0,3

Also T(2n+1,n+1), T given by A027935. Also first row of Inverse Stolarsky array.

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

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

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

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)

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.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 12.

C. Kimberling, "Interspersions and dispersions," Proceedings of the American Mathematical Society 117 (1993) 313-321.

T. D. Noe, Table of n, a(n) for n=0..200

Index entries for sequences related to linear recurrences with constant coefficients

C. Kimberling, Interspersions

N. J. A. Sloane, Classic Sequences

a(n)=sum(i=1, n, binomial(n+i, n-i)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 15 2002

G.f.: sum(k>=1, x^k/(1-x)^(2k+1)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 21 2003

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

a(n) = sum(k=1, n, F(2k)), i.e. partial sums of A001906. - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 27 2003

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

G.f.: 1/((1-x)*(1-3*x+x^2))= 1/(1-4*x+4*x^2-x^3).

a(n)= 4*a(n-1)-4*a(n-2)+a(n-3), n>=2, a(-1)=0, a(0)=1, a(1)=4.

a(n)= 3*a(n-1)-a(n-2)+1, n>=1, a(-1)=0, a(0)=1.

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

with(combinat): seq(fibonacci(2*n+1)-1, n=1..27); (Deutsch)

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

Table[Fibonacci[2*n+1]-1, {n, 0, 17}] (Vladimir Orlovsky, Jul 21 2008)

Cf. A000045, A035507, A001906.

Cf. A006318.

Cf. A000032 = Lucas numbers.

Sequence in context: A066536 A104747 A070050 this_sequence A135254 A000754 A119683

Adjacent sequences: A027938 A027939 A027940 this_sequence A027942 A027943 A027944

nonn,easy,nice

Clark Kimberling (ck6(AT)evansville.edu)

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 08 2000