A047404 - OEIS

%I #31 Sep 22 2025 16:00:32

%S 1,2,3,6,9,10,11,14,17,18,19,22,25,26,27,30,33,34,35,38,41,42,43,46,

%T 49,50,51,54,57,58,59,62,65,66,67,70,73,74,75,78,81,82,83,86,89,90,91,

%U 94,97,98,99,102,105,106,107,110,113,114,115,118,121,122,123

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

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

%F a(n) = A056594(n) + 2*n-2. - _Zerinvary Lajos_, Jul 06 2008

%F G.f.: x*(1+x)*(2*x^2-x+1)/((x^2+1)*(x-1)^2). - _R. J. Mathar_, Oct 08 2011

%F From _Wesley Ivan Hurt_, May 30 2016: (Start)

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

%F a(n) = (4*n-4+i^(1-n)-i^(1+n))/2 where i = sqrt(-1).

%F a(2k) = A016825(k-1) k>0, a(2k-1) = A047471(k). (End)

%F E.g.f.: 2 + sin(x) + 2*(x - 1)*exp(x). - _Ilya Gutkovskiy_, May 30 2016

%F Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/8 + log(2)/4. - _Amiram Eldar_, Dec 23 2021

%p A047404:=n->(4*n-4+I^(1-n)-I^(1+n))/2: seq(A047404(n), n=1..100); # _Wesley Ivan Hurt_, May 30 2016

%t Table[(4n-4+I^(1-n)-I^(1+n))/2, {n, 80}] (* _Wesley Ivan Hurt_, May 30 2016 *)

%t LinearRecurrence[{2,-2,2,-1},{1,2,3,6},70] (* _Harvey P. Dale_, Sep 15 2024 *)

%o (SageMath) [lucas_number1(n,0,1)+2*n-2 for n in range(1,56)] # _Zerinvary Lajos_, Jul 06 2008

%o (PARI) a(n)=(n-1)\4*8+[6,1,2,3][n%4+1] \\ _Charles R Greathouse IV_, Jun 11 2015

%o (Magma) [n : n in [0..150] | n mod 8 in [1, 2, 3, 6]]; // _Wesley Ivan Hurt_, May 30 2016

%Y Cf. A016825, A047471, A056594.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_