

A008611


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


43



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
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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;
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
The sequence is the interleaving of three sequences: the positive integers (A000027), the nonnegative integers (A001477), and the positive integers, in that order.  Guenther Schrack, Nov 07 2020
a(n) is the least number of football games a team has to play to be able to get n1 points, where a win is 3 points, a draw is 1 point, and a loss is 0 points.  Sigurd Kittilsen, Dec 01 2022


REFERENCES

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


LINKS



FORMULA

a(n) = a(n3) + 1.
a(n) = (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) = 4*sqrt(3)*cos(2*Pi*n/3 + Pi/6)/9 + (n+1)/3. (End)
G.f.: (1  x + x^2)/((1 + x + x^2)*(x1)^2);
a(n) = Sum_{k=0..floor(n/2)} binomial(nk, k)*A078008(n2k)*(1)^k. (End)
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) = (3*n + 3 + 2*(w^(2*n)*(1  w) + w^n*(2 + w)))/9, where w = (1 + sqrt(3))/2, a primitive third root of unity;
a(n) = (2*n  A330396(n1))/3. (End)
E.g.f.: (3*exp(x)*(1 + x) + exp(x/2)*(6*cos(sqrt(3)*x/2)  2*sqrt(3)*sin(sqrt(3)*x/2)))/9.  Stefano Spezia, May 06 2022
Sum_{n>=2} (1)^n/a(n) = 3*log(2)  1.  Amiram Eldar, Sep 10 2023


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:


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 *)
a[ n_] := Quotient[n  1, 3] + Mod[n + 2, 3]; (* Michael Somos, Jan 23 2014 *)


PROG

(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)
(PARI) {a(n) = (n1) \ 3 + (n+2) % 3}; /* Michael Somos, Jan 23 2014 */


CROSSREFS

Cf. A000027, A001477, A008619, A047878, A049310, A049347, A053737, A053824, A058207, A058788, A072528, A078008, A194960, A257075, A330396.


KEYWORD

nonn,easy,nice


AUTHOR



STATUS

approved



