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).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

• 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).

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).

Programs

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

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 *)

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

Formula

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

Extensions

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

Edited by M. F. Hasler, Apr 20 2018