cp's OEIS Frontend

A047431 Numbers that are congruent to {1, 4, 5, 6} mod 8.

%I A047431 #32 Sep 08 2022 08:44:57
%S A047431 1,4,5,6,9,12,13,14,17,20,21,22,25,28,29,30,33,36,37,38,41,44,45,46,
%T A047431 49,52,53,54,57,60,61,62,65,68,69,70,73,76,77,78,81,84,85,86,89,92,93,
%U A047431 94,97,100,101,102,105,108,109,110,113,116,117,118,121,124
%N A047431 Numbers that are congruent to {1, 4, 5, 6} mod 8.
%H A047431 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).
%F A047431 G.f.: x*(1+2*x-x^2+2*x^3)/((x^2+1)*(x-1)^2). - _R. J. Mathar_, Oct 08 2011
%F A047431 a(n) = (-2-(-i)^n-i^n+4n)/2 where i=sqrt(-1). - _Colin Barker_, Jun 06 2012
%F A047431 From _Wesley Ivan Hurt_, May 30 2016: (Start)
%F A047431 a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
%F A047431 a(2k) = A047406(k), a(2k-1) = A016813(k-1) k>0. (End)
%F A047431 E.g.f.: 2 - cos(x) - (1 - 2*x)*exp(x). - _Ilya Gutkovskiy_, May 30 2016
%F A047431 Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi/16 + 3*log(2)/8. - _Amiram Eldar_, Dec 24 2021
%p A047431 A047431:=n->(4*n-2-(-I)^n-I^n)/2: seq(A047431(n), n=1..100); # _Wesley Ivan Hurt_, May 30 2016
%t A047431 Table[(4n-2-(-I)^n-I^n)/2, {n, 80}] (* _Wesley Ivan Hurt_, May 30 2016 *)
%t A047431 LinearRecurrence[{2,-2,2,-1},{1,4,5,6},70] (* _Harvey P. Dale_, Dec 04 2018 *)
%o A047431 (Sage) [lucas_number1(n,0,1)+2*n+1 for n in range(0,56)] # _Zerinvary Lajos_, Jul 06 2008
%o A047431 (Magma) [n : n in [0..150] | n mod 8 in [1, 4, 5, 6]]; // _Wesley Ivan Hurt_, May 30 2016
%Y A047431 Cf. A016813, A047404, A047406, A047546, A056594.
%K A047431 nonn,easy
%O A047431 1,2
%A A047431 _N. J. A. Sloane_