0,4

Number of elements in all subsets of {1,2,...,n-1} with no consecutive integers. Example: a(5)=10 because the subsets of {1,2,3,4} that have no consecutive elements, i.e. {},{1},{2},{3},{4},{1,3},{1,4},{2,4}, the total number of elements is 10. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 10 2003

If g is either of the real solutions to x^2-x-1=0, g'=1-g is the other one and phi is any 2 X 2-matricial solution to the same equation, not of the form gI or g'I, then Sum'_{i+j=n-1}g^i phi^j=F_n+(A001629(n)-A001629(n-1)g')(phi-g'I), where i,j>=0,F_n is the n-th Fibonacci number and I is the 2 X 2 identity matrix... - Michele Dondi (blazar(AT)lcm.mi.infn.it), Apr 06 2004

Number of 3412-avoiding involutions containing exactly one subsequence of type 321.

Number of binary sequences of length n with exactly one pair of consecutive 1's. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 02 2004

For this sequence the n-th term is given by (nF(n+1)-F(n)+nF(n-1))/5 where F(n) is the n-th Fibonacci number. - Mrs J. P. Shiwalkar and M. N. Deshpande (dpratap_ngp(AT)sancharnet.in), Apr 20 2005

If an unbiased coin is tossed n times then there are 2^n possible strings of H and T.Out of these, number of strings with exactly one 'HH'is given by a(n)where a(n) denotes n-th term of this sequence - Mrs J. P. Shiwalkar and M. N. Deshpande (dpratap_ngp(AT)sancharnet.in), May 04 2005

a(n) = half the number of horizontal dominos in all domino tilings of a horizontally aligned 2 X n rectangle; a(n+1) = the number of vertical dominos in all domino tilings of a horizontally aligned 2 X n rectangle; thus 2*a(n)+a(n+1)=n*F(n+1) = the number of dominos in all domino tilings of a 2 X n rectangle, where F=A000045, the Fibonacci sequence. - Roberto Tauraso (tauraso(AT)mat.uniroma2.it), May 02 2005; Graeme McRae (g_m(AT)mcraefamily.com), Jun 02 2006

a(n+1)=((-I)^(n-1))*diff(S(n,x),x)|_{x=I}, n>=1. First derivative of Chebyshev S-polynomials evaluated at x=I (imaginary unit) multiplied by (-I)^(n-1). See A049310 for the S-polynomials. W. Lang, Apr 04 2007.

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.

V. E. Hoggatt, Jr. and M. Bicknell-Johnson, Fibonacci convolution sequences, Fib. Quart., 15 (1977), 117-122.

Thomas Koshy, Fibonacci and Lucas Numbers with Applications, Chapter 15, page 187, "Hosoya's Triangle"

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 101.

S. Vajda, Fibonacci and Lucas numbers, and the Golden Section, Ellis Horwood Ltd., Chichester, 1989, p. 183, Nr.(98).

T. D. Noe, Table of n, a(n) for n=0..500

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.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

E. S. Egge, Restricted 3412-Avoiding Involutions, p. 16.

T. Mansour, Generalization of some identities involving the Fibonacci numbers

P. Moree, Convoluted convolved Fibonacci numbers

Pieter Moree, Convoluted Convolved Fibonacci Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.

G.f.: x^2/(1-x-x^2)^2; a(n)=2*a(n-1)+a(n-2)-2*a(n-3)-a(n-4), n>3; a(n)=sum(F(k)F(n-k)), k=0..n where F=A000045 (the Fibonacci sequence).

a(n+1) = sum(A007895(i), 0 <= i <= F(n)), where F = A000045, the Fibonacci sequence - Claude Lenormand (claude.lenormand(AT)free.fr), Feb 04 2001

a(n)=sum((k+1)*binomial(n-k-1, k+1), k=0..floor(n/2)-1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 15 2001

a(n)=floor( (1/5)*(n-1/sqrt(5))*phi^n + 1/2 ) where phi=(1+sqrt(5))/2 is the golden ratio. - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 05 2003

a(n)=a(n-1)+A010049(n-1) for n>0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 10 2003

a(n)=sum{k=0..floor((n-2)/2), (n-k-1)binomial(n-k-2, k)} - Paul Barry (pbarry(AT)wit.ie), Jan 25 2005

a(n)= ((n-1)*F(n)+2*n*F(n-1))/5, F(n)=A000045(n) (see Vajda reference)

F'(n, 1), the first derivative of the n-th Fibonacci polynomial evaluated at 1. - T. D. Noe (noe(AT)sspectra.com), Jan 18 2006

a(n)=a(n-1)+a(n-2)+F(n-1), where F=A000045, the Fibonacci sequence. - Graeme McRae (g_m(AT)mcraefamily.com), Jun 02 2006

a(n)=(1/5)(n-1/sqrt(5))((1+sqrt(5))/2)^n + (1/5)(n+1/sqrt(5))((1-sqrt(5))/2)^n - Graeme McRae (g_m(AT)mcraefamily.com), Jun 02 2006

a(n) = A055244(n-1) - F(n-2). Example: a(6) = 20 = A055244(5) - F(3) = (23 - 3). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 27 2007

a(n) = sum of (n-1)-th row terms of triangle A134510; e.g., a(6) = 20 = (8 + 5 + 5 + 1 + 1). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 28 2007

Starting (1, 2, 5, 10, 20, 38,...), = row sums of triangle A134836. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 12 2007

A001629:=1/(z**2+z-1)**2; [Conjectured by S. Plouffe in his 1992 dissertation.]

a(n)= A037027(n-1, 1), n >= 1, (Fibonacci convolution triangle). Cf. A000045, A001628.

Row sums of triangle A058071.

Cf. A010049.

First differences of A006478.

Cf. A055244.

Cf. A134510.

Cf. A134836.

Adjacent sequences: A001626 A001627 A001628 this_sequence A001630 A001631 A001632

Sequence in context: A000712 A032442 A102688 this_sequence A068034 A084215 A024810

nonn,easy,nice

njas