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Numbers that are congruent to {1, 2, 3, 4, 6} mod 8.
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%I #21 Sep 08 2022 08:44:57

%S 1,2,3,4,6,9,10,11,12,14,17,18,19,20,22,25,26,27,28,30,33,34,35,36,38,

%T 41,42,43,44,46,49,50,51,52,54,57,58,59,60,62,65,66,67,68,70,73,74,75,

%U 76,78,81,82,83,84,86,89,90,91,92,94,97,98,99,100,102

%N Numbers that are congruent to {1, 2, 3, 4, 6} mod 8.

%H Vincenzo Librandi, <a href="/A047419/b047419.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).

%F G.f.: x*(1+x)*(2*x^4+x^2+1) / ( (x^4+x^3+x^2+x+1)*(x-1)^2 ). - _R. J. Mathar_, Dec 05 2011

%F From _Wesley Ivan Hurt_, Aug 08 2016: (Start)

%F a(n) = a(n-1) + a(n-5) - a(n-6) for n > 6.

%F a(n) = a(n-5) + 8 for n > 5.

%F a(n) = (40*n - 40 - 2*(n mod 5) + 3*((n+1) mod 5) + 3*((n+2) mod 5) + 3*((n+3) mod 5) - 7*((n+4) mod 5))/25.

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

%F a(n) = (2/25)*(20*n-20+5*cos(2*Pi*(n-1)/5)-2*cos(2*Pi*n/5)-2*cos(4*Pi*n/5)- 4*cos(2*Pi*(n+1)/5)-2*cos(Pi*(2*n+1)/5)+2*cos(2*Pi*(2*n+1)/5)-5*cos(Pi*(4*n+1)/5)+sin(Pi*(4*n+3)/10)+4*sin(Pi*(8*n+3)/10)-sin(Pi*(8*n+1)/10)). - _Wesley Ivan Hurt_, Oct 10 2018

%p A047419:=n->8*floor(n/5)+[(1, 2, 3, 4, 6)][(n mod 5)+1]: seq(A047419(n), n=0..100); # _Wesley Ivan Hurt_, Aug 08 2016

%t Select[Range[0, 100], MemberQ[{1, 2, 3, 4, 6}, Mod[#, 8]] &] (* _Wesley Ivan Hurt_, Aug 08 2016 *)

%t LinearRecurrence[{1, 0, 0, 0, 1, -1}, {1, 2, 3, 4, 6, 9}, 100] (* _Vincenzo Librandi_, Aug 08 2016 *)

%o (Magma) [n : n in [0..150] | n mod 8 in [1, 2, 3, 4, 6]]; // _Wesley Ivan Hurt_, Aug 08 2016

%o (GAP) Filtered([1..103],n->n mod 8 = 1 or n mod 8 = 2 or n mod 8 = 3 or n mod 8 = 4 or n mod 8 = 6); # _Muniru A Asiru_, Oct 23 2018

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_