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

Author

N. J. A. Sloane, Jul 11 2015

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A109377, A259968, A259969.

Programs

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

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

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 *)

PARI

x='x+O('x^50); Vec((x-1)*(x-3)/(1-2*x+x^2-x^3)) \\ G. C. Greubel, May 24 2017

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