OFFSET
0,6
COMMENTS
a(n) is the 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
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 [Pólya-Szegő]. - N. J. A. Sloane, Jul 28 2012
a(n+4) is the number of 0-1 sequences of length n that avoid 1111. - David Callan, Jul 19 2004
a(n) is the 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
Number of permutations satisfying -k <= p(i)-i <= r, i=1..n-3, with k = 1, r = 3. - Vladimir Baltic, Jan 17 2005
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
a(n) is the number of ways to tile a hexagonal double-strip (two rows of adjacent hexagons) containing (n-4) cells with hexagons and double-hexagons (two adjacent hexagons). - Ziqian Jin, Jul 28 2019
The term "tetranacci number" was coined by Mark Feinberg (1963; see A000073). - Amiram Eldar, Apr 16 2021
a(n) is the number of ways to tile a skew double-strip of n-3 cells using squares and all possible "dominos", as seen in Ziqian Jin's article, below. Here is the skew double-strip corresponding to n=15, with 12 cells:
___ ___ ___ ___ ___ ___
| | | | | | |
_|___|___|___|___|_ _|___|
| | | | | | |
|___|___|___|___|___|___|,
and here are the three possible "domino" tiles:
___ ___
| | | |
_| _| |_ |_ _______
| | | | | |
|___|, |___|, |_______|.
As an example, here is one of the a(15) = 1490 ways to tile the skew double-strip of 12 cells:
___ ___ _______ _______
| | | | | | |
_|___|_ |___|_ _|_ _| _|
| | | | | |
|_______|___|___|___|___|. - Greg Dresden, Jun 05 2024
REFERENCES
Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1, Problems 3 and 4.
J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, Princeton, NJ, 1978.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..3505 (terms 0..200 from T. D. Noe)
Abdullah Açikel, Amrouche Said, Hacene Belbachir, and Nurettin Irmak, On k-generalized Lucas sequence with its triangle, Turkish J. Math. (2023) Vol. 47, No. 4, Art. 6, 1129-1143. See p. 1130.
Isha Agarwal, Matvey Borodin, Aidan Duncan, Kaylee Ji, Tanya Khovanova, Shane Lee, Boyan Litchev, Anshul Rastogi, Garima Rastogi, and Andrew Zhao, From Unequal Chance to a Coin Game Dance: Variants of Penney's Game, arXiv:2006.13002 [math.HO], 2020.
Tomás Aguilar-Fraga, Jennifer Elder, Rebecca E. Garcia, Kimberly P. Hadaway, Pamela E. Harris, Kimberly J. Harry, Imhotep B. Hogan, Jakeyl Johnson, Jan Kretschmann, Kobe Lawson-Chavanu, J. Carlos Martínez Mori, Casandra D. Monroe, Daniel Quiñonez, Dirk Tolson III, and Dwight Anderson Williams II, Interval and L-interval Rational Parking Functions, arXiv:2311.14055 [math.CO], 2023.
Kassie Archer and Aaron Geary, Powers of permutations that avoid chains of patterns, arXiv:2312.14351 [math.CO], 2023. See p. 15.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 307-309.
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics 4(1) (2010), 119-135.
Elena Barcucci, Antonio Bernini, Stefano Bilotta, and Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016.
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, J. Integer Seq. 18 (2015), Article 15.4.5.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integer Seq. 3 (2000), Article 00.1.5.
S. A. Corey and Otto Dunkel, Problem 2803, Amer. Math. Monthly 33 (1926), 229-232.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, J. Integer Seq. 19 (2016), Article 16.1.1.
Emeric Deutsch, Problem 1613: A recursion in four parts, Math. Mag. 75(1) (2002), 64-65.
Ömür Deveci, Zafer Adıgüzel and Taha Doğan, On the Generalized Fibonacci-circulant-Hurwitz numbers, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 1, 179-190.
G. P. B. Dresden and Z. Du, A Simplified Binet Formula for k-Generalized Fibonacci Numbers, J. Integer Seq. 17 (2014), Article 14.4.7.
M. Feinberg, Fibonacci-Tribonacci, Fib. Quart. 1(3) (1963), 71-74.
Taras Goy and Mark Shattuck, Some Toeplitz-Hessenberg determinant identities for the tetranacci numbers, J. Integer Seq. 23 (2020), Article 20.6.8.
Petros Hadjicostas, Cyclic compositions of a positive integer with parts avoiding an arithmetic sequence, J. Integer Seq. 19 (2016), Article 16.8.2.
Petros Hadjicostas, Generalized colored circular palindromic compositions, Moscow Journal of Combinatorics and Number Theory, 9(2) (2020), 173-186.
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.
Russell Jay Hendel, A Method for Uniformly Proving a Family of Identities, arXiv:2107.03549 [math.CO], 2021.
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quarterly 49 (2011), 231-242.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 11
Ziqian Jin, Tetrancci Identities With Squares, Dominoes, And Hexagonal Double Strips, arXiv:1907.09935 [math.GM], 2019.
Omar Khadir, László Németh, and László Szalay, Tiling of dominoes with ranked colors, Results in Math. (2024) Vol. 79, Art. No. 253. See p. 2.
Sergey Kirgizov, Q-bonacci words and numbers, arXiv:2201.00782 [math.CO], 2022.
W. C. Lynch, The t-Fibonacci numbers and polyphase sorting, Fib. Quart., 8 (1970), 6-22.
T. Mansour and M. Shattuck, A monotonicity property for generalized Fibonacci sequences, arXiv:1410.6943 [math.CO], 2014.
