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A001648
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Tetranacci numbers A073817 without the leading term 4.
(Formerly M2648 N1055)
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18
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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
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OFFSET
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1,2
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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MAPLE
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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
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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 *)
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PROG
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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))
(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 */
(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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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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