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A001631
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Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4), with initial conditions a(0..3) = (0, 0, 1, 0).
(Formerly M1081 N0410)
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41
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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
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OFFSET
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0,6
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COMMENTS
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The "standard" Tetranacci numbers with initial terms (0,0,0,1) are listed in A000078.
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
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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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Harvey P. Dale, Table of n, a(n) for n = 0..1000
Matthias Beck and Neville Robbins, Variations on a Generating Function Theme: Enumerating Compositions with Parts Avoiding an Arithmetic Sequence, arXiv:1403.0665 [math.NT], 2014.
Petros Hadjicostas, Cyclic compositions of a positive integer with parts avoiding an arithmetic sequence, Journal of Integer Sequences, 19 (2016), #16.8.2.
W. C. Lynch, The t-Fibonacci numbers and polyphase sorting, Fib. Quart., 8 (1970), pp. 6-22.
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
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).
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FORMULA
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G.f.: ((x-1)*x^2)/(x^4+x^3+x^2+x-1). - Harvey P. Dale, Oct 21 2011
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; 1, 1, 1, 1]^n)[1, 3] \\ Charles R Greathouse IV, Apr 08 2016, simplified by M. F. Hasler, Apr 20 2018
(PARI) x='x+O('x^30); concat([0, 0], Vec(((x-1)*x^2)/(x^4+x^3+x^2+x-1))) \\ G. C. Greubel, Jan 09 2018
(Magma) I:=[0, 0, 1, 0]; [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, Jan 09 2018
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CROSSREFS
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Absolute values of first differences of standard Tetranacci numbers A000078.
Cf. A000288 (variant: starting with 1, 1, 1, 1).
Cf. A000336 (variant: sum replaced by product).
Sequence in context: A347783 A079968 A280194 * A108758 A018085 A167751
Adjacent sequences: A001628 A001629 A001630 * A001632 A001633 A001634
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Jul 31 2000
Edited by M. F. Hasler, Apr 20 2018
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STATUS
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approved
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