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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
a(n) is the number of ways to tile a skew double-strip of n-2 cells using squares and all possible "dominos", as seen in the comments in A000078, but with the added provision that the first tile (in the lower left corner) must be a domino. For reference, here is the skew double-strip corresponding to n=14, with 12 cells:
___ ___ ___ ___ ___ ___
| | | | | | |
_|___|___|___|___|_ _|___|
| | | | | | |
|___|___|___|___|___|___|,
and here are the three possible "domino" tiles:
___ ___
| | | |
_| _| |_ |_ _______
| | | | | |
|___|, |___|, |_______|. - Greg Dresden and Ruotong Li, Jun 05 2024
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
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
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
KEYWORD
nonn,easy,changed
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 June 14 21:14 EDT 2024. Contains 373401 sequences. (Running on oeis4.)