2323, 2343, 3212, 3232, 3234, 3432, 3434, 4321, 4323, 4343.
%p A102699 p:= 0; paths := proc(m, n, s, t) global p; if(((t+1) <= m) and s <= (n)) 
               then paths(m,n,s+1,t+1); end if; if(((t-1) > 0) and s <= (n)) then 
               paths(m,n,s+1,t-1); end if; if(s = n) then p:=p+1; end if; end proc; 
               sumpaths:=proc(j) global p; p:=0; sp:=0; for h from 1 to j do p:=0; 
               paths(j,j,1,h); sp:=sp+ p ; end do; sp; end proc; for l from 1 to 
               50 do sumpaths(l); end do; - Ben Thurston (benthurston27(AT)yahoo.com), 
               Oct 04 2006
%Y A102699 Sequence in context: A158920 A143123 A152089 this_sequence A156664 A025169 
               A111282
%Y A102699 Adjacent sequences: A102696 A102697 A102698 this_sequence A102700 A102701 
               A102702
%K A102699 nonn
%O A102699 1,2
%A A102699 Don Rogers (donrogers42(AT)aol.com), Feb 07 2005
%E A102699 More terms from Ben Thurston (benthurston27(AT)yahoo.com), Oct 04 2006
%E A102699 a(20) onwards from David Wasserman (dwasserm(AT)earthlink.net), Apr 26 
               2008
%E A102699 Edited by N. J. A. Sloane (njas(AT)research.att.com), Jan 03 2009