A259967 a(n) = a(n-1) + a(n-2) + a(n-4).

3, 2, 2, 5, 10, 17, 29, 51, 90, 158, 277, 486, 853, 1497, 2627, 4610, 8090, 14197, 24914, 43721, 76725, 134643, 236282, 414646, 727653, 1276942, 2240877, 3932465, 6900995, 12110402, 21252274, 37295141, 65448410, 114853953, 201554637, 353703731, 620706778

Offset

0,1

Comments

Also the number of maximal independent vertex sets (and minimal vertex covers) in the n-gear graph. - Eric W. Weisstein, May 25 2017

Also the number of chordless cycles in the n-antiprism graph for n >= 4. - Eric W. Weisstein, Jan 02 2018

References

R. K. Guy, Letter to N. J. A. Sloane, Feb 05 1986.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Nazim Fatès, Biswanath Sethi, Sukanta Das, On the reversibility of ECAs with fully asynchronous updating: the recurrence point of view, Preprint, hal-01571847, 2017.

R. K. Guy, Letter to N. J. A. Sloane, Feb 1986

Bojan Vučković and Miodrag Živković, Row Space Cardinalities Above 2^(n - 2) + 2^(n - 3), ResearchGate, January 2017, p. 3.

Eric Weisstein's World of Mathematics, Antiprism Graph

Eric Weisstein's World of Mathematics, Chordless Cycle

Eric Weisstein's World of Mathematics, Gear Graph

Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set

Eric Weisstein's World of Mathematics, Minimal Vertex Cover

Index entries for linear recurrences with constant coefficients, signature (2,-1,1).

Formula

G.f.: (x-1)*(x-3) / (1 -2*x +x^2 -x^3). - R. J. Mathar, Jul 15 2015

a(n) = -4*A005314(n) +3*A005314(n+1) +A005314(n-1). - R. J. Mathar, Jul 15 2015

a(n) = Sum_{i=1..3} r_i^n where r_i are the roots of x^3-2*x^2+x-1. - Robert Israel, Jul 18 2016

a(n) = A109377(n-2) for n > 1. - Georg Fischer, Oct 09 2018

Maple

f:= gfun:-rectoproc({-a(n+3)+2*a(n+2)-a(n+1)+a(n), a(0) = 3, a(1) = 2, a(2) = 2}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Jul 18 2016

Mathematica

Abs @ CoefficientList[Series[(x - 1) (x - 3)/(-1 + 2 x - x^2 + x^3), {x, 0, 36}], x] (* Michael De Vlieger, Jul 18 2016 *)
LinearRecurrence[{2, -1, 1}, {2, 2, 5}, 20] (* Eric W. Weisstein, May 25 2017 *)
Table[RootSum[-1 + # - 2 #^2 + #^3 &, #^n &], {n, 20}] (* Eric W. Weisstein, May 25 2017 *)
RootSum[-1 + # - 2 #^2 + #^3 &, #^Range[0, 20] &] (* Eric W. Weisstein, Jan 02 2018 *)

Prog

(Haskell)
a259967 n = a259967_list !! n
a259967_list = 3 : 2 : 2 : 5 : zipWith3 (((+) .) . (+))
   a259967_list (drop 2 a259967_list) (drop 3 a259967_list)
-- Reinhard Zumkeller, Jul 12 2015
(PARI) x='x+O('x^50); Vec((x-1)*(x-3)/(1-2*x+x^2-x^3)) \\ G. C. Greubel, May 24 2017
(Magma) I:=[3, 2, 2, 5]; [n le 4 select I[n] else Self(n-1)+Self(n-2)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Sep 26 2017

Crossrefs

Cf. A109377, A259968, A259969.

Sequence in context: A058608 A112196 A021035 * A007567 A093055 A285733

Adjacent sequences: A259964 A259965 A259966 * A259968 A259969 A259970

Keyword

nonn,easy

Author

N. J. A. Sloane, Jul 11 2015

Status

approved