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A129710 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 01 subwords (0 <= k <= floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword. 2
1, 2, 2, 1, 2, 3, 2, 5, 1, 2, 7, 4, 2, 9, 9, 1, 2, 11, 16, 5, 2, 13, 25, 14, 1, 2, 15, 36, 30, 6, 2, 17, 49, 55, 20, 1, 2, 19, 64, 91, 50, 7, 2, 21, 81, 140, 105, 27, 1, 2, 23, 100, 204, 196, 77, 8, 2, 25, 121, 285, 336, 182, 35, 1, 2, 27, 144, 385, 540, 378, 112, 9, 2, 29, 169, 506 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Also number of Fibonacci binary words of length n and having k 10 subwords.

Row n has 1+floor(n/2) terms.

Row sums are the Fibonacci numbers (A000045).

T(n,0)=2 for n >= 1.

Sum_{k>=0} k*T(n,k) = A023610(n-2).

Triangle, with zeros omitted, given by (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 14 2012

Riordan array ((1+x)/(1-x), x^2/(1-x)), zeros omitted. - Philippe Deléham, Jan 14 2012

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..10200 (rows 0 <= n <= 200, flattened.)

FORMULA

T(n,k) = binomial(n-k,k) + binomial(n-k-1,k) for n >= 1 and 0 <= k <= floor(n/2).

G.f. = G(t,z) = (1+z)/(1-z-tz^2).

Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A078050(n), A057079(n), A040000(n), A000045(n+2), A000079(n), A006138(n), A026597(n), A133407(n), A133467(n), A133469(n), A133479(n), A133558(n), A133577(n), A063092(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively. - Philippe Deléham, Jan 14 2012

T(n,k) = T(n-1,k) + T(n-2,k-1) with T(0,0)=1, T(1,0)=2, T(1,1)=0 and T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Jan 14 2012

EXAMPLE

T(5,2)=4 because we have 10101, 01101, 01010 and 01011.

Triangle starts:

  1;

  2;

  2, 1;

  2, 3;

  2, 5, 1;

  2, 7, 4;

  2, 9, 9, 1;

Triangle (2, -1, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) begins:

  1;

  2, 0;

  2, 1, 0;

  2, 3, 0, 0;

  2, 5, 1, 0, 0;

  2, 7, 4, 0, 0, 0;

  2, 9, 9, 1, 0, 0, 0;

MAPLE

T:=proc(n, k) if n=0 and k=0 then 1 elif k<=floor(n/2) then binomial(n-k, k)+binomial(n-k-1, k) else 0 fi end: for n from 0 to 18 do seq(T(n, k), k=0..floor(n/2)) od; # yields sequence in triangular form

MATHEMATICA

MapAt[# - 1 &, #, 1] &@ Table[Binomial[n - k, k] + Binomial[n - k - 1, k], {n, 0, 16}, {k, 0, Floor[n/2]}] // Flatten (* Michael De Vlieger, Nov 15 2019 *)

CROSSREFS

Cf. A000045, A023610.

Cf. A029635, A029653.

Columns: A040000, A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.

Sequence in context: A263073 A133091 A112204 * A298857 A070680 A054711

Adjacent sequences:  A129707 A129708 A129709 * A129711 A129712 A129713

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, May 12 2007

STATUS

approved

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Last modified February 18 02:57 EST 2020. Contains 332006 sequences. (Running on oeis4.)