OFFSET
0,7
COMMENTS
Number of permutations satisfying -k <= p(i) - i <= r, i=1..n-4, with k=1, r=4. - Vladimir Baltic, Jan 17 2005
a(n) is the number of compositions of n-4 with no part greater than 5. - Vladimir Baltic, Jan 17 2005
The pentanomial (A035343(n)) transform of a(n) is a(5n+4), n >= 0. - Bob Selcoe, Jun 10 2014
a(n) is the number of ways to tile a strip of length n-4 with squares, dominoes, trominoes (of length 3), and rectangles with length 4 (tetrominoes) and length 5 (pentominoes). - Wajdi Maaloul, Jun 21 2022
REFERENCES
Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
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
T. D. Noe, Table of n, a(n) for n = 0..200
Abdullah Açikel, Amrouche Said, Hacene Belbachir, and Nurettin Irmak, On k-generalized Lucas sequence with its triangle, Turkish J. Math. 47(4) (2023), Art. 6, 1129-1143. See p. 1130.
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. See p. 14.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 307-309.
Félix Balado and Guénolé C. M. Silvestre, Systematic Enumeration of Fundamental Quantities Involving Runs in Binary Strings, arXiv:2602.10005 [math.CO], 2026. See p. 19.
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Appl. Anal. Disc. Math. 4(1) (April, 2010), 119-135.
Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, J. Int. Seq. 18 (2015), Art. 15.4.5.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Int. Seq. 3 (2000), Art. 00.1.5.
Ömür Deveci, Yesim Akuzum, Erdal Karaduman, and Ozgur Erdag, The Cyclic Groups via Bezout Matrices, J. Math. Res. 7(2) (2015), 34-41.
Ömür Deveci, Zafer Adıgüzel, and Taha Doğan, On the Generalized Fibonacci-circulant-Hurwitz numbers, Notes Num. Theory Disc. Math. 26(1) (2020), 179-190.
Gregory P. B. Dresden and Zhaohui Du, A Simplified Binet Formula for k-Generalized Fibonacci Numbers, J. Int. Seq. 17 (2014), Art. 14.4.7.
Ivan Flores, k-Generalized Fibonacci numbers, Fib. Quart. 5 (1967), 258-266.
Taras Goy and Mark Shattuck, Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers, J. Int. Seq. 23 (2020), Art. 20.6.8.
Elizabeth Hartung, Brett Stevens, and Aaron Williams, Pyramid Schemes for Eating M&Ms: Enumeration, Generation, and Gray Codes, LIPIcs Vol. 366, 13th Int'l Conf. Fun Algor. (FUN 2026), Art. 23. See p. 20 (Table 5).
Tian-Xiao He, Impulse Response Sequences and Construction of Number Sequence Identities, J. Int. Seq. 16 (2013), Art. 13.8.2.
V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart. 7 (1969), 341-358, 393.
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fib. Quart. 49(3) (2011), 231-243.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 12
Omar Khadir, László Németh, and László Szalay, Tiling of dominoes with ranked colors, Results in Math. 79 (2024), Art. 253. See p. 2.
Sergey Kirgizov, Q-bonacci words and numbers, arXiv:2201.00782 [math.CO], 2022.
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Michel Mollard, p-th order generalized Fibonacci cubes and maximal cubes in Fibonacci p-cubes, arXiv:2507.16387 [math.CO], 2025. See p. 3.
Elisa Heinrich Mora and Noah A. Rosenberg, An nth-cousin mating model and the n-anacci numbers, arXiv:2506.16577 [q-bio.PE], 2025.
Elisa Heinrich Mora and Noah A. Rosenberg, An nth-cousin mating model and the n-anacci numbers. Fibonacci Quart. 64 (2026), 270-280. See Table 1.
László Németh and László Szalay, Explicit solution of system of two higher-order recurrences, arXiv:2408.12196 [math.NT], 2024. See p. 10.
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. Int. Seq. 8 (2005), Art. 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.
Helmut Prodinger, Counting Palindromes According to r-Runs of Ones Using Generating Functions, J. Int. Seq. 17 (2014), Art. 14.6.2, odd length middle 0, r=4.
B. Sivakumar and V. James, A Notes on Matrix Sequence of Pentanacci Numbers and Pentanacci Cubes, Comm. Math. Appl. 13(2) (2022), 603-611.
Yüksel Soykan, On A Generalized Pentanacci Sequence, Asian Res. J. Math. 14(3) (2019), 1-9.
Yüksel Soykan, Sum Formulas for Generalized Fifth-Order Linear Recurrence Sequences, J. Adv. Math. Comp. Sci. 34(5) (2019), 1-14.
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, Pentanacci Number.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1).
FORMULA
G.f.: x^4/(1 - x - x^2 - x^3 - x^4 - x^5). - Simon Plouffe in his 1992 dissertation.
