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. (Formerly M0795 N0301)

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

0,4

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

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, 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

H. 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).

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]

Example

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 + ...

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 *)

Prog

(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
(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

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.

Sequence in context: A019138 A154324 A338218 * A293363 A326021 A164363

Adjacent sequences: A001627 A001628 A001629 * A001631 A001632 A001633

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved