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By expanding on the first row we see that the sequence of T(n)'s is the Fibonacci sequence without the initial stammer on the 1's.- Larry Gerstein (gerstein(AT)edu), Mar 30 2007Suppose psi=log(phi).

Also, a(n 1) is the number of compositions where there is a drop between every second pair of parts, starting with the second and third part; see example.

- Joerg Arndt, May 21 2013a(n) are the pentagon (not pentagonal) numbers because the algebraic degree 2 number rho(5) = 2*cos(Pi/5) = phi (golden section), the length ratio diagonal/side in a pentagon, has minimal polynomial C(5,x) = x^2 - x - 1 (see A187360, n=5), hence rho(5)^n = a(n-1)*1 a(n)*rho(5), n For any prime p there is an infinite periodic subsequence within F(n) divisible by p, that begins at index n = 0 with value 0, and its first nonzero term at n = 0 such that the n-th prime divides Fibonacci(m)."Any consecutive pair (m, k) of the Fibonacci sequence a(n) illustrates a fair equivalence between m miles and k kilometers. -Lekraj Beedassy, Oct 06 2014a(n 1) counts closed walks on K_2, containing one loop on the other vertex.

- Hieronymus Fischer, Apr 18 2007 0, floor((1/2)*log_phi(5*F(n)*F(n 1))) = n.

Extension valid for integer n, except n=0,-1: floor((1/2)*sign(F(n)*F(n 1))*log_phi|5*F(n)*F(n 1)|) = n (where sign(x) = sign of x). Also phi = 1/1 1/2 1/(2*5) 1/(5*13) 1/(13*34) 1/(34*89) ... Adamson, Dec 15 2007a(n) = the number of different ways to run up a staircase with n steps, taking steps of odd sizes where the order is relevant and there is no other restriction on the number or the size of each step taken. Azarian, May 21 2008F(n) is the number of possible binary sequences of length n that obey the sequential construction rule: if last symbol is 0, add the complement (1); else add 0 or 1.

inverse is Fib^(-1)(x) = -C[Pinv(-x)] = -BTC(-x) and Fib(x) = -BTC^(-1)(-x).

Generalizing to P(x,t) = x /(1 t*x) and Pinv(x,t) = x /(1 - t*x) = -P(-x,t) gives other relations to lattice paths, such as the o.g.f.- Ross Drewe, Oct 05 2008F(n 1) = number of Motzkin paths of length n having exactly one weak ascent.A Motzkin path of length n is a lattice path from (0,0) to (n,0) consisting of U=(1,1), D=(1,-1) and H=(1,0) steps and never going below the x-axis.0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155F(n 1) = number of matchings (i.e., Hosoya index) in a path graph on n vertices: F(5)=5 because the matchings of the path graph on the vertices A, B, C, D are the empty set, , , and .- Emeric Deutsch, Jun 18 2001F(n) = number of binary words of length n beginning with 0 and having all runlengths odd; e.g., F(6) counts 010101, 010111, 010001, 011101, 011111, 000101, 000111, 000001.for A091867, C[P[x,1-t, and that for A104597, Pinv[Cinv(x),t 1].

||Generalizing to P(x,t) = x /(1 t*x) and Pinv(x,t) = x /(1 - t*x) = -P(-x,t) gives other relations to lattice paths, such as the o.g.f.

- Ross Drewe, Oct 05 2008F(n 1) = number of Motzkin paths of length n having exactly one weak ascent.

A Motzkin path of length n is a lattice path from (0,0) to (n,0) consisting of U=(1,1), D=(1,-1) and H=(1,0) steps and never going below the x-axis.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155F(n 1) = number of matchings (i.e., Hosoya index) in a path graph on n vertices: F(5)=5 because the matchings of the path graph on the vertices A, B, C, D are the empty set, , , and .

- Emeric Deutsch, Jun 18 2001F(n) = number of binary words of length n beginning with 0 and having all runlengths odd; e.g., F(6) counts 010101, 010111, 010001, 011101, 011111, 000101, 000111, 000001.

for A091867, C[P[x,1-t]], and that for A104597, Pinv[Cinv(x),t 1].

||Generalizing to P(x,t) = x /(1 t*x) and Pinv(x,t) = x /(1 - t*x) = -P(-x,t) gives other relations to lattice paths, such as the o.g.f.

- Ross Drewe, Oct 05 2008F(n 1) = number of Motzkin paths of length n having exactly one weak ascent.

A Motzkin path of length n is a lattice path from (0,0) to (n,0) consisting of U=(1,1), D=(1,-1) and H=(1,0) steps and never going below the x-axis.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155F(n 1) = number of matchings (i.e., Hosoya index) in a path graph on n vertices: F(5)=5 because the matchings of the path graph on the vertices A, B, C, D are the empty set, , , and .

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