A001630 Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4), with a(0)=a(1)=0, a(2)=1, a(3)=2.

0, 0, 1, 2, 3, 6, 12, 23, 44, 85, 164, 316, 609, 1174, 2263, 4362, 8408, 16207, 31240, 60217, 116072, 223736, 431265, 831290, 1602363, 3088654, 5953572, 11475879, 22120468, 42638573, 82188492, 158423412, 305370945, 588621422, 1134604271, 2187020050

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also (with a different offset), coordination sequence for (4,infinity,infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015

Apparently for n>=2 the number of 1-D walks of length n-2 using steps +1, +3 and -1, avoiding consecutive -1 steps. - David Scambler, Jul 15 2013

From Elkhan Aday and Greg Dresden, Jun 24 2024: (Start)

For n > 1, a(n) is the number of ways to tile a skew double-strip of n-1 cells with one extra initial cell, using squares and all possible "dominoes". Here is the skew double-strip corresponding to n=12, with 11 cells:

_ ___ _ ___ _

| | | | | |

_ _|_|___|_|___|_|

| | | | | | |

|_|___|_|___|_|___|,

and here are the three possible "domino" tiles:

_ _

| | | |

| | | | _____

| | | | | |

|_|, |_|, |_____|.

As an example, here is one of the a(12) = 609 ways to tile the skew double-strip of 11 cells:

_ _______ _____

| | | | | |

___|_ |_|_ | _| _|

| | | | | |

|_____|___|_|___|_|. (End)

Examples

			G.f. = x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 12*x^6 + 23*x^7 + 44*x^8 + 85*x^9 + ...
		

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

• Indranil Ghosh, Table of n, a(n) for n = 0..3503 (terms 0..500 from T. D. Noe)
• Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
• J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
• W. C. Lynch, The t-Fibonacci numbers and polyphase sorting, Fib. Quart., 8 (1970), pp. 6ff.
• 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
• Helmut Prodinger, Counting Palindromes According to r-Runs of Ones Using Generating Functions, J. Int. Seq. 17 (2014) # 14.6.2, even length, r=3.
• Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

Crossrefs

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

Cf. A000032.

Programs

Magma

I:=[0, 0, 1, 2]; [n le 4 select I[n] else Self(n-1)+ Self(n-2) + Self(n-3) + Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jan 29 2013

Maple

a:= proc(n) option operator; local M; M := Matrix(4, (i,j)-> if (i=j-1) or j=1 then 1 else 0 fi)^n; M[1,4]+M[1,3] end; seq (a(n), n=0..34); # Alois P. Heinz, Aug 01 2008

Mathematica

a=0; b=0; c=1; d=2; lst={a, b, c, d}; Do[e=a+b+c+d; AppendTo[lst, e]; a=b; b=c; c=d; d=e, {n, 4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 30 2008 *)
RecurrenceTable[{a[0] == a[1] == 0, a[2] == 1, a[3] == 2, a[n] == a[n - 1] + a[n - 2] + a[n - 3] + a[n - 4]}, a, {n, 35}] (* or *) a = {0, 0, 1, 2}; Do[AppendTo[a, a[[-1]] + a[[-2]] + a[[-3]] + a[[-4]]], {35}]; a (* Bruno Berselli, Jan 29 2013 *)
CoefficientList[Series[- x^2 * (1 + x)/(- 1 + x + x^2 + x^3 + x^4), {x, 0, 35}], x] (* Vincenzo Librandi, Jan 29 2013 *)
LinearRecurrence[{1,1,1,1},{0,0,1,2},40] (* Harvey P. Dale, Aug 25 2013 *)

PARI

concat([0, 0], Vec(-x^2*(1+x)/(-1+x+x^2+x^3+x^4) + O(x^50))) \\ Michel Marcus, Dec 30 2015

Formula

G.f.: -x^2*(1+x)/(-1+x+x^2+x^3+x^4). [Simon Plouffe in his 1992 dissertation]

a(n) = A000078(n) + A000078(n+1) = a(n-1) + A000078(n+1) - A000078(n-1). - Henry Bottomley

a(n) = 2*a(n-1) - a(n-5) with n>4, a(0)=a(1)=0, a(2)=1, a(3)=2, a(4)=3. - Vincenzo Librandi, Dec 21 2010

G.f.: x^2 + x^3*G(0) where G(k) = 2 + x*(1 + x + x^2 + (1+x)*(1+x^2)*G(k+1)). - Sergei N. Gladkovskii, Jan 27 2013 [Edited by Michael Somos, Nov 12 2013]