G.f.: Sum_{n >= 0} x^(n+4) * (Product_{k = 1..n} (k + k*x + k*x^2 + k*x^3 + x^4)/(1 + k*x + k*x^2 + k*x^3 + k*x^4)). - Peter Bala, Jan 04 2015
Another form of the g.f.: f(z) = (z^4-z^5)/(1-2*z+z^6); then a(n) = Sum_{i=0..floor((n-4)/6)} ((-1)^i*binomial(n-4-5*i,i)*2^(n-4-6*i)) - Sum_{i=0..floor((n-5)/6)} ((-1)^i*binomial(n-5-5*i,i)*2^(n-5-6*i)) with convention Sum_{i=m..n} alpha(i) = 0 for m > n. - Richard Choulet, Feb 22 2010
a(n) = Sum_{k=1..n} (Sum_{r=0..k} (binomial(k,r) * Sum_{m=0..r} (binomial(r,m) * Sum_{j=0..m} (binomial(m,j)*binomial(j,n-m-k-j-r))))), n > 0. - Vladimir Kruchinin, Aug 30 2010
Sum_{k=0..4*n} a(k+b)*A035343(n,k) = a(5*n+b), b >= 0.
a(n) = 2*a(n-1) - a(n-6). - Vincenzo Librandi, Dec 19 2010
a(n) = (Sum_{i=0..n-1} a(i)*A074048(n-i))/(n-4) for n > 4. - Greg Dresden and Advika Srivastava, Oct 01 2019
For k>0 and n>0, a(n+5*k) = A074048(k)*a(n+4*k) - A123127(k-1)*a(n+3*k) + A123126(k-1)*a(n+2*k) - A074062(k)*a(n+k) + a(n). - Kai Wang, Sep 06 2020
lim n->oo a(n)/a(n-1) = A103814. - R. J. Mathar, Mar 11 2024
EXAMPLE
n=2: a(14) = (1*1 + 2*1 + 3*2 + 4*4 + 5*8 + 4*16 + 3*31 + 2*61 + 1*120) = 464. - Bob Selcoe, Jun 10 2014
G.f. = x^4 + x^5 + 2*x^6 + 4*x^7 + 8*x^8 + 16*x^9 + 31*x^10 + 120*x^11 + ...
MAPLE
g:=1/(1-z-z^2-z^3-z^4-z^5): gser:=series(g, z=0, 49): seq((coeff(gser, z, n)), n=-4..32); # Zerinvary Lajos, Apr 17 2009
# Alternative:
a:= n-> (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>, <0|0|0|0|1>, <1|1|1|1|1>>^n)[1, 5]:
seq(a(n), n=0..44); # Alois P. Heinz, Apr 09 2021
MATHEMATICA
CoefficientList[Series[x^4/(1 - x - x^2 - x^3 - x^4 - x^5), {x, 0, 50}], x]
(* Alternative: *)
a[0] = a[1] = a[2] = a[3] = 0; a[4] = a[5] = 1; a[n_] := a[n] = 2 a[n - 1] - a[n - 6]; Array[a, 37, 0]
(* Alternative: *)
LinearRecurrence[{1, 1, 1, 1, 1}, {0, 0, 0, 0, 1}, 50] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
PROG
(PARI) a=vector(100); a[4]=a[5]=1; for(n=6, #a, a[n]=a[n-1]+a[n-2]+a[n-3]+a[n-4]+a[n-5]); concat(0, a) \\ Charles R Greathouse IV, Jul 15 2011
(PARI) A001591(n, m=5)=(matrix(m, m, i, j, i==j-1||i==m)^n)[1, m] \\ M. F. Hasler, Apr 20 2018
(PARI) a(n)= {my(x='x, p=polrecip(1 - x - x^2 - x^3 - x^4 - x^5)); polcoef(lift(Mod(x, p)^n), 4); }
vector(41, n, a(n-1)) \\ Joerg Arndt, May 16 2021
(Maxima) a(n):=mod(floor(10^((n-4)*(n+1))*10^(5*(n+1))*(10^(n+1)-1)/(10^(6*(n+1))-2*10^(5*(n+1))+1)), 10^n); /* Tani Akinari, Apr 10 2014 */
(Magma) a:=[0, 0, 0, 0, 1]; [n le 5 select a[n] else Self(n-1) + Self(n-2) + Self(n-3) + Self(n-4) + Self(n-5): n in [1..40]]; // Marius A. Burtea, Oct 03 2019
(Python)
def pentanacci():
a, b, c, d, e = 0, 0, 0, 0, 1
while True:
yield a
a, b, c, d, e = b, c, d, e, a + b + c + d + e
f = pentanacci()
print([next(f) for _ in range(100)]) # Reza K Ghazi Apr 09 2021
KEYWORD
nonn,easy,changed
AUTHOR
STATUS
approved