A004773 Numbers congruent to {0, 1, 2} mod 4: a(n) = floor(4*n/3).

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, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 62, 64, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 89, 90

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The sequence b(n) = floor((4/3)*(n+2)) appears as an upper bound in Fijavz and Wood.

Binary expansion does not end in 11.

From Guenther Schrack, May 04 2023: (Start)

The sequence is the interleaving of the sequences A008586, A016813, A016825, in that order.

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)

Links

• Daniel Starodubtsev, Table of n, a(n) for n = 0..10000
• Gasper Fijavz and David R. Wood, Graph Minors and Minimum Degree, arXiv:0812.1064 [math.CO], 2008.
• Niall Graham and Frank Harary, Edge Sums of Hypercubes, Bull. Irish Math. Soc., Vol. 21 (1988), pp. 8-12.
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A177702 (first differences), A000969 (partial sums).

Cf. A032766, this sequence, A001068, A047226, A047368, A004777.

Cf. similar sequences with formula n+i*floor(n/3) listed in A281899.

Cf. A002264, A004396, A004523, A004767, A004772, A008586, A010872, A016813, A016825, A042965, A042968.

Programs

Magma

[n: n in [0..100] | n mod 4 in [0..2]]; // Vincenzo Librandi, Dec 23 2010

Maple

seq(floor(n/3)+n,n=0..68); # Gary Detlefs, Mar 20 2010

Mathematica

f[n_] := Floor[4 n/3]; Array[f, 69, 0] (* Robert G. Wilson v, Dec 24 2010 *)
fQ[n_] := Mod[n, 4] != 3; Select[ Range[0, 90], fQ] (* Robert G. Wilson v, Dec 24 2010 *)
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 *)
CoefficientList[ Series[x (1 + x + 2 x^2)/((1 - x) (1 - x^3)), {x, 0, 68}], x] (* Robert G. Wilson v, Dec 24 2010 *)

PARI

a(n)=4*n\3 \\ Charles R Greathouse IV, Sep 27 2012

Formula

G.f.: x*(1+x+2*x^2)/((1-x)*(1-x^3)).

a(0) = 0, a(n+1) = a(n) + a(n) mod 4 + 0^(a(n) mod 4). - Reinhard Zumkeller, Mar 23 2003

a(n) = A004396(n) + A004523(n); complement of A004767. - Reinhard Zumkeller, Aug 29 2005

a(n) = floor(n/3) + n. - Gary Detlefs, Mar 20 2010

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

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

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

From Guenther Schrack, May 04 2023: (Start)

a(n) = (12*n - 3 + w^(2*n)*(w + 2) - w^n*(w - 1))/9 where w = (-1 + sqrt(-3))/2.

a(n) = 2*floor(n/3) + floor((n+1)/3) + floor((n+2)/3).

a(n) = (4*n - n mod 3)/3.

a(n) = a(n-3) + 4.

a(n) = a(n-1) + a(n-3) - a(n-4).

a(n) = 4*A002264(n) + A010872(n).

a(n) = A042968(n+1) - 1.

(End)