A001648 Tetranacci numbers A073817 without the leading term 4.
1, 3, 7, 15, 26, 51, 99, 191, 367, 708, 1365, 2631, 5071, 9775, 18842, 36319, 70007, 134943, 260111, 501380, 966441, 1862875, 3590807, 6921503, 13341626, 25716811, 49570747, 95550687, 184179871, 355018116, 684319421, 1319068095, 2542585503, 4900991135
Offset: 1
References
- 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 = 1..3501 (terms 1..200 from T. D. Noe)
- Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.
- 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
- Shingo Saito, Tatsushi Tanaka, and Noriko Wakabayashi, Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values, J. Int. Seq. 14 (2011) # 11.2.4, Table 3.
- Eric Weisstein's World of Mathematics, Lucas n-Step Number
- Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).
Programs
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Magma
I:=[1,3,7,15]; [n le 4 select I[n] else Self(n-1) + Self(n-2) + Self(n-3) + Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 18 2017
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Maple
A001648:=-(1+2*z+3*z**2+4*z**3)/(-1+z+z**2+z**3+z**4); # conjectured by Simon Plouffe in his 1992 dissertation
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Mathematica
Rest@ CoefficientList[ Series[(4 - 3 x - 2 x^2 - x^3)/(1 - x - x^2 - x^3 - x^4), {x, 0, 40}], x] (* Or *) a[0] = 4; a[1] = 1; a[2] = 3; a[3] = 7; a[4] = 15; a[n_] := 2*a[n - 1] - a[n - 5]; Array[a, 33] (* Robert G. Wilson v *) LinearRecurrence[{1, 1, 1, 1}, {1, 3, 7, 15}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *) -
Maxima
a(n):=n*sum(sum((-1)^i*binomial(k,k-i)*binomial(n-i*4-1,k-1),i,0,((n-k)/4))/k,k,ceiling(n/5),n); /* Vladimir Kruchinin, Jan 20 2012 */
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PARI
a(n)=if(n<0,0,polcoeff(x*(1+2*x+3*x^2+4*x^3)/(1-x-x^2-x^3-x^4)+x*O(x^n),n))
Formula
G.f.: x*(1+2*x+3*x^2+4*x^3)/(1-x-x^2-x^3-x^4).
a(n) = trace of M^n, where M = the 4 X 4 matrix [ 0 1 0 0 / 0 0 1 0 / 0 0 0 1 / 1 1 1 1]. E.g., the trace (sum of diagonal terms) of M^12 = a(12) = 2631 = (108 + 316 + 717 + 1490). - Gary W. Adamson, Feb 22 2004
a(n) = n*Sum_{k=ceiling(n/5)..n} Sum_{i=0..(n-k)/4} (-1)^i*binomial(k,k-i)*binomial(n-i*4-1,k-1)/k, n>0. - Vladimir Kruchinin, Jan 20 2012