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%I M1122 N0429 #229 Mar 11 2024 05:20:14
%S 0,0,0,0,1,1,2,4,8,16,31,61,120,236,464,912,1793,3525,6930,13624,
%T 26784,52656,103519,203513,400096,786568,1546352,3040048,5976577,
%U 11749641,23099186,45411804,89277256,175514464,345052351,678355061,1333610936,2621810068
%N 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.
%C Number of permutations satisfying -k <= p(i) - i <= r, i=1..n-4, with k=1, r=4. - _Vladimir Baltic_, Jan 17 2005
%C a(n) is the number of compositions of n-4 with no part greater than 5. - _Vladimir Baltic_, Jan 17 2005
%C The pentanomial (A035343(n)) transform of a(n) is a(5n+4), n >= 0. - _Bob Selcoe_, Jun 10 2014
%C 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 (tetraminoes) and length 5 (pentaminoes). - _Wajdi Maaloul_, Jun 21 2022
%D Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
%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).
%H T. D. Noe, <a href="/A001591/b001591.txt">Table of n, a(n) for n = 0..200</a>
%H Abdullah Açikel, Amrouche Said, Hacene Belbachir, and Nurettin Irmak, <a href="https://doi.org/10.55730/1300-0098.3416">On k-generalized Lucas sequence with its triangle</a>, Turkish J. Math. (2023) Vol. 47, No. 4, Art. 6, 1129-1143. See p. 1130.
%H 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, <a href="https://arxiv.org/abs/2311.14055">Interval and L-interval Rational Parking Functions</a>, arXiv:2311.14055 [math.CO], 2023. See p. 14.
%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://dx.doi.org/10.2298/AADM1000008B">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 Paul Barry, <a href="https://arxiv.org/abs/1804.05027">The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays</a>, arXiv:1804.05027 [math.CO], 2018.
%H Martin Burtscher, Igor Szczyrba, and 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 O. Deveci, Y. Akuzum, E. Karaduman, and O. Erdag, <a href="http://dx.doi.org/10.5539/jmr.v7n2p34">The Cyclic Groups via Bezout Matrices</a>, Journal of Mathematics Research, Vol. 7, No. 2, 2015, pp. 34-41.
%H Ömür Deveci, Zafer Adıgüzel, and Taha Doğan, <a href="https://doi.org/10.7546/nntdm.2020.26.1.179-190">On the Generalized Fibonacci-circulant-Hurwitz numbers</a>, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 1, 179-190.
%H G. P. B. Dresden and 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 I. Flores, <a href="http://www.fq.math.ca/Scanned/5-3/flores.pdf">k-Generalized Fibonacci numbers</a>, Fib. Quart., 5 (1967), 258-266.
%H Taras Goy and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Shattuck/shattuck20.html">Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.6.8.
%H T.-X. He, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/He/he13.html">Impulse Response Sequences and Construction of Number Sequence Identities</a>, J. Int. Seq. 16 (2013) #13.8.2.
%H V. E. Hoggatt, Jr. and M. Bicknell, <a href="http://www.fq.math.ca/Scanned/7-4/hoggatt-a.pdf">Diagonal sums of generalized Pascal triangles</a>, Fib. Quart., 7 (1969), 341-358, 393.
%H F. T. Howard and Curtis Cooper, <a href="http://www.fq.math.ca/Papers1/49-3/HowardCooper.pdf">Some identities for r-Fibonacci numbers</a>, Fibonacci Quart. 49 (2011), no. 3, 231-243.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=12">Encyclopedia of Combinatorial Structures 12</a>
%H Sergey Kirgizov, <a href="https://arxiv.org/abs/2201.00782">Q-bonacci words and numbers</a>, arXiv:2201.00782 [math.CO], 2022.
%H Vladimir Victorovich Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%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="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Helmut 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=4.
%H B. Sivakumar and V. James, <a href="https://doi.org/10.26713/cma.v13i2.1725">A Notes on Matrix Sequence of Pentanacci Numbers and Pentanacci Cubes</a>, Communications in Mathematics and Applications (2022) Vol. 13, Iss. 2, 603-611.
%H Yüksel Soykan, <a href="https://doi.org/10.9734/ARJOM/2019/v14i330129">On A Generalized Pentanacci Sequence</a>, Asian Research Journal of Mathematics (2019) Vol. 14, No. 3, 1-9.
%H Yüksel Soykan, <a href="https://doi.org/10.9734/JAMCS/2019/v34i530224">Sum Formulas for Generalized Fifth-Order Linear Recurrence Sequences</a>, Journal of Advances in Mathematics and Computer Science (2019) Vol. 34, No. 5, 1-14.
%H Kai Wang, <a href="https://www.researchgate.net/publication/344295426_IDENTITIES_FOR_GENERALIZED_ENNEANACCI_NUMBERS">Identities for generalized enneanacci numbers</a>, Generalized Fibonacci Sequences (2020).
%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/PentanacciNumber.html">Pentanacci Number</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1,1).
%F G.f.: x^4/(1 - x - x^2 - x^3 - x^4 - x^5). - _Simon Plouffe_ in his 1992 dissertation.
%F 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
%F 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
%F 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
%F Sum_{k=0..4*n} a(k+b)*A035343(n,k) = a(5*n+b), b >= 0.
%F a(n) = 2*a(n-1) - a(n-6). - _Vincenzo Librandi_, Dec 19 2010
%F 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
%F 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
%F lim n->oo a(n)/a(n-1) = A103814. - _R. J. Mathar_, Mar 11 2024
%e 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
%e G.f. = x^4 + x^5 + 2*x^6 + 4*x^7 + 8*x^8 + 16*x^9 + 31*x^10 + 120*x^11 + ...
%p 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
%p # second Maple program:
%p 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]:
%p seq(a(n), n=0..44); # _Alois P. Heinz_, Apr 09 2021
%t CoefficientList[Series[x^4/(1 - x - x^2 - x^3 - x^4 - x^5), {x, 0, 50}], x]
%t 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]
%t LinearRecurrence[{1, 1, 1, 1, 1}, {0, 0, 0, 0, 1}, 50] (* _Vladimir Joseph Stephan Orlovsky_, May 25 2011 *)
%o (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
%o (PARI) A001591(n,m=5)=(matrix(m,m,i,j,i==j-1||i==m)^n)[1,m] \\ _M. F. Hasler_, Apr 20 2018
%o (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); }
%o vector(41, n, a(n-1)) \\ _Joerg Arndt_, May 16 2021
%o (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 */
%o (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
%o (Python)
%o def pentanacci():
%o a, b, c, d, e = 0, 0, 0, 0, 1
%o while True:
%o yield a
%o a, b, c, d, e = b, c, d, e, a + b + c + d + e
%o f = pentanacci()
%o print([next(f) for _ in range(100)]) # _Reza K Ghazi_ Apr 09 2021
%Y Row 5 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
%Y Cf. A106303 (Pisano period lengths).
%Y Cf. A035343 (pentanomial coefficients).
%Y Cf. A074048, A123127, A123126, A074062.
%K nonn,easy
%O 0,7
%A _N. J. A. Sloane_
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