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%I #106 May 24 2023 10:58:34
%S 0,1,2,4,5,6,8,9,10,12,13,14,16,17,18,20,21,22,24,25,26,28,29,30,32,
%T 33,34,36,37,38,40,41,42,44,45,46,48,49,50,52,53,54,56,57,58,60,61,62,
%U 64,65,66,68,69,70,72,73,74,76,77,78,80,81,82,84,85,86,88,89,90
%N Numbers congruent to {0, 1, 2} mod 4: a(n) = floor(4*n/3).
%C The sequence b(n) = floor((4/3)*(n+2)) appears as an upper bound in Fijavz and Wood.
%C Binary expansion does not end in 11.
%C From _Guenther Schrack_, May 04 2023: (Start)
%C The sequence is the interleaving of the sequences A008586, A016813, A016825, in that order.
%C Let S(n) = a(n) + a(n+1) + a(n+2). Then floor(S(n)/3) = A042968(n+1), round(S(n)/3) = a(n+1), ceiling(S(n)/3) = A042965(n+2). (End)
%H Daniel Starodubtsev, <a href="/A004773/b004773.txt">Table of n, a(n) for n = 0..10000</a>
%H Gasper Fijavz and David R. Wood, <a href="http://arxiv.org/abs/0812.1064">Graph Minors and Minimum Degree</a>, arXiv:0812.1064 [math.CO], 2008.
%H Niall Graham and Frank Harary, <a href="http://www.maths.tcd.ie/pub/ims/bull21/bull21_8-12.pdf">Edge Sums of Hypercubes</a>, Bull. Irish Math. Soc., Vol. 21 (1988), pp. 8-12.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F G.f.: x*(1+x+2*x^2)/((1-x)*(1-x^3)).
%F a(0) = 0, a(n+1) = a(n) + a(n) mod 4 + 0^(a(n) mod 4). - _Reinhard Zumkeller_, Mar 23 2003
%F a(n) = A004396(n) + A004523(n); complement of A004767. - _Reinhard Zumkeller_, Aug 29 2005
%F a(n) = floor(n/3) + n. - _Gary Detlefs_, Mar 20 2010
%F a(n) = (12*n-3+3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9. - _Wesley Ivan Hurt_, Sep 30 2017
%F E.g.f.: (3*exp(x)*(4*x - 1) + exp(-x/2)*(3*cos((sqrt(3)*x)/2) + sqrt(3)*sin((sqrt(3)*x)/2)))/9. - _Stefano Spezia_, Jun 09 2021
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/8 + sqrt(2)*log(sqrt(2)+2)/4 + (2-sqrt(2))*log(2)/8. - _Amiram Eldar_, Dec 05 2021
%F From _Guenther Schrack_, May 04 2023: (Start)
%F a(n) = (12*n - 3 + w^(2*n)*(w + 2) - w^n*(w - 1))/9 where w = (-1 + sqrt(-3))/2.
%F a(n) = 2*floor(n/3) + floor((n+1)/3) + floor((n+2)/3).
%F a(n) = (4*n - n mod 3)/3.
%F a(n) = a(n-3) + 4.
%F a(n) = a(n-1) + a(n-3) - a(n-4).
%F a(n) = 4*A002264(n) + A010872(n).
%F a(n) = A042968(n+1) - 1.
%F (End)
%p seq(floor(n/3)+n,n=0..68); # _Gary Detlefs_, Mar 20 2010
%t f[n_] := Floor[4 n/3]; Array[f, 69, 0] (* _Robert G. Wilson v_, Dec 24 2010 *)
%t fQ[n_] := Mod[n, 4] != 3; Select[ Range[0, 90], fQ] (* _Robert G. Wilson v_, Dec 24 2010 *)
%t a[0] = 0; a[n_] := a[n] = a[n - 1] + 2 - If[ Mod[a[n - 1], 4] < 2, 1, 0]; Array[a, 69, 0] (* _Robert G. Wilson v_, Dec 24 2010 *)
%t CoefficientList[ Series[x (1 + x + 2 x^2)/((1 - x) (1 - x^3)), {x, 0, 68}], x] (* _Robert G. Wilson v_, Dec 24 2010 *)
%o (Magma) [n: n in [0..100] | n mod 4 in [0..2]]; // _Vincenzo Librandi_, Dec 23 2010
%o (PARI) a(n)=4*n\3 \\ _Charles R Greathouse IV_, Sep 27 2012
%Y Cf. A177702 (first differences), A000969 (partial sums).
%Y Cf. A032766, this sequence, A001068, A047226, A047368, A004777.
%Y Cf. similar sequences with formula n+i*floor(n/3) listed in A281899.
%Y Cf. A002264, A004396, A004523, A004767, A004772, A008586, A010872, A016813, A016825, A042965, A042968.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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