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A001648 Tetranacci numbers A073817 without the leading term 4.
(Formerly M2648 N1055)
8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

Fielder, Daniel C.; Special integer sequences controlled by three parameters. Fibonacci Quart 6 1968 64-70.

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=1..200

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric Weisstein's World of Mathematics, Lucas n-Step Number

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. [From Vladimir Kruchinin, Jan 20 2012]

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.]

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 *)

PROG

(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); [From Vladimir Kruchinin, Jan 20 2012]

CROSSREFS

Cf. A000288, A073817.

Sequence in context: A078869 A011890 A131076 * A051054 A001649 A213215

Adjacent sequences:  A001645 A001646 A001647 * A001649 A001650 A001651

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified October 21 14:25 EDT 2014. Contains 248377 sequences.