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 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. (Formerly M1108 N0423) 76

%I M1108 N0423

%S 0,0,0,1,1,2,4,8,15,29,56,108,208,401,773,1490,2872,5536,10671,20569,

%T 39648,76424,147312,283953,547337,1055026,2033628,3919944,7555935,

%U 14564533,28074040,54114452,104308960,201061985,387559437,747044834,1439975216,2775641472

%N 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 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_, Mar 10 2004

%C In other words, a(n) is the number of ways of putting stamps in one row on an envelope using stamps of denominations 1, 2, 3 and 4 cents so as to total n-3 cents [Polya-Szego]. - _N. J. A. Sloane_, Jul 28 2012

%C a(n+4) = number of 0-1 sequences of length n that avoid 1111. - _David Callan_, Jul 19 2004

%C 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_, Dec 25 2004

%C Number of permutations satisfying -k<=p(i)-i<=r, i=1..n-3, with k=1, r=3. - _Vladimir Baltic_, Jan 17 2005

%C For n>=0, a(n+4) is the number of palindromic compositions of 2*n+1 into an odd number of parts that are not multiples of 4. In addition, a(n+4) is also the number of Sommerville symmetric cyclic compositions (= bilaterally symmetric cyclic compositions) of 2*n+1 into an odd number of parts that are not multiples of 4. - _Petros Hadjicostas_, Mar 10 2018

%D P. Hadjicostas and L. Zhang, Sommerville's symmetrical cyclic compositions of a positive integer with parts avoiding multiples of an integer, Fibonacci Quarterly 55 (2017), 54-73.

%D Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.

%D G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1, Problems 3 and 4.

%D J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, Princeton, NJ, 1978.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%D L. Zhang and P. Hadjicostas, On sequences of independent Bernoulli trials avoiding the pattern '11..1', Math. Scientist, 40 (2015), 89-96.

%H Indranil Ghosh, <a href="/A000078/b000078.txt">Table of n, a(n) for n = 0..3505</a> (terms 0..200 from T. D. Noe)

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, pp.307-309

%H Vladimir Baltic, <a href="http://pefmath.etf.rs/vol4num1/AADM-Vol4-No1-119-135.pdf">On the number of certain types of strongly restricted permutations</a>, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135

%H Elena Barcucci, Antonio Bernini, Stefano Bilotta, Renzo Pinzani, <a href="http://arxiv.org/abs/1601.07723">Non-overlapping matrices</a>, arXiv:1601.07723 [cs.DM], 2016.

%H Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, <a href="http://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

%H S. A. Corey and Otto Dunkel, <a href="http://www.jstor.org/stable/2299550">Problem 2803</a>, Amer. Math. Monthly, 33 (1926), 229-232.

%H F. Michel Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Dekking/dekk4.html">Morphisms, Symbolic Sequences, and Their Standard Forms</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.

%H E. Deutsch, <a href="http://www.jstor.org/stable/3219192">Problem 1613: A recursion in four parts</a>, Math. Mag., 75, No. 1, 64-64.

%H G. P. B. Dresden, Z. Du, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Dresden/dresden6.html">A Simplified Binet Formula for k-Generalized Fibonacci Numbers</a>, J. Int. Seq. 17 (2014) # 14.4.7

%H M. Feinberg, <a href="http://www.fq.math.ca/Scanned/1-3/feinberg.pdf">Fibonacci-Tribonacci</a>, Fib. Quart. 1(#3) (1963), 71-74.

%H Petros Hadjicostas, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Hadjicostas/hadji2.html">Cyclic compositions of a positive integer with parts avoiding an arithmetic sequence</a>, Journal of Integer Sequences, 19 (2016), Article 16.8.2.

%H F. T. Howard and Curtis Cooper, <a href="http://www.math-cs.ucmo.edu/~curtisc/articles/howardcooper/genfib4.pdf">Some identities for r-Fibonacci numbers</a>.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=11">Encyclopedia of Combinatorial Structures 11</a>

%H W. C. Lynch, <a href="http://www.fq.math.ca/Scanned/8-1/lynch.pdf">The t-Fibonacci numbers and polyphase sorting</a>, Fib. Quart., 8 (1970), pp. 6ff.

