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

K. Edwards, A Pascal-like triangle related to the tribonacci numbers, Fib. Q., 46/47 (2008/2009), 18-25.

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

Index entries for two-way infinite sequences

The 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.

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.rs), 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).

Sequence in context: A076922 A157679 A057602 this_sequence A074677 A101756 A096398

Adjacent sequences: A006495 A006496 A006497 this_sequence A006499 A006500 A006501

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

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

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