A047513 Numbers that are congruent to {0, 1, 2, 4, 6, 7} mod 8.

0, 1, 2, 4, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 20, 22, 23, 24, 25, 26, 28, 30, 31, 32, 33, 34, 36, 38, 39, 40, 41, 42, 44, 46, 47, 48, 49, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 70, 71, 72, 73, 74, 76, 78, 79, 80, 81, 82, 84, 86, 87, 88

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The product of any two terms belongs to the sequence and therefore also a(n)^2, a(n)^3, a(n)^4, etc. - Bruno Berselli, Nov 28 2012

Nonnegative m such that floor(k*(m/4)^2) = k*floor((m/4)^2), where k = 2 or 3. - Bruno Berselli, Dec 03 2015

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1).

Crossrefs

Cf. A047498, A047581.

Programs

Magma

[n : n in [0..100] | n mod 8 in [0, 1, 2, 4, 6, 7]]; // Wesley Ivan Hurt, May 29 2016

Maple

A047513:=n->(24*n-24-2*sqrt(3)*(cos((1-4*n)*Pi/6)-3*cos((2*n+1)*Pi/6)))/18: seq(A047513(n), n=1..100); # Wesley Ivan Hurt, Jun 15 2016

Mathematica

Select[Range[0, 75], Function[k, Mod[#, 8] == k] /@ Nor[3, 5] &] (* or *) Select[Range[0, 75], Function[k, Floor[k (#/4)^2] == k Floor[(#/4)^2]] /@ Or[2, 3] &] (* Michael De Vlieger, Dec 03 2015 *)
Select[Range[0,100], MemberQ[{0,1,2,4,6,7}, Mod[#,8]]&] (* Harvey P. Dale, Apr 26 2016 *)
LinearRecurrence[{2, -2, 2, -2, 2, -1}, {0, 1, 2, 4, 6, 7}, 50] (* G. C. Greubel, May 29 2016 *)

Formula

From Chai Wah Wu, May 29 2016: (Start)

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6), n>6.

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

From Wesley Ivan Hurt, Jun 15 2016: (Start)

a(n) = (24*n-24-2*sqrt(3)*(cos((1-4*n)*Pi/6)-3*cos((2*n+1)*Pi/6)))/18.

a(6k) = 8k-1, a(6k-1) = 8k-2, a(6k-2) = 8k-4, a(6k-3) = 8k-6, a(6k-4) = 8k-7, a(6k-5) = 8k-8. (End)

Sum_{n>=2} (-1)^n/a(n) = (4-sqrt(2))*log(2)/8 + sqrt(2)*log(sqrt(2)+2)/4. - Amiram Eldar, Dec 27 2021