0,4

Number of ordered partitions of n into 1's, 3's and 4's. - Len Smiley (smiley(AT)math.uaa.alaska.edu), May 08 2001

The sum of any two alternating terms (terms separated by one term) produces a number from the Fibonacci sequence. (e.g. 4+9=13, 9+25=34, 6+15=21, etc.) Taking square roots starting from the first term and every other term after, we get the Fibonacci sequence. - Sreyas Srinivasan (sreyas_srinivasan(AT)hotmail.com), May 02 2002

(1 + x + 2*x^2 + x^3)/(1 - x - x^3 - x^4) = 1 + 2*x + 4*x^2 + 6*x^3 + 9*x^4 + 15*x^5 + 25*x^6 + 40*x^7 + ... is the g.f. for the number of binary strings of length where neither 101 nor 111 occur. [Lozansky and Rousseau]

a(n) is the number of words written with the letters "a" and "b", with the following restriction: any "b" must be followed by at least two letters, the second of which being an "a" - Bruno Petazzoni (bpetazzoni(AT)ac-creteil.fr), Oct 31 2005

G.f. -(1+z+2*z**2+z**3)/((z**2+z-1)*(1+z**2)) for the truncated version 1, 2, 4, 6, 9, 15, 25, 40,... is given in the S. Plouffe thesis of 1992.

G. E. Bergum and V. E. Hoggatt, Jr., A combinatorial problem involving recursive sequences and tridiagonal matrices, Fib. Quart., 16 (1978), 113-118.

E. Lozansky and C. Rousseau, Winning Solutions, Springer, 1996; see pp. 157 and 172.

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

Joerg Arndt, Fxtbook

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.

Index entries for sequences related to Chebyshev polynomials.

a(n) = round((-1/5*sqrt(5)-1/5)*(-2*1/(-sqrt(5)+1))^n/(-sqrt(5)+1)+(1/5*sqrt(5)-1/5)*(-2*1/( sqrt(5)+1))^n/(sqrt(5)+1)). G.f.: 1/(1-x-x^2)/(1+x^2). - Vladeta Jovovic (vladeta(AT)Eunet.yu), May 03 2002

a(n)=(-i)^n*sum{k=0..n, U(n-2k, i/2)} with i^2=-1. - Paul Barry (pbarry(AT)wit.ie), Nov 15 2003

G.f.: 1/((1-x-x^2)(1+x^2)). a(2n)=F(n+1)^2, a(2n-1)=F(n+1)F(n). a(n)=a(-4-n)(-1)^n - Michael Somos Mar 10 2004

a(n)=sum{k=0..floor(n/2), (-1)^k*F(n-2k+1)}; - Paul Barry (pbarry(AT)wit.ie), Oct 12 2007

(PARI) a(n)=fibonacci((n+2)\2)*fibonacci((n+3)\2) - Michael Somos Mar 10 2004

Cf. A060945 (for 1's, 2's and 4's). Essentially the same as A074677.

Diagonal sums of number triangle A059259.

A001654(n)=a(2n-1), A007598(n+1)=a(2n).

Adjacent sequences: A006495 A006496 A006497 this_sequence A006499 A006500 A006501

Sequence in context: A024787 A076922 A057602 this_sequence A074677 A101756 A096398

nonn,easy,nice,new

njas

More terms from Larry Reeves (larryr(AT)acm.org), May 09 2000

Plouffe g.f. edited by R. J. Mathar, May 13 2008