login
A259967
a(n) = a(n-1) + a(n-2) + a(n-4).
5
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
Nazim Fatès, Biswanath Sethi, Sukanta Das, On the reversibility of ECAs with fully asynchronous updating: the recurrence point of view, Preprint, hal-01571847, 2017.
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
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
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jul 11 2015
STATUS
approved