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. Integer Seq. 8 (2005), Article 05.4.4.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
H. Prodinger, Counting Palindromes According to r-Runs of Ones Using Generating Functions, J. Integer Seq. 17 (2014), Article 14.6.2; odd length middle 0, r=3.
Helmut Prodinger and Sarah J. Selkirk, Sums of squares of Tetranacci numbers: A generating function approach, arXiv:1906.08336 [math.NT], 2019.
Lan Qi and Zhuoyu Chen, Identities Involving the Fourth-Order Linear Recurrence Sequence, Symmetry 11(12) (2019), 1476, 8pp.
J. L. Ramírez and V. F. Sirvent, A Generalization of the k-Bonacci Sequence from Riordan Arrays, Electron. J. Combin. 22(1) (2015), #P1.38.
O. Turek, Abelian Complexity Function of the Tribonacci Word, J. Integer Seq. 18 (2015), Article 15.3.4.
Kai Wang, Identities for generalized enneanacci numbers, Generalized Fibonacci Sequences (2020).
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Eric Weisstein's World of Mathematics, Tetranacci Number.
L. Zhang and P. Hadjicostas, On sequences of independent Bernoulli trials avoiding the pattern '11..1', Math. Scientist, 40 (2015), 89-96.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).
FORMULA
a(n) = A001630(n) - a(n-1). - Henry Bottomley
G.f.: x^3/(1 - x - x^2 - x^3 - x^4). - Simon Plouffe in his 1992 dissertation
G.f.: x^3 / (1 - x / (1 - x / (1 + x^3 / (1 + x / (1 - x / (1 + x)))))). - Michael Somos, May 12 2012
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
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
Another form of the g.f.: f(z) = (z^3 - z^4)/(1 - 2*z + z^5), then a(n) = Sum_{i=0..floor((n-3)/5)} (-1)^i*binomial(n-3-4*i, i)*2^(n - 3 - 5*i) - Sum_{i=0..floor((n-4)/5)} (-1)^i*binomial(n-4-4*i, i)*2^(n - 4 - 5*i) with natural convention Sum_{i=m..n} alpha(i) = 0 for m > n. - Richard Choulet, Feb 22 2010
a(n+3) = Sum_{k=1..n} Sum_{i=k..n} [(5*k-i mod 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 [Edited by Petros Hadjicostas, Jul 26 2020, so that the formula agrees with the offset of the sequence]
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
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
Starting (1, 2, 4, 8, ...) = the INVERT transform of (1, 1, 1, 1, 0, 0, 0, ...). - Gary W. Adamson, May 13 2013
a(n) ~ c*r^n, where c = 0.079077767399388561146007... and r = 1.92756197548292530426195... = A086088 (One of the roots of the g.f. denominator polynomial is 1/r.). - Fung Lam, Apr 29 2014
a(n) = 2*a(n-1) - a(n-5), n >= 5. - Bob Selcoe, Jul 06 2014
From Ziqian Jin, Jul 28 2019: (Start)
a(2*n+5) = a(n+4)^2 + a(n+3)^2 + a(n+2)^2 + 2*a(n+3)*(a(n+2) + a(n+1)).
a(n) - 1 = a(n-2) + 2*a(n-3) + 3*(a(n-4) + a(n-5) + ... + a(2) + a(1)), n >= 4. (End)
a(n) = (Sum_{i=0..n-1} a(i)*A073817(n-i))/(n-3) for n > 3. - Greg Dresden and Advika Srivastava, Sep 28 2019
a(n) = Sum_{r root of x^4-x^3-x^2-x-1} r^n/(4*r^3-3*r^2-2*r-1). - Fabian Pereyra, Dec 06 2024
EXAMPLE
From Petros Hadjicostas, Mar 10 2018: (Start)
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.
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.
As David 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. (End)
MAPLE
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
MATHEMATICA
CoefficientList[Series[x^3/(1-x-x^2-x^3-x^4), {x, 0, 50}], x]
LinearRecurrence[{1, 1, 1, 1}, {0, 0, 0, 1}, 50] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
(* From Eric W. Weisstein, Nov 09 2017 *)
Table[RootSum[-1 -# -#^2 -#^3 +#^4 &, 10#^n +157#^(n+1) -103 #^(n+2) +16#^(n+3) &]/563, {n, 0, 40}]
Table[RootSum[#^4 -#^3 -#^2 -# -1 &, #^(n-2)/(-#^3 +6# -1) &], {n, 0, 40}] (* End *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( x^3 / (1 - x - x^2 - x^3 - x^4) + x * O(x^n), n))}
(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 */
(Haskell)
import Data.List (tails, transpose)
a000078 n = a000078_list !! n
a000078_list = 0 : 0 : 0 : f [0, 0, 0, 1] where
f xs = y : f (y:xs) where
y = sum $ head $ transpose $ take 4 $ tails xs
-- Reinhard Zumkeller, Jul 06 2014, Apr 28 2011
(Python)
A000078 = [0, 0, 0, 1]
for n in range(4, 100):
# Chai Wah Wu, Aug 20 2014
(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
(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
CROSSREFS
First differences are in A001631.
KEYWORD
nonn,easy,nice,changed
AUTHOR
EXTENSIONS
Definition augmented (with 4 initial terms) by Daniel Forgues, Dec 02 2009
Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021
STATUS
approved