A047490 Numbers that are congruent to {0, 1, 2, 3, 5, 7} mod 8.
0, 1, 2, 3, 5, 7, 8, 9, 10, 11, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 39, 40, 41, 42, 43, 45, 47, 48, 49, 50, 51, 53, 55, 56, 57, 58, 59, 61, 63, 64, 65, 66, 67, 69, 71, 72, 73, 74, 75, 77, 79, 80, 81, 82, 83, 85, 87, 88
Offset: 1
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10001
- Giulio Cerbai, Pattern-avoiding modified ascent sequences, arXiv:2401.10027 [math.CO], 2024. See p. 28.
- Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1).
Programs
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Magma
[n : n in [0..100] | n mod 8 in [0, 1, 2, 3, 5, 7]]; // Wesley Ivan Hurt, Jun 16 2016
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Maple
A047490:=n->(24*n-30+6*sqrt(3)*cos((1-2*n)*Pi/6)+2*sqrt(3)*cos((1+4*n)*Pi/6))/18: seq(A047490(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016
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Mathematica
Select[Range[0, 100], MemberQ[{0, 1, 2, 3, 5, 7}, Mod[#, 8]] &] (* Wesley Ivan Hurt, Jun 16 2016 *)
Formula
G.f.: x^2*(x^4+x^3+x^2+1)/((x-1)^2*(x^2-x+1)*(x^2+x+1)). - Colin Barker, Jun 22 2012
From Wesley Ivan Hurt, Jun 16 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) for n>6.
a(n) = (24*n-30+6*sqrt(3)*cos((1-2n)*Pi/6)+2*sqrt(3)*cos((1+4n)*Pi/6))/18.
a(6k) = 8k-1, a(6k-1) = 8k-3, a(6k-2) = 8k-5, a(6k-3) = 8k-6, a(6k-4) = 8k-7, a(6k-5) = 8k-8. (End)
Sum_{n>=2} (-1)^n/a(n) = (2*sqrt(2)-3)*Pi/16 + (5-sqrt(2))*log(2)/8 + sqrt(2)*log(sqrt(2)+2)/4. - Amiram Eldar, Dec 26 2021