A000336 -id:A000336 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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).

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).

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

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

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 *)

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

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

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

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

Edited by M. F. Hasler, Apr 20 2018

A251656 4-step Fibonacci sequence starting with 1,0,1,0.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 6, 11, 22, 42, 81, 156, 301, 580, 1118, 2155, 4154, 8007, 15434, 29750, 57345, 110536, 213065, 410696, 791642, 1525939, 2941342, 5669619, 10928542, 21065442, 40604945, 78268548, 150867477, 290806412, 560547382, 1080489819

Offset: 0

Arie Bos, Dec 06 2014

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

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251654, A251655, A251703, A251704, A251705.

Cf. A000336.

NB. see A251655 for the program and apply it to 1,0,1,0.

LinearRecurrence[Table[1, {4}], {1, 0, 1, 0}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n)+a(n+1)+a(n+2)+a(n+3).
G.f.: (-1+x+2*x^3)/(-1+x+x^2+x^3+x^4) . - R. J. Mathar, Mar 28 2025
a(n) = A000078(n+3)-A000078(n+2)-2*A000078(n). - R. J. Mathar, Mar 28 2025

A000308 a(n) = a(n-1)a(n-2)a(n-3) with a(1)=1, a(2)=2 and a(3)=3.

Original entry on oeis.org

1, 2, 3, 6, 36, 648, 139968, 3265173504, 296148833645101056, 135345882205792807436868315512832, 130876399105969522361889021452224949874232743897657526714368

Offset: 1

N. J. A. Sloane

nonn

			a(6)=36*6*3=648.

John Cerkan, Table of n, a(n) for n = 1..15

Cf. A000304, A000336, A063401.

A000308 := proc(n) option remember; if n <=3 then n else A000308(n-1)*A000308(n-2)*A000308(n-3); fi; end;

a(n) = 2^A001590(n-1)*3^A000073(n-1). - Henry Bottomley, Jul 16 2001

A299399 a(n) = a(n-1)a(n-2)a(n-3)*a(n-4); a(0..3) = (1, 1, 2, 3).

Original entry on oeis.org

1, 1, 2, 3, 6, 36, 1296, 839808, 235092492288, 9211413321697223245824, 2356948205087252000835395074931259831484416, 4286423488783965214900384842824017360544199884413056912194095171350270745233063936

Offset: 0

M. F. Hasler, Apr 22 2018

nonn

A variant of A000336 which uses initial values (1,2,3,4).

A multiplicative variant of the tetranacci sequences A000078, A001631 and other variants.

Seiichi Manyama, Table of n, a(n) for n = 0..14

Cf. A000336 (variant starting 1,2,3,4).
Cf. A000301 (order 2 variant), A000308 (order 3 variant).
Subsequence of A003586 (3-smooth numbers).
Cf. A000078, A001631 (additive variants).

nxt[{a_,b_,c_,d_}]:={b,c,d,a b c d}; NestList[nxt,{1,1,2,3},13][[All,1]] (* Harvey P. Dale, Jun 09 2022 *)

A299399(n,a=[1,1,2,3,6])={for(n=5,n,a[n%#a+1]=a[(n-1)%#a+1]^2\a[n%#a+1]);a[n%#a+1]}

a(n) = a(n-1)^2 / a(n-5) for n > 4.

a(n) = 2^A001631(n)*3^A000078(n).

cp's OEIS Frontend

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).

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A251656 4-step Fibonacci sequence starting with 1,0,1,0.

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A000308 a(n) = a(n-1)*a(n-2)*a(n-3) with a(1)=1, a(2)=2 and a(3)=3.

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A299399 a(n) = a(n-1)*a(n-2)*a(n-3)*a(n-4); a(0..3) = (1, 1, 2, 3).

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A000308 a(n) = a(n-1)a(n-2)a(n-3) with a(1)=1, a(2)=2 and a(3)=3.

A299399 a(n) = a(n-1)a(n-2)a(n-3)*a(n-4); a(0..3) = (1, 1, 2, 3).