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Numbers that are congruent to {0, 1, 3} (mod 8).
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%I #22 Aug 31 2024 17:02:45

%S 0,1,3,8,9,11,16,17,19,24,25,27,32,33,35,40,41,43,48,49,51,56,57,59,

%T 64,65,67,72,73,75,80,81,83,88,89,91,96,97,99,104,105,107,112,113,115,

%U 120,121,123,128,129,131,136,137,139,144,145,147,152,153,155

%N Numbers that are congruent to {0, 1, 3} (mod 8).

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).

%F Equals partial sums of (0, 1, 2, 5, 1, 2, 5, 1, 2, 5, ...). - _Gary W. Adamson_, Jun 19 2008

%F From _Colin Barker_, Jan 26 2012: (Start)

%F G.f.: x^2*(1+2*x+5*x^2)/(1-x-x^3+x^4).

%F a(n) = a(n-1) + a(n-3) - a(n-4) for n>4. (End)

%F From _Wesley Ivan Hurt_, Jun 09 2016: (Start)

%F a(n) = 8*n/3 - 4 - cos(2*n*Pi/3) + 5*sin(2*n*Pi/3)/(3*sqrt(3)).

%F a(3k) = 8k-5, a(3k-1) = 8k-7, a(3k-2) = 8k-8. (End)

%p A047472:=n->8*n/3-4-cos(2*n*Pi/3)+5*sin(2*n*Pi/3)/(3*sqrt(3)): seq(A047472(n), n=1..100); # _Wesley Ivan Hurt_, Jun 09 2016

%t Select[Range[0, 150], MemberQ[{0, 1, 3}, Mod[#, 8]] &] (* _Wesley Ivan Hurt_, Jun 09 2016 *)

%t LinearRecurrence[{1,0,1,-1},{0,1,3,8},60] (* _Harvey P. Dale_, Aug 31 2024 *)

%o (Magma) [n : n in [0..150] | n mod 8 in [0, 1, 3]]; // _Wesley Ivan Hurt_, Jun 09 2016

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_