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A047541 Numbers that are congruent to {1, 2, 4, 7} mod 8.

%I A047541 #23 Sep 08 2022 08:44:57
%S A047541 1,2,4,7,9,10,12,15,17,18,20,23,25,26,28,31,33,34,36,39,41,42,44,47,
%T A047541 49,50,52,55,57,58,60,63,65,66,68,71,73,74,76,79,81,82,84,87,89,90,92,
%U A047541 95,97,98,100,103,105,106,108,111,113,114,116,119,121,122,124
%N A047541 Numbers that are congruent to {1, 2, 4, 7} mod 8.
%H A047541 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).
%F A047541 From _Wesley Ivan Hurt_, Jun 04 2016: (Start)
%F A047541 G.f.: x*(1+2*x^2+x^3)/(x-1)^2*(1+x^2).
%F A047541 a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
%F A047541 a(n) = (1+i)*(n*(4-4*i)+3*i-3+i^(-n)-i^(1+n))/4 where i=sqrt(-1).
%F A047541 a(2k) = A047524(k), a(2k-1) = A047461(k). (End)
%F A047541 E.g.f.: (2 + sin(x) + cos(x) + (4*x - 3)*exp(x))/2. - _Ilya Gutkovskiy_, Jun 04 2016
%F A047541 Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)+1)*Pi/16 - log(2)/8. - _Amiram Eldar_, Dec 24 2021
%p A047541 A047541:=n->(1+I)*(n*(4-4*I)+3*I-3+I^(-n)-I^(1+n))/4: seq(A047541(n), n=1..100); # _Wesley Ivan Hurt_, Jun 04 2016
%t A047541 Table[(1+I)*(n*(4-4*I)+3*I-3+I^(-n)-I^(1+n))/4, {n, 80}] (* _Wesley Ivan Hurt_, Jun 04 2016 *)
%t A047541 Select[Range[200],MemberQ[{1,2,4,7},Mod[#,8]]&] (* or  *) LinearRecurrence[ {2,-2,2,-1},{1,2,4,7},70] (* _Harvey P. Dale_, Jul 09 2020 *)
%o A047541 (PARI) a(n)=n\4*8+[-1,1,2,4][n%4+1] \\ _Charles R Greathouse IV_, Nov 04 2011
%o A047541 (Magma) [n : n in [0..150] | n mod 8 in [1, 2, 4, 7]]; // _Wesley Ivan Hurt_, Jun 04 2016
%Y A047541 Cf. A047461, A047524.
%K A047541 nonn,easy
%O A047541 1,2
%A A047541 _N. J. A. Sloane_