%I A000078 M1108 N0423
%S A000078 0,0,0,1,1,2,4,8,15,29,56,108,208,401,773,1490,2872,5536,10671,20569,
%T A000078 39648,76424,147312,283953,547337,1055026,2033628,3919944,7555935,
%U A000078 14564533,28074040,54114452,104308960,201061985,387559437,747044834
%N A000078 Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) with a(0)=a(1)=a(2)=0,
a(3)=1.
%C A000078 a(n)=number of compositions of n-3 with no part greater than 4. Example:
a(7)=8 because we have 1+1+1+1=2+1+1=1+2+1=3+1=1+1+2=2+2=1+3=4. -
Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 10 2004
%C A000078 a(n+4)=number of 0-1 sequences of length n that avoid 1111. - David Callan
(callan(AT)stat.wisc.edu), Jul 19 2004
%C A000078 a(n)=number of matchings in the graph obtained by a zig-zag triangulation
of a convex (n-3)-gon. Example: a(8)=15 because in the triangulation
of the convex pentagon ABCDEA with diagonals AD and AC we have 15
matchings: the empty set, seven singletons and {AB,CD},{AB,DE},{BC,
AD},{BC,DE},{BC,EA},{CD,EA} and {DE,AC}. - Emeric Deutsch (deutsch(AT)duke.poly.edu),
Dec 25 2004
%C A000078 Number of permutations satisfying -k<=p(i)-i<=r, i=1..n-3, with k=1,
r=3. - Vladimir Baltic (baltic(AT)matf.bg.ac.yu), Jan 17 2005
%D A000078 E. Deutsch, Problem 1613, Math. Mag., 75, No. 1, 64-64.
%D A000078 M. Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(#3) (1963), 71-74.
%D A000078 W. C. Lynch, The t-Fibonacci numbers and polyphase sorting, Fib. Quart.,
8 (1970), pp. 6ff.
%D A000078 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000078 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000078 Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas
n-step Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article
05.4.4.
%D A000078 Problem 2803, Amer. Math. Monthly, 33 (1926), 229-232.
%D A000078 J. Riordan, An Introduction to Combinatorial Analysis, Princeton University
Press, Princeton, NJ, 1978.
%H A000078 T. D. Noe, <a href="b000078.txt">Table of n, a(n) for n = 0..200</a>
%H A000078 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A000078 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Sequences realized by oligomorphic permutation groups</a>, J. Integ.
Seqs. Vol. 3 (2000), #00.1.5.
%H A000078 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=11">
Encyclopedia of Combinatorial Structures 11</a>
%H A000078 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A000078 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000078 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Fibonaccin-StepNumber.html">Fibonacci n-Step Number.</a>
%H A000078 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
TetranacciNumber.html">Tetranacci Number.</a>
%H A000078 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%F A000078 a(n) =A001630(n)-a(n-1) - Henry Bottomley
%F A000078 G.f.: x^3/(1 - x - x^2 - x^3 - x^4).
%F A000078 a(n) = term (1,4) in the 4x4 matrix [1,1,0,0; 1,0,1,0; 1,0,0,1; 1,0,0,
0]^n. - Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jun 12 2008
%F A000078 G.f.: 1/(1-z-z^2-z^3-z^4). (S. Plouffe) [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 17 2009]
%e A000078 sage: taylor( mul(x/(1-x-x^2-x^3-x^4) for i in xrange(1,2)),x,0,33)#solution>
> x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 15*x^6 + 29*x^7 +....+ 201061985*x^31
+ 387559437*x^32 + 747044834*x^33+etc... [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jun 02 2009]
%p A000078 A000078:=-1/(-1+z+z**2+z**3+z**4); [S. Plouffe in his 1992 dissertation.]
%p A000078 a := n -> (Matrix([[1,1,0,0], [1,0,1,0], [1,0,0,1], [1,0,0,0]])^n)[1,
4]; seq ((a(n)), n=0..50); - Alois P. Heinz (heinz(AT)hs-heilbronn.de),
Jun 12 2008
%p A000078 g:=1/(1-z-z^2-z^3-z^4): gser:=series(g, z=0, 49): seq((coeff(gser, z,
n)), n=-3..32);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Apr 17 2009]
%t A000078 CoefficientList[Series[x^3/(1 - x - x^2 - x^3 - x^4), {x, 0, 50}], x]
%o A000078 (PARI) a(n)=if(n<0,0,polcoeff(x^3/(1-x-x^2-x^3-x^4)+x*O(x^n),n))
%o A000078 Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Apr 29
2009: (Start)
%o A000078 (PARI) /* Simple Generation with 5 variables*/
%o A000078 g(n) =
%o A000078 {
%o A000078 local(a1=0,a2=0,a3=0,a4=1);
%o A000078 print1(a1","a2","a3","a4",");
%o A000078 for(x=5,n,a5=a1+a2+a3+a4;
%o A000078 print1(a5",");
%o A000078 a1=a2;a2=a3;a3=a4;a4=a5;
%o A000078 )
%o A000078 }
%o A000078 (End)
%o A000078 (Other) sage: taylor( mul(x/(1-x-x^2-x^3-x^4) for i in xrange(1,2)),x,
0,33)# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 02
2009]
%Y A000078 Row 4 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
%Y A000078 First differences are in A001631.
%K A000078 nonn,easy,nice,new
%O A000078 0,6
%A A000078 N. J. A. Sloane (njas(AT)research.att.com).
%E A000078 More terms from Henry Bottomley (se16(AT)btinternet.com), Oct 09 2000
%E A000078 Definition augmented (with 4 initial terms) by Daniel Forgues (squid(AT)zensearch.com),
Dec 02 2009
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