%I A027941
%S A027941 0,1,4,12,33,88,232,609,1596,4180,10945,28656,75024,196417,514228,
%T A027941 1346268,3524577,9227464,24157816,63245985,165580140,433494436,
%U A027941 1134903169,2971215072,7778742048,20365011073,53316291172,139583862444
%N A027941 Fibonacci(2n+1)-1.
%C A027941 Also T(2n+1,n+1), T given by A027935. Also first row of Inverse Stolarsky 
               array.
%C A027941 Number of Schroeder paths of length 2(n+1) having exactly one up step 
               starting at an even height (a Schroeder path is a lattice path starting 
               from (0,0), ending at a point on the x-axis, consisting only of steps 
               U=(1,1) (up steps), D=(1,-1) (down steps) and H=(2,0) (level steps) 
               and never going below the x-axis. Schroeder paths are counted by 
               the large Schroeder numbers (A006318). Example. a(1)=4 because among 
               the six Schroeder paths of length 4 only the paths (U)HD, (U)UDD, 
               H(U)D, (U)DH have exactly one U step that starts at an even height 
               (shown between parentheses). - Emeric Deutsch, Dec 19 2004
%C A027941 Also: smallest number not writeable as the sum of n or fewer Fibonacci 
               numbers. E.g. a(4)=88 because it is the smallest number that needs 
               at least 5 Fibonacci numbers: 88=55+21+8+3+1 - Johan Claes (johan.claes(AT)luc.ac.be), 
               Apr 19 2005
%C A027941 Except for first term, numbers a(n) that set a new record in the number 
               of Fibonacci numbers needed to sum up to n. Position of records in 
               sequence A007895. - R. Stephan, May 15 2005
%C A027941 Successive extremal petal bends beta(n) = a(n-2). See the Ring Lemma 
               of Rodin and Sullivan in K. Stephenson, Introduction to Circle Packing 
               (Cambridge U. P., 2005), pp. 73-74 and 318-321. - David W. Cantrell 
               (DWCantrell(AT)sigmaxi.net)
%C A027941 a(n+1)= AAB^(n)(1), n>=1, with compositions of Wythoff's complementary 
               A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link 
               under A135817 for the Wythoff representation of numbers (with A as 
               1 and B as 0 and the argument 1 omitted). E.g. 4=`110`, 12=`1100`, 
               33=`11000`, 88=`110000,..., in Wythoff code. AA(1)=1=a(1) but for 
               uniqueness reason 1=A(1) in Wythoff code.
%D A027941 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, id. 12.
%D A027941 C. Kimberling, "Interspersions and dispersions," Proceedings of the American 
               Mathematical Society 117 (1993) 313-321.
%H A027941 T. D. Noe, <a href="b027941.txt">Table of n, a(n) for n=0..200</a>
%H A027941 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A027941 C. Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/intersp.html">
               Interspersions</a>
%H A027941 N. J. A. Sloane, <a href="classic.html#WYTH">Classic Sequences</a>
%F A027941 a(n)=sum(i=1, n, binomial(n+i, n-i)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Oct 15 2002
%F A027941 G.f.: sum(k>=1, x^k/(1-x)^(2k+1)). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 21 2003
%F A027941 Third diagonal of array defined by T(i, 1)=T(1, j)=1, T(i, j)=Max(T(i-1, 
               j)+T(i-1, j-1); T(i-1, j-1)+T(i, j-1)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Aug 05 2003
%F A027941 a(n) = sum(k=1, n, F(2k)), i.e. partial sums of A001906. - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Oct 27 2003
%F A027941 a(n)=sum{k=0..n, U(k, 3/2)}=sum{k=0..n, S(k, 3)}, S(k, 3):=A001906(k+1) 
               - Paul Barry (pbarry(AT)wit.ie), Nov 14 2003
%F A027941 G.f.: 1/((1-x)*(1-3*x+x^2))= 1/(1-4*x+4*x^2-x^3).
%F A027941 a(n)= 4*a(n-1)-4*a(n-2)+a(n-3), n>=2, a(-1)=0, a(0)=1, a(1)=4.
%F A027941 a(n)= 3*a(n-1)-a(n-2)+1, n>=1, a(-1)=0, a(0)=1.
%F A027941 a(n) = Sum[ F(k)*L(k), {k,1,n} ], where L(k) = Lucas(k) = A000032(k) 
               = F(k-1) + F(k+1). - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               May 18 2007
%p A027941 with(combinat): seq(fibonacci(2*n+1)-1,n=1..27); (Deutsch)
%p A027941 a:=n->sum(binomial(n+k+1,2*k), k=0..n): seq(a(n), n=-1..26); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Oct 02 2007
%t A027941 Table[Fibonacci[2*n+1]-1,{n,0,17}] (Vladimir Orlovsky, Jul 21 2008)
%Y A027941 Cf. A000045, A035507, A001906.
%Y A027941 Cf. A006318.
%Y A027941 Cf. A000032 = Lucas numbers.
%Y A027941 Sequence in context: A066536 A104747 A070050 this_sequence A135254 A000754 
               A119683
%Y A027941 Adjacent sequences: A027938 A027939 A027940 this_sequence A027942 A027943 
               A027944
%K A027941 nonn,easy,nice
%O A027941 0,3
%A A027941 Clark Kimberling (ck6(AT)evansville.edu)
%E A027941 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 08 2000