A047493 - OEIS

%I #20 Sep 08 2022 08:44:57

%S 1,4,5,7,9,12,13,15,17,20,21,23,25,28,29,31,33,36,37,39,41,44,45,47,

%T 49,52,53,55,57,60,61,63,65,68,69,71,73,76,77,79,81,84,85,87,89,92,93,

%U 95,97,100,101,103,105,108,109,111,113,116,117,119,121,124

%N Numbers that are congruent to {1, 4, 5, 7} mod 8.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).

%F G.f.: x*(1+3*x+x^2+2*x^3+x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - _R. J. Mathar_, Nov 06 2015

%F From _Wesley Ivan Hurt_, May 26 2016: (Start)

%F a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.

%F a(n) = (8*n-3+i^(2*n)-i^(-n)-i^n)/4 where i=sqrt(-1).

%F a(2k) = A047535(k), a(2k-1) = A016813(k-1) for n>0. (End)

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

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+3)*Pi/16 + log(2)/4 + sqrt(2)*log(sqrt(2)-1)/8. - _Amiram Eldar_, Dec 24 2021

%p A047493:=n->(8*n-3+I^(2*n)-I^(-n)-I^n)/4: seq(A047493(n), n=1..100); # _Wesley Ivan Hurt_, May 26 2016

%t Table[(8n-3+I^(2n)-I^(-n)-I^n)/4, {n, 80}] (* _Wesley Ivan Hurt_, May 26 2016 *)

%t LinearRecurrence[{1,0,0,1,-1},{1,4,5,7,9},80] (* _Harvey P. Dale_, May 05 2018 *)

%o (Magma) [n : n in [0..150] | n mod 8 in [1, 4, 5, 7]]; // _Wesley Ivan Hurt_, May 26 2016

%Y Cf. A016813, A047535.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_