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A047569
Numbers that are congruent to {0, 1, 4, 5, 6, 7} mod 8.
2
0, 1, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 36, 37, 38, 39, 40, 41, 44, 45, 46, 47, 48, 49, 52, 53, 54, 55, 56, 57, 60, 61, 62, 63, 64, 65, 68, 69, 70, 71, 72, 73, 76, 77, 78, 79, 80, 81, 84, 85, 86, 87, 88
OFFSET
1,3
FORMULA
G.f.: x^2*(1+3*x+x^2+x^3+x^4+x^5) / ((1-x)^2*(1+x)*(1-x+x^2)*(1+x+x^2)). - Colin Barker, Jan 09 2016
From Wesley Ivan Hurt, Jun 16 2016: (Start)
a(n) = a(n-1) + a(n-6) - a(n-7) for n>7.
a(n) = (24*n-15-3*cos(n*Pi)+4*sqrt(3)*cos((1-4*n)*Pi/6)-12*sin((1+2*n)*Pi/6))/18.
a(6k) = 8k-1, a(6k-1) = 8k-2, a(6k-2) = 8k-3, a(6k-3) = 8k-4, a(6k-4) = 8k-7, a(6k-5) = 8k-8. (End)
Sum_{n>=2} (-1)^n/a(n) = (2-sqrt(2))*Pi/16 + (14-sqrt(2))*log(2)/16 + sqrt(2)*log(sqrt(2)+2)/8. - Amiram Eldar, Dec 27 2021
MAPLE
A047569:=n->(24*n-15-3*cos(n*Pi)+4*sqrt(3)*cos((1-4*n)*Pi/6)-12*sin((1+2*n)*Pi/6))/18: seq(A047569(n), n=1..100); # Wesley Ivan Hurt, Jun 16 2016
MATHEMATICA
Select[Range[0, 100], MemberQ[{0, 1, 4, 5, 6, 7}, Mod[#, 8]] &] (* Wesley Ivan Hurt, Jun 16 2016 *)
LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 1, 4, 5, 6, 7, 8}, 80] (* Harvey P. Dale, Feb 15 2024 *)
PROG
(PARI) concat(0, Vec(x^2*(1+3*x+x^2+x^3+x^4+x^5)/((1-x)^2*(1+x)*(1-x+x^2)*(1+x+x^2)) + O(x^100))) \\ Colin Barker, Jan 09 2016
CROSSREFS
Sequence in context: A016070 A299536 A321025 * A039062 A067187 A321353
KEYWORD
nonn,easy
STATUS
approved