%I A000126 M1103 N0421
%S A000126 1,2,4,8,15,27,47,80,134,222,365,597,973,1582,2568,4164,6747,10927,
%T A000126 17691,28636,46346,75002,121369,196393,317785,514202,832012,1346240,
%U A000126 2178279,3524547,5702855,9227432,14930318,24157782,39088133,63245949
%N A000126 A nonlinear binomial sum.
%C A000126 a(n)-1 counts ternary numbers with no 0 digit (A007931) and at least 
               one 2 digit, where the total of ternary digits is <= n. E.g. a(4)-1 
               = 7: 2 12 21 22 112 121 211. - Frank Ellermann (frank.ellermann(AT)t-online.de), 
               Dec 02, 2001
%C A000126 A107909(a(n-1)) = A000079(n-1) = 2^(n-1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               May 28 2005
%C A000126 a(n) is the permanent of the n X n 0-1 matrix whose (i,j) entry is 1 
               iff i=1 or j=n or |i-j|<=1. For example, a(5)=15 is per([[1, 1, 1, 
               1, 1], [1, 1, 1, 0, 1], [0, 1, 1, 1, 1], [0, 0, 1, 1, 1], [0, 0, 
               0, 1, 1]]). - David Callan (callan(AT)stat.wisc.edu), Jun 07 2006
%C A000126 Conjecture. Let S(1)={1} and, for n>1, let S(n) be the smallest set containing 
               x+1 and 2x+1 for each element x in S(n-1). Then a(n) is the sum of 
               the elements in S(n). (See A122554 for a sequence defined in this 
               way.) - John W. Layman (layman(AT)math.vt.edu), Nov 21 2007
%C A000126 a(n+1) indexes the corner blocks on the Fibonacci spiral built from blocks 
               of unit area (using F(1) and F(2) as the sides of the first block). 
               - Paul Barry (pbarry(AT)wit.ie), Mar 06 2008
%D A000126 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000126 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000126 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques 
               Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%D A000126 D. A. Lind, On a class of nonlinear binomial sums, Fib. Quart., 3 (1965), 
               292-298.
%H A000126 T. D. Noe, <a href="b000126.txt">Table of n, a(n) for n=1..201</a>
%H A000126 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000126 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A000126 G.f.: -(1 - x + x^3 ) / (( x^2 + x - 1 )*( x - 1 )^2 ).
%F A000126 a(n) = Fib(n+4)-(n+1) = a(n-1)+a(n-2)+n-2 = A001924(n-1)+1 = A065220(n+3)+2. 
               - Henry Bottomley (se16(AT)btinternet.com), Oct 22 2001
%F A000126 a(n)=2*a(n-1)-a(n-3)+1 - Frank Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jan 13 2006
%F A000126 a(n+1)=1+sum{k=0..n, F(k+2)-1}=sum{k=0..n, F(k+2)}-n=F(n+4)-n-2; - Paul 
               Barry (pbarry(AT)wit.ie), Mar 06 2008
%p A000126 A000126:=-(1-z+z**3)/(z**2+z-1)/(z-1)**2; [Conjectured by S. Plouffe 
               in his 1992 dissertation.]
%p A000126 a:= n-> (Matrix([[1,1,1,2]]). Matrix(4, (i,j)-> if (i=j-1) then 1 elif 
               j=1 then [3,-2,-1,1][i] else 0 fi)^n)[1,2]; seq (a(n), n=1..36); 
               [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 26 2008]
%Y A000126 Heap-transform of A000071 - John Layman.
%Y A000126 Cf. A066067, A001924, A065220.
%Y A000126 Cf. A007931: binary strings with leading 0's, or ternary strings without 
               0's.
%Y A000126 Differences are A000071.
%Y A000126 Cf. A122554.
%Y A000126 Sequence in context: A125513 A054174 A001523 this_sequence A143281 A098057 
               A074029
%Y A000126 Adjacent sequences: A000123 A000124 A000125 this_sequence A000127 A000128 
               A000129
%K A000126 nonn,easy
%O A000126 1,2
%A A000126 N. J. A. Sloane (njas(AT)research.att.com).