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Numbers that are congruent to {1, 4, 6, 7} mod 8.
1

%I #16 Sep 08 2022 08:44:57

%S 1,4,6,7,9,12,14,15,17,20,22,23,25,28,30,31,33,36,38,39,41,44,46,47,

%T 49,52,54,55,57,60,62,63,65,68,70,71,73,76,78,79,81,84,86,87,89,92,94,

%U 95,97,100,102,103,105,108,110,111,113,116,118,119,121,124

%N Numbers that are congruent to {1, 4, 6, 7} mod 8.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).

%F From _Wesley Ivan Hurt_, May 27 2016: (Start)

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

%F a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.

%F a(n) = (1+i)*(4*n-4*n*i+i-1-i^(-n)+i^(1+n))/4 where i=sqrt(-1).

%F a(2k) = A047535(k), a(2k-1) = A047452(k). (End)

%F E.g.f.: (2 - sin(x) - cos(x) + (4*x - 1)*exp(x))/2. - _Ilya Gutkovskiy_, May 27 2016

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)+1)*Pi/16 + log(2)/8. - _Amiram Eldar_, Dec 24 2021

%p A047508:=n->(1+I)*(4*n-4*n*I+I-1-I^(-n)+I^(1+n))/4: seq(A047508(n), n=1..100); # _Wesley Ivan Hurt_, May 27 2016

%t Table[(1+I)*(4n-4*n*I+I-1-I^(-n)+I^(1+n))/4, {n, 80}] (* _Wesley Ivan Hurt_, May 27 2016 *)

%o (Magma) [n : n in [0..150] | n mod 8 in [1, 4, 6, 7]]; // _Wesley Ivan Hurt_, May 27 2016

%Y Cf. A047452, A047535.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_