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A006498
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a(n) = a(n-1)+a(n-3)+a(n-4).
(Formerly M1005)
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22
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1, 1, 1, 2, 4, 6, 9, 15, 25, 40, 64, 104, 169, 273, 441, 714, 1156, 1870, 3025, 4895, 7921, 12816, 20736, 33552, 54289, 87841, 142129, 229970, 372100, 602070, 974169, 1576239, 2550409, 4126648, 6677056, 10803704, 17480761, 28284465, 45765225
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Number of compositions 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] Or, equivalently, where neither 000 nor 010 occur.
Equivalently, a(n+2) is the number of length-n binary strings with no two set bits with distance 2; see fxtbook link. [Joerg Arndt, Jul 10 2011]
a(n) is the number of words written with the letters "a" and "b", with the following restriction: any "a" must be followed by at least two letters, the second of which is a "b" - Bruno Petazzoni (bpetazzoni(AT)ac-creteil.fr), Oct 31 2005. This is also equivalent to the previous two conditions.
Let a(0) = 1, then A006498 = partial products of Product_{n=2..inf.(F(n)/F(n-1))^2 = 1*1*2*2*(3/2)*(3/2)*(5/3)*(5/3)*(8/5)*(8/5)*.... E.g. a(7) = 15 = 1*1*1*2*2*(3/2)*(3/2)*(5/3). [Gary W. Adamson, Dec 13 2009]
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REFERENCES
| 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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..500
Joerg Arndt, Fxtbook, section 14.10.1, p.320
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
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FORMULA
| G.f.: 1/((1-x-x^2)*(1+x^2)); a(2*n)=F(n+1)^2, a(2*n-1)=F(n+1)*F(n). a(n)=a(-4-n)*(-1)^n. - Michael Somos Mar 10 2004
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,... was 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, Nov 15 2003
a(n)=sum{k=0..floor(n/2), (-1)^k*F(n-2k+1)}; - Paul Barry, Oct 12 2007
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MATHEMATICA
| a=b=c=0; d=1; lst={d}; Do[AppendTo[lst, e=a+b+d]; a=b; b=c; c=d; d=e, {n, 0, 5!}]; lst [From Vladimir Orlovsky, May 28 2010]
LinearRecurrence[{1, 0, 1, 1}, {1, 1, 1, 2}, 50] (* From Harvey P. Dale, Jul 13 2011 *)
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PROG
| (PARI) a(n)=fibonacci((n+2)\2)*fibonacci((n+3)\2) - Michael Somos Mar 10 2004
(PARI) Vec(1/(1-x-x^3-x^4)+O(x^66))
(MAGMA) [ n eq 1 select 1 else n eq 2 select 1 else n eq 3 select 1 else n eq 4 select 2 else Self(n-1)+Self(n-3)+ Self(n-4): n in [1..40] ]; // Vincenzo Librandi, Aug 20 2011
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CROSSREFS
| 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: A057602 A171646 * A074677 A179997 A101756 A173241
Adjacent sequences: A006495 A006496 A006497 * A006499 A006500 A006501
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), May 09 2000
Plouffe g.f. edited by R. J. Mathar, May 13 2008
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