%H T. Mansour, M. Shattuck, <a href="http://arxiv.org/abs/1410.6943">A monotonicity property for generalized Fibonacci sequences</a>, arXiv:1410.6943 [math.CO], 2014.

%H Tony D. Noe and Jonathan Vos Post, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Noe/noe5.html">Primes in Fibonacci n-step and Lucas n-step Sequences,</a> J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H H. Prodinger, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Prodinger2/prod31.html">Counting Palindromes According to r-Runs of Ones Using Generating Functions</a>, J. Int. Seq. 17 (2014) # 14.6.2, odd length middle 0, r=3.

%H J. L. Ramírez, V. F. Sirvent, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i1p38">A Generalization of the k-Bonacci Sequence from Riordan Arrays</a>, The Electronic Journal of Combinatorics, 22(1) (2015), #P1.38.

%H O. Turek, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Turek/turek3.html">Abelian Complexity Function of the Tribonacci Word</a>, J. Int. Seq. 18 (2015) # 15.3.4

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number.</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TetranacciNumber.html">Tetranacci Number.</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1).

%F a(n) = A001630(n) - a(n-1). - _Henry Bottomley_

%F G.f.: x^3/(1 - x - x^2 - x^3 - x^4).

%F G.f.: x^3 / (1 - x / (1 - x / (1 + x^3 / (1 + x / (1 - x / (1 + x)))))). - _Michael Somos_, May 12 2012

%F G.f.: sum {n >= 0} x^(n+3) *[ product {k = 1..n} (k + k*x + k*x^2 + x^3)/(1 + k*x + k*x^2 + k*x^3) ] . - _Peter Bala_, Jan 04 2015

%F a(n) = term (1,4) in the 4 X 4 matrix [1,1,0,0; 1,0,1,0; 1,0,0,1; 1,0,0,0]^n. - _Alois P. Heinz_, Jun 12 2008

%F Another form of the g.f.: f(z)=(z^3-z^4)/(1-2*z+z^5) then a(n)=sum((-1)^i*binomial(n-3-4*i,i)*2^(n-3-5*i),i=0..floor((n-3)/5))-sum((-1)^i*binomial(n-4-4*i,i)*2^(n-4-5*i),i=0..floor((n-4)/5)) with natural convention sum(alpha(i),i=m..n)=0 for m>n. - _Richard Choulet_, Feb 22 2010

%F a(n) = sum(k=1..n, sum(i=k..n mod(5*k-i,4)=0 binomial(k,(5*k-i)/4)*(-1)^((i-k)/4)*binomial(n-i+k-1,k-1))), n>0. - _Vladimir Kruchinin_, Aug 18 2010

%F sum_{k=0..3*n} a(k+b) * A008287(n,k) = a(4*n+b), b>=0 ("quadrinomial transform"). - _N. J. A. Sloane_, Nov 10 2010.

%F G.f.: x^3*(1 + x*(G(0)-1)/(x+1)) where G(k) = 1 + (1+x+x^2+x^3)/(1-x/(x+1/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 26 2013

%F Starting (1, 2, 4, 8, ...) = the INVERT transform of (1, 1, 1, 1, 0, 0, 0, ...). - _Gary W. Adamson_, May 13 2013

%F a(n) ~ c*r^n, where c=0.079077767399388561146007, and r=1.92756197548292530426195 (One of the roots of the g.f. denominator polynomial is 1/r). - _Fung Lam_, Apr 29 2014

%F a(n) = 2*a(n-1) - a(n-5), n>=5. - _Bob Selcoe_, Jul 06 2014

%e From _Petros Hadjicostas_, Mar 10 2018: (Start)

%e For n=3, we get a(3+4) = a(7) = 8 palindromic compositions of 2*n+1 = 7 into an odd number of parts that are not a multiple of 4. They are the following: 7 = 1+5+1 = 3+1+3 = 2+3+2 = 1+2+1+2+1 = 2+1+1+1+2 = 1+1+3+1+1 = 1+1+1+1+1+1+1. If we put these compositions on a circle, they become bilaterally symmetric cyclic compositions of 2*n+1 = 7.

