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A001631 Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4).
(Formerly M1081 N0410)
37
0, 0, 1, 0, 1, 2, 4, 7, 14, 27, 52, 100, 193, 372, 717, 1382, 2664, 5135, 9898, 19079, 36776, 70888, 136641, 263384, 507689, 978602, 1886316, 3635991, 7008598, 13509507, 26040412, 50194508, 96753025, 186497452, 359485397, 692930382, 1335666256, 2574579487 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Starting (1, 2, 4,...) is the INVERT transform of the cyclic sequence (1, 1, 1, 0, (repeat)...); equivalent to the statement that (1, 2, 4,...) corresponding to n = (1, 2, 3,...) represents the numbers of ordered compositions of n using terms in the set "not multiples of four". - Gary W. Adamson, May 13 2013

a(n+4) equals the number of n-length binary words avoiding runs of zeros of lengths 4i+3, (i=0,1,2,...). - Milan Janjic, Feb 26 2015

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

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Matthias Beck, Neville Robbins, Variations on a Generatingfunctional Theme: Enumerating Compositions with Parts Avoiding an Arithmetic Sequence, arXiv:1403.0665 [math.NT] (2014)

W. C. Lynch, The t-Fibonacci numbers and polyphase sorting, Fib. Quart., 8 (1970), pp. 6ff.

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.

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

FORMULA

G.f.: ((x-1)*x^2)/(x^4+x^3+x^2+x-1). - Harvey P. Dale, Oct 21 2011

MAPLE

A001631:=(-1+z)/(-1+z+z**2+z**3+z**4); # [Conjectured by Simon Plouffe in his 1992 dissertation.]

a:= n-> (Matrix([[0, -1, 2, -1]]). Matrix(4, (i, j)-> `if`(i=j-1 or j=1, 1, 0))^n)[1, 1]: seq(a(n), n=0..35); # Alois P. Heinz, Aug 01 2008

MATHEMATICA

LinearRecurrence[{1, 1, 1, 1}, {0, 0, 1, 0}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)

CoefficientList[Series[((-1+x) x^2)/(-1+x+x^2+x^3+x^4), {x, 0, 50}], x] (* Harvey P. Dale, Oct 21 2011 *)

PROG

(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; 1, 1, 1, 1]^n*[0; 0; 1; 0])[1, 1] \\ Charles R Greathouse IV, Apr 08 2016

CROSSREFS

First differences of A000078.

Cf. A000336.

Sequence in context: A123196 A079968 A280194 * A108758 A018085 A167751

Adjacent sequences:  A001628 A001629 A001630 * A001632 A001633 A001634

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Jul 31 2000

STATUS

approved

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Last modified August 21 00:45 EDT 2017. Contains 290855 sequences.