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A001630
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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)
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49
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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
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OFFSET
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0,4
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COMMENTS
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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
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:
___ ___ ___ ___ ___
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___ _|___|___|___|___|___|
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|___|___|___|___|___|___|,
and here are the three possible "domino" tiles:
___ ___
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_| _| |_ |_ _______
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|___|, |___|, |_______|.
As an example, here is one of the a(12) = 609 ways to tile the skew double-strip of 11 cells:
___ _______ _______
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_____|_ |___|_ _|_ _| _|
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|_______|___|___|___|___|. (End)
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REFERENCES
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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).
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LINKS
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FORMULA
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G.f.: -x^2*(1+x)/(-1+x+x^2+x^3+x^4). [Simon Plouffe in his 1992 dissertation]
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
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EXAMPLE
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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 + ...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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