Search: id:A006498
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%I A006498 M1005
%S A006498 1,1,1,2,4,6,9,15,25,40,64,104,169,273,441,714,1156,1870,3025,4895,7921,
%T A006498 12816,20736,33552,54289,87841,142129,229970,372100,602070,974169,
%U A006498 1576239,2550409,4126648,6677056,10803704,17480761,28284465,45765225
%N A006498 a(n) = a(n-1)+a(n-3)+a(n-4).
%C A006498 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
%C A006498 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
%C A006498 (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]
%C A006498 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
%C A006498 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 13 2009:
(Start)
%C A006498 Let a(0) = 1, then A006498 = partial products, Product_{n=2..inf.}
%C A006498 (Fn/F(n-1)*Fn/F(n-1) = 1*1*2*2*(3/2)*(3/2)*(5/3)*(5/3)*(8/5)*(8/5)*...;
%C A006498 e.g. a(7) = 15 = 1*1*1*2*2*(3/2)*(3/2)*(5/3). (End)
%D A006498 G. E. Bergum and V. E. Hoggatt, Jr., A combinatorial problem involving
recursive sequences and tridiagonal matrices, Fib. Quart., 16 (1978),
113-118.
%D A006498 K. Edwards, A Pascal-like triangle related to the tribonacci numbers,
Fib. Q., 46/47 (2008/2009), 18-25.
%D A006498 E. Lozansky and C. Rousseau, Winning Solutions, Springer, 1996; see pp.
157 and 172.
%D A006498 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%H A006498 T. D. Noe, Table of n, a(n) for n=0..500
%H A006498 Joerg Arndt, Fxtbook
%H A006498 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.
%H A006498 S. Plouffe,
1031 Generating Functions and Conjectures, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A006498 Index entries for sequences related to
Chebyshev polynomials.
%H A006498 Index entries for two-way infinite sequences
%F A006498 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.
%F A006498 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
%F A006498 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
%F A006498 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
%F A006498 a(n)=sum{k=0..floor(n/2), (-1)^k*F(n-2k+1)}; - Paul Barry (pbarry(AT)wit.ie),
Oct 12 2007
%o A006498 (PARI) a(n)=fibonacci((n+2)\2)*fibonacci((n+3)\2) - Michael Somos Mar
10 2004
%Y A006498 Cf. A060945 (for 1's, 2's and 4's). Essentially the same as A074677.
%Y A006498 Diagonal sums of number triangle A059259.
%Y A006498 A001654(n)=a(2n-1), A007598(n+1)=a(2n).
%Y A006498 Sequence in context: A157679 A057602 A171646 this_sequence A074677 A101756
A096398
%Y A006498 Adjacent sequences: A006495 A006496 A006497 this_sequence A006499 A006500
A006501
%K A006498 nonn,easy,nice,new
%O A006498 0,4
%A A006498 N. J. A. Sloane (njas(AT)research.att.com).
%E A006498 More terms from Larry Reeves (larryr(AT)acm.org), May 09 2000
%E A006498 Plouffe g.f. edited by R. J. Mathar, May 13 2008
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