%I A002083 M0787 N0297
%S A002083 1,1,1,2,3,6,11,22,42,84,165,330,654,1308,2605,5210,10398,20796,
%T A002083 41550,83100,166116,332232,664299,1328598,2656866,5313732,10626810,
%U A002083 21253620,42505932,85011864,170021123,340042246,680079282,1360158564
%N A002083 Narayana-Zidek-Capell numbers: a(2n)=2a(2n-1), a(2n+1)=2a(2n)-a(n).
%C A002083 Number of words beginning with 1, with sum of integers = n, in the sequence 
               1, 11, 111, 112, 1111, 1112, 1113, 1121, 1122, 1123, 1124, 11111, 
               11112, 11113, 11114, 11121, 11122, 11123, 11124, 11125, 11131, 11132, 
               11133, 11134, 11135, 11136, where any positive integer, in any word, 
               is <= the sum of the preceding integers - Claude Lenormand (claude.lenormand(AT)free.fr), 
               Jan 29 2001
%C A002083 a(n) = number of sequences (b(1),b(2),...) of unspecified length satisfying 
               b(1) = 1, 1 <= b(i) <= 1 + Sum[b(j),{j,i-1}] for i>=2, Sum[b(i)] 
               = n-1. For example, a(5) = 3 counts (1, 1, 1, 1), (1, 2, 1), (1, 
               1, 2). These sequences are generated by the Mathematica code below. 
               - David Callan (callan(AT)stat.wisc.edu), Nov 02 2005
%C A002083 a(n+1) is the number of padded compositions (d_1,d_2,...,d_n) of n satisfying 
               d_i <= i for all i. A padded composition of n is obtained from an 
               ordinary composition (c_1,c_2,...,c_r) of n (r >= 1, each c_i >= 
               1, sum(c_i,i=1..r) = n) by inserting c_i - 1 zeros immediately after 
               each c_i. Thus (3,1,2) -> (3,0,0,1,2,0) is a padded composition of 
               6 and a padded composition of n has length n. For example, with n=4, 
               a(5) counts (1,1,1,1), (1,1,2,0), (1,2,0,1). - David Callan (callan(AT)stat.wisc.edu), 
               Feb 04 2006
%C A002083 Further comments from David Callan (callan(AT)stat.wisc.edu), Sep 25 
               2006: a(n) = # ordered trees on n edges in which (i) every node (= 
               non-root non-leaf vertex) has at least 2 children and (ii) each leaf 
               is either the leftmost or rightmost child of its parent. For example, 
               a(4)=2 counts
%C A002083 ./\.../\
%C A002083 /\...../\,
%C A002083 and a(5)=3 counts
%C A002083 .|.......|....../|\
%C A002083 / \...../ \...../ \
%C A002083 ../\.../\.
%C A002083 First column of A155092. [From Mats Granvik (mats.granvik(AT)abo.fi), 
               Jan 20 2009]
%C A002083 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 15 2009: 
               (Start)
%C A002083 Starting with offset 2 = eigensequence of triangle A101688 and row sums 
               of
%C A002083 triangle A167948 (End)
%D A002083 Capell, P.; Narayana, T. V.; On knock-out tournaments. Canad. Math. Bull. 
               13 1970 105-109.
%D A002083 Nathaniel D. Emerson, A Family of Meta-Fibonacci Sequences Defined by 
               Variable-Order Recursions, Journal of Integer Sequences, Vol. 9 (2006), 
               Article 06.1.8.
%D A002083 Kreweras, G.; Sur quelques problemes relatifs au vote pondere, [ Some 
               problems of weighted voting ] Math. Sci. Humaines No. 84 (1983), 
               45-63.
%D A002083 Narayana, T. V.; Quelques resultats relatifs aux tournois "knock-out" 
               et leurs applications aux comparaisons aux paires. C. R. Acad. Sci. 
               Paris, Series A-B 267 1968 A32-A33.
%D A002083 T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. 
               Univ. Toronto Press, 1979, pp. 100-101.
%D A002083 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002083 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002083 M. Torelli, Increasing integer sequences and Goldbach's conjecture, preprint, 
               1996.
%H A002083 T. D. Noe, <a href="b002083.txt">Table of n, a(n) for n = 1..200</a>
%H A002083 Thomas Wieder, <a href="http://www.thomas-wieder.privat.t-online.de/">
               HomePage</a>.
%H A002083 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A002083 a(1)=1, else a(n) is sum of [ n/2 ] previous terms.
%F A002083 Conjecture: lim n->inf a(n)/2^(n-3) = a constant ~ .633368 - Gerald McGarvey 
               (Gerald.McGarvey(AT)comcast.net), Jul 18 2004
%F A002083 a(n) is the permanent of the (n-1) X (n-1) matrix with (i, j) entry = 
               1 if i-1 <= j <= 2*i-1 and = 0 otherwise. - David Callan (callan(AT)stat.wisc.edu), 
               Nov 02 2005
%F A002083 a(n)=sum(K(n-k+1, k)*a(n - k),k=1..n), where K(n,k) = 1 if 0 <= k AND 
               k <= n and K(n,k)=0 else. (Several arguments to the K-coefficient 
               K(n,k) can lead to the same sequence. For example, we get A002083 
               also from a(n)=sum(K((n - k)!,k!)*a(n - k),k=1..n), where K(n,k) 
               = 1 if 0 <= k <= n and 0 else. See also the comment to a similar 
               formula for A002487.) - Thomas Wieder (thomas.wieder(AT)t-online.de), 
               Jan 13 2008
%p A002083 A002083 := proc(n) option remember; if n<=3 then 1 elif n mod 2 = 0 then 
               2*A002083(n-1) else 2*A002083(n-1)-A002083((n-1)/2); fi; end;
%p A002083 a := proc(n::integer) # A002083 Narayana-Zidek-Capell numbers: a(2n)=2a(2n-1), 
               a(2n+1)=2a(2n)-a(n). local k; option remember; if n = 0 then 1 else 
               add(K(n-k+1, k)*procname(n - k), k = 1 .. n); #else add(K((n-k)!, 
               k!)*procname(n - k), k = 1 .. n); end if end proc; K := proc(n::integer, 
               k::integer) local KC; if 0 <= k and k <= n then KC := 1 else KC := 
               0 end if; end proc; - Thomas Wieder (thomas.wieder(AT)t-online.de), 
               Jan 13 2008
%t A002083 a[1] = 1; a[n_] := a[n] = Sum[a[k], {k, n/2, n-1} ]; Table[ a[n], {n, 
               2, 70, 2} ] - from Robert G. Wilson v Apr 22 2001
%t A002083 bSequences[1]={ {1} }; bSequences[n_]/;n>=2 := bSequences[n] = Flatten[Table[Map[Join[ 
               #, {i}]&, bSequences[n-i]], {i, Ceiling[n/2]}], 1] (Callan)
%o A002083 (PARI) a(n)=if(n<3,n>0,2*a(n-1)-(n%2)*a(n\2))
%Y A002083 Cf. A045690. A058222 gives sums of words.
%Y A002083 Cf. A001045, A002487.
%Y A002083 Sequence in context: A123341 A141072 A043328 this_sequence A124973 A043327 
               A005578
%Y A002083 Cf. A101688, A167948 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 
               15 2009]
%Y A002083 Adjacent sequences: A002080 A002081 A002082 this_sequence A002084 A002085 
               A002086
%K A002083 easy,core,nonn,nice
%O A002083 1,4
%A A002083 N. J. A. Sloane (njas(AT)research.att.com).