A047237 Numbers that are congruent to {0, 1, 2, 4} mod 6.
0, 1, 2, 4, 6, 7, 8, 10, 12, 13, 14, 16, 18, 19, 20, 22, 24, 25, 26, 28, 30, 31, 32, 34, 36, 37, 38, 40, 42, 43, 44, 46, 48, 49, 50, 52, 54, 55, 56, 58, 60, 61, 62, 64, 66, 67, 68, 70, 72, 73, 74, 76, 78, 79, 80, 82, 84, 85, 86, 88, 90, 91, 92, 94, 96, 97
Offset: 1
Links
- Guenther Schrack, Table of n, a(n) for n = 1..10002
- Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
Programs
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GAP
Filtered([0..100],n->n mod 6 = 0 or n mod 6 = 1 or n mod 6 = 2 or n mod 6 = 4); # Muniru A Asiru, Feb 19 2019
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Magma
[n : n in [0..110] | n mod 6 in [0, 1, 2, 4]]; // G. C. Greubel, Feb 16 2019
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Maple
A047237:=n->(6*n-8+I^(1-n)-I^(1+n))/4: seq(A047237(n), n=1..100); # Wesley Ivan Hurt, May 21 2016
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Mathematica
Table[(6n-8+I^(1-n)-I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *) LinearRecurrence[{2,-2,2,-1},{0,1,2,4},120] (* Harvey P. Dale, Jan 21 2018 *) -
PARI
my(x='x+O('x^70)); concat([0], Vec(x^2*(1+2*x^2)/((1+x^2)*(1-x)^2))) \\ G. C. Greubel, Feb 16 2019 -
Sage
a=(x^2*(1+2*x^2)/((1+x^2)*(1-x)^2)).series(x, 72).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019
Formula
Starting (1, 2, 4, 6, ...) = partial sums of (1, 1, 2, 2, 1, 1, 2, 2, ...). - Gary W. Adamson, Jun 19 2008
G.f.: x^2*(1+2*x^2) / ((1+x^2)*(1-x)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(n) = (6*n - 8 + i^(1-n) - i^(1+n))/4 where i=sqrt(-1).
From Guenther Schrack, Feb 11 2019: (Start)
a(n) = (6*n - 8 + (1 - (-1)^n)*(-1)^(n*(n-1)/2))/4.
a(n) = a(n-4) + 6, a(1)=0, a(2)=1, a(3)=2, a(4)=4, for n > 4.
a(-n) = -A047262(n+2).
a(n) = A118286(n-1)/2 for n > 1.
a(n) = A047255(n) - 1. (End)
Sum_{n>=2} (-1)^n/a(n) = sqrt(3)*Pi/36 + log(2)/3 + log(3)/4. - Amiram Eldar, Dec 16 2021
Extensions
More terms from Wesley Ivan Hurt, May 21 2016
Comments