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A047514 Numbers that are congruent to {3, 4, 6, 7} mod 8.

%I A047514 #22 Sep 08 2022 08:44:57
%S A047514 3,4,6,7,11,12,14,15,19,20,22,23,27,28,30,31,35,36,38,39,43,44,46,47,
%T A047514 51,52,54,55,59,60,62,63,67,68,70,71,75,76,78,79,83,84,86,87,91,92,94,
%U A047514 95,99,100,102,103,107,108,110,111,115,116,118,119,123,124
%N A047514 Numbers that are congruent to {3, 4, 6, 7} mod 8.
%H A047514 Vincenzo Librandi, <a href="/A047514/b047514.txt">Table of n, a(n) for n = 1..1000</a>
%H A047514 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A047514 From _Wesley Ivan Hurt_, May 27 2016: (Start)
%F A047514 G.f.: x*(3+x+2*x^2+x^3+x^4) / ((x-1)^2*(1+x+x^2+x^3)).
%F A047514 a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
%F A047514 a(n) = (1+i)*(4*n-4*n*i+(i-1)*i^(2*n)+i^(1-n)-i^n)/4 where i=sqrt(-1).
%F A047514 a(2k) = A047535(k), a(2k-1) = A047398(k). (End)
%F A047514 E.g.f.: (2 + sin(x) - cos(x) + (4*x + 1)*sinh(x) + (4*x - 1)*cosh(x))/2. - _Ilya Gutkovskiy_, May 27 2016
%F A047514 From _Wesley Ivan Hurt_, Aug 10 2016: (Start)
%F A047514 a(n) = a(n-4) + 8 for n > 4.
%F A047514 a(4*k) = 8*k-1, a(4*k-1) = 8*k-2, a(4*k-2) = 8*k-4, a(4*k-3) = 8*k-5. (End)
%F A047514 Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/16 - (sqrt(2)-1)*log(2)/8 + sqrt(2)*log(2-sqrt(2))/4. - _Amiram Eldar_, Dec 26 2021
%p A047514 A047514:=n->(1+I)*(4*n-4*n*I+(I-1)*I^(2*n)+I^(1-n)-I^n)/4: seq(A047514(n), n=1..100); # _Wesley Ivan Hurt_, May 27 2016
%t A047514 Table[(1+I)*(4n-4n*I+(I-1)*I^(2n)+I^(1-n)-I^n)/4, {n, 80}] (* _Wesley Ivan Hurt_, May 27 2016 *)
%t A047514 LinearRecurrence[{1, 0, 0, 1, -1}, {3, 4, 6, 7, 11}, 100] (* _Vincenzo Librandi_, Aug 11 2016 *)
%o A047514 (Magma) [n : n in [0..150] | n mod 8 in [3, 4, 6, 7]]; // _Wesley Ivan Hurt_, May 27 2016
%Y A047514 Cf. A047398, A047535.
%K A047514 nonn,easy
%O A047514 1,1
%A A047514 _N. J. A. Sloane_