A047517 Numbers that are congruent to {0, 1, 3, 4, 6, 7} mod 8.
0, 1, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 16, 17, 19, 20, 22, 23, 24, 25, 27, 28, 30, 31, 32, 33, 35, 36, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 51, 52, 54, 55, 56, 57, 59, 60, 62, 63, 64, 65, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 84, 86, 87, 88, 89
Offset: 1
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).
Programs
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Magma
[n: n in [0..110] | n mod 8 in [0,1,3,4,6,7]]; // Vincenzo Librandi, May 30 2016
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Maple
A047424:=n->(24*n-21-3*cos(n*Pi)+2*sqrt(3)*cos((1+4*n)*Pi/6)+6*sin((1-2*n)* Pi/6))/18: seq(A047424(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016
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Mathematica
LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 1, 3, 4, 6, 7, 8}, 50] (* G. C. Greubel, May 30 2016 *) Select[Range[0,200], MemberQ[{0, 1, 3, 4, 6, 7}, Mod[#, 8]] &] (* Vincenzo Librandi, May 30 2016 *) -
PARI
my(x='x+O('x^50)); concat([0], Vec(x^2*(x^5 + x^4 + 2*x^3 + x^2 + 2*x + 1)/(x^7 - x^6 - x + 1))) \\ G. C. Greubel, Oct 29 2017
Formula
From Chai Wah Wu, May 30 2016: (Start)
a(n) = a(n-1) + a(n-6) - a(n-7), for n > 7.
G.f.: x^2*(x^5 + x^4 + 2*x^3 + x^2 + 2*x + 1)/(x^7 - x^6 - x + 1). (End)
From Wesley Ivan Hurt, Jun 16 2016: (Start)
a(n) = (24*n-21-3*cos(n*Pi)+2*sqrt(3)*cos((1+4*n)*Pi/6)+6*sin((1-2*n)*Pi/6))/18.
a(6k) = 8k-1, a(6k-1) = 8k-2, a(6k-2) = 8k-4, a(6k-3) = 8k-5, a(6k-4) = 8k-7, a(6k-5) = 8k-8. (End)
Sum_{n>=2} (-1)^n/a(n) = (2-sqrt(2))*Pi/16 + (6-3*sqrt(2))*log(2)/16 + 3*sqrt(2)*log(sqrt(2)+2)/8. - Amiram Eldar, Dec 27 2021