

A008611


a(n) = a(n3) + 1, with a(0)=a(2)=1, a(1)=0.


39



1, 0, 1, 2, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9, 8, 9, 10, 9, 10, 11, 10, 11, 12, 11, 12, 13, 12, 13, 14, 13, 14, 15, 14, 15, 16, 15, 16, 17, 16, 17, 18, 17, 18, 19, 18, 19, 20, 19, 20, 21, 20, 21, 22, 21, 22, 23, 22, 23, 24, 23, 24, 25, 24, 25, 26, 25, 26, 27, 26, 27, 28
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,4


COMMENTS

Molien series of 2dimensional representation of cyclic group of order 3 over GF(2).
One step back, two steps forward.
The crossing number of the graph C(n, {1,3}), n >= 8, is [n/3] + n mod 3, which gives this sequence starting at the first 4. [Yang Yuansheng et al.]
A Chebyshev transform of A078008. The g.f. is the image of (1x)/(1x2x^2) (g.f. of A078008) under the Chebyshev transform A(x)> 1/(1+x^2))A(x/(1+x^2)).  Paul Barry, Oct 15 2004
A047878 is an essentially identical sequence.  Anton Chupin, Oct 24 2009
Rhyme scheme of Dante Alighieri's "Divine Comedy."  David Gaita, Feb 11 2011
A194960 results from deleting the first four terms of A008611. Note that deleting the first term or first four terms of A008611 leaves a concatenation of segments (n, n+1, n+2); for related concatenations, see
A008619, (n,n+1) after deletion of first term;
A053737, (n,n+1,n+2,n+3) beginning with n=0;
A053824, (n to n+4) beginning with n=0.  Clark Kimberling, Sep 07 2011
It appears that a(n) is the number of roots of x^(n+1) + x + 1 inside the unit circle.  Michel Lagneau, Nov 02 2012
Also apparently for n>=2: a(n) is the largest remainder r that results from dividing n+2 by 1...n+2 more than once, i.e., a(n) = max(i, A072528(n+2,i)>1).  Ralf Stephan, Oct 21 2013
Number of nelement subsets of [n+1] whose sum is a multiple of 3. a(4) = 1: {1,2,4,5}.  Alois P. Heinz, Feb 06 2017
It appears that a(n) is the number of roots of the Fibonacci polynomial F(n+2,x) strictly inside the unit circle of the complex plane.  Michel Lagneau, Apr 07 2017
For the proof of the preceding conjecture see my comments under A008615 and A049310. Chebyshev S(n,x) = i^n*F(n+1,i*x), with i = sqrt(1).  Wolfdieter Lang, May 06 2017


REFERENCES

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 103.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 447
G. P. Michon, Counting Polyhedra
Yang Yuansheng et al., The crossing number of C(n; {1,3}), Discr. Math. 289 (2004), 107118.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).


FORMULA

a(n) = a(n3) + 1 = (n1)  2*floor[(n1)/3]. G.f.: (1 + x^2 + x^4)/(1  x^3)^2.
After the initial term, has form {n, n+1, n+2} for n=0, 1, 2, ...
a(n) = Sum_{k=0..n} (1)^floor(2(k2)/3); a(n) = 4sqrt(3)cos(2*Pi*n/3 + Pi/6)/9 + (n+1)/3.  Paul Barry, Mar 18 2004
G.f.: (1x+x^2)/( (1+x+x^2)*(x1)^2); a(n) = sum{k=0..floor(n/2), binomial(nk, k)*A078008(n2k)*(1)^k}.  Paul Barry, Oct 15 2004
a(n) = a(2n) for all n in Z.
Euler transform of length 6 sequence [ 0, 1, 2, 0, 0, 1].  Michael Somos, Jan 23 2014
a(n) = ((n1) mod 3) + floor((n1)/3).  Wesley Ivan Hurt, May 18 2014
PSUM transform of A257075.  Michael Somos, Apr 15 2015
a(n) = A194960(n3), n >= 0, with extended A194960. See the a(n) formula two lines above.  Wolfdieter Lang, May 06 2017


EXAMPLE

G.f. = 1 + x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + 2*x^7 + 3*x^8 + 4*x^9 + ...


MAPLE

with(numtheory): for n from 1 to 70 do:it:=0:
y:=[fsolve(x^n+x+1, x, complex)] : for m from 1 to nops(y) do : if abs(y[m])< 1 then it:=it+1:else fi:od: printf(`%d, `, it):od:
A008611:=n>(n1)2*floor((n1)/3); seq(A008611(n), n=0..50); # Wesley Ivan Hurt, May 18 2014


MATHEMATICA

With[{nn=30}, Riffle[Riffle[Range[nn], Range[0, nn1]], Range[nn], 3]] (* or *) RecurrenceTable[{a[0]==a[2]==1, a[1]==0, a[n]==a[n3]+1}, a, {n, 90}] (* Harvey P. Dale, Nov 06 2011 *)
LinearRecurrence[{1, 0, 1, 1}, {1, 0, 1, 2}, 100] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
a[ n_] := Quotient[n  1, 3] + Mod[n + 2, 3]; (* Michael Somos, Jan 23 2014 *)


PROG

(MAGMA) [(n1)2*Floor((n1)/3): n in [0..90]]; // Vincenzo Librandi, Aug 21 2011
(Haskell)
a008611 n = n' + mod r 2 where (n', r) = divMod (n + 1) 3
a008611_list = f [1, 0, 1] where f xs = xs ++ f (map (+ 1) xs)
 Reinhard Zumkeller, Nov 25 2013
(PARI) {a(n) = (n1) \ 3 + (n+2) % 3}; /* Michael Somos, Jan 23 2014 */


CROSSREFS

Cf. A058207, A058788, A194960, A257075.
Sequence in context: A246017 A116939 A253174 * A025798 A161064 A070086
Adjacent sequences: A008608 A008609 A008610 * A008612 A008613 A008614


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane, Mar 15 1996


STATUS

approved



