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 A001591 Pentanacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5), a(0)=a(1)=a(2)=a(3)=0, a(4)=1. (Formerly M1122 N0429) 57
 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513, 400096, 786568, 1546352, 3040048, 5976577, 11749641, 23099186, 45411804, 89277256, 175514464, 345052351, 678355061, 1333610936, 2621810068 (list; graph; refs; listen; history; text; internal format)
 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 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 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 Vol. 4, No 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, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5. P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. O. Deveci, Y. Akuzum, E. Karaduman, and O. Erdag, The Cyclic Groups via Bezout Matrices, Journal of Mathematics Research, Vol. 7, No. 2, 2015, pp. 34-41. 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. Int. Seq. 17 (2014) # 14.4.7. I. 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., Vol. 23 (2020), Article 20.6.8. T.-X. He, Impulse Response Sequences and Construction of Number Sequence Identities, J. Int. Seq. 16 (2013) #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, Fibonacci Quart. 49 (2011), no. 3, 231-243. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 12 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. Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 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. Int. Seq. 17 (2014) # 14.6.2, odd length middle 0, r=4. Yüksel Soykan, On A Generalized Pentanacci Sequence, Asian Research Journal of Mathematics (2019) Vol. 14, No. 3, 1-9. Yüksel Soykan, Sum Formulas for Generalized Fifth-Order Linear Recurrence Sequences, Journal of Advances in Mathematics and Computer Science (2019) Vol. 34, No. 5, 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 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 # second Maple program: 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] 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] 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 CROSSREFS Row 5 of arrays A048887 and A092921 (k-generalized Fibonacci numbers). Cf. A106303 (Pisano period lengths). Cf. A035343 (pentanomial coefficients). Cf. A074048, A123127, A123126, A074062. Sequence in context: A128761 A332726 A239557 * A194628 A003240 A280543 Adjacent sequences:  A001588 A001589 A001590 * A001592 A001593 A001594 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified May 18 21:30 EDT 2022. Contains 353825 sequences. (Running on oeis4.)