A004772 Numbers that are not congruent to 1 (mod 4).

0, 2, 3, 4, 6, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 26, 27, 28, 30, 31, 32, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 74, 75, 76, 78, 79, 80, 82, 83, 84, 86, 87, 88, 90

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Numbers whose binary expansion does not end in 01.

Equals partial sums of 0 together with 2, 1, 1, 2, 1, 1, ... (repeated, that is A131534 without the first term). - Bruno Berselli, Dec 06 2016

It seems that the numbers m = 3*(a(n)+1)= 1, 7, 10, 13, 19, 22, 25, 31, 34 ... are exactly the indices of Fibonacci numbers F_m that start 5 consecutive pairwise coprime F_m, F_{m+1}, ..., F_{m+4}. E.g. for m=13 the numbers {233,377,610,987,1597} are coprime. - R. J. Mathar, Mar 04 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A016813 (complement), A042965, A131534.

Programs

Magma

[n: n in [0..100] | not n  mod 4 eq 1 ]; // Vincenzo Librandi, Mar 09 2014

Magma

[(4*n-2) div 3: n in [1..100]]; // Bruno Berselli, Dec 06 2016

Maple

seq(seq(4*i+j,j=[0,2,3]),i=0..100); # Robert Israel, Sep 01 2015

Mathematica

LinearRecurrence[{1,0,1,-1},{0,2,3,4},68] (* Ant King, Oct 19 2012 *)
DeleteCases[Range[0,90],?(Mod[#,4]==1&)] (* _Harvey P. Dale, Jun 11 2013 *)
CoefficientList[Series[x (2 + x + x^2)/((1 + x + x^2) (x - 1)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 08 2014 *)

PARI

a(n) = (4*n-2)\3; \\ Michel Marcus, Sep 03 2015

Formula

G.f.: x^2*(2 + x + x^2)/((1 + x + x^2)*(x - 1)^2). - R. J. Mathar, Oct 08 2011

a(n) = floor((4*n-2)/3). - Gary Detlefs, Jan 02 2012

a(n) = n + ceiling((n-1)/3) - 1. - Arkadiusz Wesolowski, Sep 18 2012

From Ant King, Oct 19 2012: (Start)

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

a(n) = (12*n -9 - 3*cos(2*(n-1)*Pi/3) + sqrt(3)*sin(2*(n-1)*Pi/3))/9. (End)

a(n) = ceiling(4*(n-1)/3). - Jean-François Alcover, Mar 07 2014

Sum_{n>=2} (-1)^n/a(n) = log(sqrt(2)+2)/(2*sqrt(2)) + (2-sqrt(2))*log(2)/8 - (sqrt(2)-1)*Pi/8. - Amiram Eldar, Dec 05 2021

a(n) = A042965(n+1)-1. - R. J. Mathar, Mar 04 2025

Extensions

Corrected by Michael Somos, Jun 08 2000