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 A001631 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) 41
 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 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 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 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). 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)[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 CROSSREFS 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 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms from Larry Reeves (larryr(AT)acm.org), Jul 31 2000 Edited by M. F. Hasler, Apr 20 2018 STATUS approved

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Last modified March 26 09:21 EDT 2023. Contains 361529 sequences. (Running on oeis4.)