%e For n=4, we get a(4+4) = a(8) = 15 palindromic compositions of 2*n+1 = 9 into an odd number of parts that are not a multiple of 4. They are the following: 9 = 3+3+3 = 2+5+2 = 1+7+1 = 1+1+5+1+1 = 2+1+3+1+2 = 1+2+3+2+1 = 1+3+1+3+1 = 3+1+1+1+3 = 2+2+1+2+2 = 2+1+1+1+1+1+2 = 1+2+1+1+1+2+1 = 1+1+2+1+2+1+1 = 1+1+1+3+1+1+1 = 1+1+1+1+1+1+1+1+1.

%e As D. Callan points out in the comments above, for n>=1, a(n+4) is also the number of 0-1 sequences of length n that avoid 1111. For example, for n=5, a(5+4) = a(9) = 29 is the number of binary strings of length n that avoid 1111. Out of the 2^5 = 32 binary strings of length n=5, the following do not avoid 1111: 11111, 01111, and 11110.

%e (End)

%p A000078:=-1/(-1+z+z**2+z**3+z**4); # _Simon Plouffe_ in his 1992 dissertation

%p a:= n-> (<<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_, Jun 12 2008

%t CoefficientList[Series[x^3/(1 - x - x^2 - x^3 - x^4), {x, 0, 50}], x]

%t LinearRecurrence[{1, 1, 1, 1}, {0, 0, 0, 1}, 50] (* _Vladimir Joseph Stephan Orlovsky_, May 25 2011 *)

%t Table[RootSum[-1 - # - #^2 - #^3 + #^4 &, 10 #^n + 157 #^(n + 1) - 103 #^(n + 2) + 16 #^(n + 3) &]/563, {n, 0, 20}] (* _Eric W. Weisstein_, Nov 09 2017 *)

%t Table[RootSum[#^4 - #^3 - #^2 - # - 1 &, #^(n - 2)/(-#^3 + 6 # - 1) &], {n, 0, 20}] (* _Eric W. Weisstein_, Nov 09 2017 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( x^3 / (1 - x - x^2 - x^3 - x^4) + x * O(x^n), n))}

%o (Maxima) a(n):=sum(sum(if mod(5*k-i,4)>0 then 0 else binomial(k,(5*k-i)/4)*(-1)^((i-k)/4)*binomial(n-i+k-1,k-1),i,k,n),k,1,n); \\ _Vladimir Kruchinin_, Aug 18 2010

%o (Haskell)

%o import Data.List (tails, transpose)

%o a000078 n = a000078_list !! n

%o a000078_list = 0 : 0 : 0 : f [0,0,0,1] where

%o f xs = y : f (y:xs) where

%o y = sum \$ head \$ transpose \$ take 4 \$ tails xs

%o -- _Reinhard Zumkeller_, Jul 06 2014, Apr 28 2011

%o (Python)

%o A000078 = [0,0,0,1]

%o for n in range(4,100):

%o ....A000078.append(A000078[n-1]+A000078[n-2]+A000078[n-3]+A000078[n-4])

%o # _Chai Wah Wu_, Aug 20 2014

%o (MAGMA) [n le 4 select Floor(n/4) else Self(n-1)+Self(n-2)+Self(n-3)+Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jan 29 2016

%o (GAP) a:=[0,0,0,1];; for n in [5..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]+a[n-4]; od; a; # _Muniru A Asiru_, Mar 11 2018

%Y Row 4 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).

%Y First differences are in A001631.

%Y Cf. A008287 (quadrinomial coefficients).

%K nonn,easy,nice

%O 0,6

%A _N. J. A. Sloane_

%E Definition augmented (with 4 initial terms) by _Daniel Forgues_, Dec 02 2009

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Last modified April 23 01:29 EDT 2018. Contains 302916 sequences. (Running on oeis4.)