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Numbers that are congruent to {1, 2, 6, 7} mod 8.
2

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

%S 1,2,6,7,9,10,14,15,17,18,22,23,25,26,30,31,33,34,38,39,41,42,46,47,

%T 49,50,54,55,57,58,62,63,65,66,70,71,73,74,78,79,81,82,86,87,89,90,94,

%U 95,97,98,102,103,105,106,110,111,113,114,118,119,121,122,126

%N Numbers that are congruent to {1, 2, 6, 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 From _Wesley Ivan Hurt_, May 29 2016: (Start)

%F G.f.: x*(1+x+4*x^2+x^3+x^4) / ((x-1)^2*(1+x+x^2+x^3)).

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

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

%F a(2k) = A047524(k), a(2k-1) = A047452(k). (End)

%F E.g.f.: (2 - sin(x) + cos(x) + (4*x - 1)*sinh(x) + (4*x - 3)*cosh(x))/2. - _Ilya Gutkovskiy_, May 30 2016

%F Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi/8 (A193887). - _Amiram Eldar_, Dec 24 2021

%p A047554:=n->2*n+(1+I)*(2*I-2-(1-I)*I^(2*n)-I^(1-n)+I^n)/4: seq(A047554(n), n=1..100); # _Wesley Ivan Hurt_, May 29 2016

%t Select[Range[120],MemberQ[{1,2,6,7},Mod[#,8]]&] (* _Harvey P. Dale_, Nov 29 2011 *)

%o (Magma) [n : n in [0..150] | n mod 8 in [1, 2, 6, 7]]; // _Wesley Ivan Hurt_, May 29 2016

%Y Cf. A047452, A047524, A193887.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_