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%I #122 Sep 10 2023 01:42:53
%S 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,
%T 11,12,11,12,13,12,13,14,13,14,15,14,15,16,15,16,17,16,17,18,17,18,19,
%U 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
%N a(n) = a(n-3) + 1, with a(0)=a(2)=1, a(1)=0.
%C Molien series of 2-dimensional representation of cyclic group of order 3 over GF(2).
%C One step back, two steps forward.
%C 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.]
%C A Chebyshev transform of A078008. The g.f. is the image of (1-x)/(1-x-2x^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
%C A047878 is an essentially identical sequence. - Anton Chupin, Oct 24 2009
%C Rhyme scheme of Dante Alighieri's "Divine Comedy." - David Gaita, Feb 11 2011
%C 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
%C A008619, (n,n+1) after deletion of first term;
%C A053737, (n,n+1,n+2,n+3) beginning with n=0;
%C A053824, (n to n+4) beginning with n=0. - _Clark Kimberling_, Sep 07 2011
%C 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
%C 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
%C Number of n-element subsets of [n+1] whose sum is a multiple of 3. a(4) = 1: {1,2,4,5}. - _Alois P. Heinz_, Feb 06 2017
%C 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
%C 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
%C 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
%C a(n) is the number of multiples of 3 between n and 2n. - _Christian Barrientos_, Dec 20 2021
%C a(n) is the least number of football games a team has to play to be able to get n-1 points, where a win is 3 points, a draw is 1 point, and a loss is 0 points. - _Sigurd Kittilsen_, Dec 01 2022
%D D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 103.
%H Vincenzo Librandi, <a href="/A008611/b008611.txt">Table of n, a(n) for n = 0..10000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=447">Encyclopedia of Combinatorial Structures 447</a>.
%H G. P. Michon, <a href="http://www.numericana.com/data/polyhedra.htm">Counting Polyhedra</a>.
%H Yang Yuansheng et al., <a href="http://dx.doi.org/10.1016/j.disc.2004.08.014">The crossing number of C(n; {1,3})</a>, Discr. Math. 289 (2004), 107-118.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>.
%F a(n) = a(n-3) + 1.
%F a(n) = (n-1) - 2*floor((n-1)/3).
%F G.f.: (1 + x^2 + x^4)/(1 - x^3)^2.
%F After the initial term, has form {n, n+1, n+2} for n=0, 1, 2, ...
%F From _Paul Barry_, Mar 18 2004: (Start)
%F a(n) = Sum_{k=0..n} (-1)^floor(2*(k-2)/3);
%F a(n) = 4*sqrt(3)*cos(2*Pi*n/3 + Pi/6)/9 + (n+1)/3. (End)
%F From _Paul Barry_, Oct 15 2004: (Start)
%F G.f.: (1 - x + x^2)/((1 + x + x^2)*(x-1)^2);
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*A078008(n-2k)*(-1)^k. (End)
%F a(n) = -a(-2-n) for all n in Z.
%F Euler transform of length 6 sequence [0, 1, 2, 0, 0, -1]. - _Michael Somos_, Jan 23 2014
%F a(n) = ((n-1) mod 3) + floor((n-1)/3). - _Wesley Ivan Hurt_, May 18 2014
%F PSUM transform of A257075. - _Michael Somos_, Apr 15 2015
%F a(n) = A194960(n-3), n >= 0, with extended A194960. See the a(n) formula two lines above. - _Wolfdieter Lang_, May 06 2017
%F From _Guenther Schrack_, Nov 07 2020: (Start)
%F 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;
%F a(n) = (n + 1 + 2*A049347(n))/3;
%F a(n) = (2*n - A330396(n-1))/3. (End)
%F 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
%F Sum_{n>=2} (-1)^n/a(n) = 3*log(2) - 1. - _Amiram Eldar_, Sep 10 2023
%e 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 + ...
%p with(numtheory): for n from 1 to 70 do:it:=0:
%p 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:
%p A008611:=n->(n-1)-2*floor((n-1)/3); seq(A008611(n), n=0..50); # _Wesley Ivan Hurt_, May 18 2014
%t With[{nn=30},Riffle[Riffle[Range[nn],Range[0,nn-1]],Range[nn],3]] (* or *) RecurrenceTable[{a[0]==a[2]==1,a[1]==0,a[n]==a[n-3]+1},a,{n,90}] (* _Harvey P. Dale_, Nov 06 2011 *)
%t LinearRecurrence[{1, 0, 1, -1}, {1, 0, 1, 2}, 100] (* _Vladimir Joseph Stephan Orlovsky_, Feb 23 2012 *)
%t a[ n_] := Quotient[n - 1, 3] + Mod[n + 2, 3]; (* _Michael Somos_, Jan 23 2014 *)
%o (Magma) [(n-1)-2*Floor((n-1)/3): n in [0..90]]; // _Vincenzo Librandi_, Aug 21 2011
%o (Haskell)
%o a008611 n = n' + mod r 2 where (n', r) = divMod (n + 1) 3
%o a008611_list = f [1,0,1] where f xs = xs ++ f (map (+ 1) xs)
%o -- _Reinhard Zumkeller_, Nov 25 2013
%o (PARI) {a(n) = (n-1) \ 3 + (n+2) % 3}; /* _Michael Somos_, Jan 23 2014 */
%Y Cf. A000027, A001477, A008619, A047878, A049310, A049347, A053737, A053824, A058207, A058788, A072528, A078008, A194960, A257075, A330396.
%K nonn,easy,nice
%O 0,4
%A _N. J. A. Sloane_, Mar 15 1996
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