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Numbers that are congruent to {1, 5, 6, 7} mod 8.
1

%I #21 Sep 08 2022 08:44:57

%S 1,5,6,7,9,13,14,15,17,21,22,23,25,29,30,31,33,37,38,39,41,45,46,47,

%T 49,53,54,55,57,61,62,63,65,69,70,71,73,77,78,79,81,85,86,87,89,93,94,

%U 95,97,101,102,103,105,109,110,111,113,117,118,119,121,125

%N Numbers that are congruent to {1, 5, 6, 7} mod 8.

%H Vincenzo Librandi, <a href="/A047576/b047576.txt">Table of n, a(n) for n = 1..1000</a>

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

%F a(2k) = A047550(k), a(2k-1) = A047452(k). (End)

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

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

%p A047576:=n->(8*n-1+I^(2*n)-(2+I)*I^(-n)-(2-I)*I^n)/4: seq(A047576(n), n=1..100); # _Wesley Ivan Hurt_, May 29 2016

%t Flatten[#+{1,5,6,7}&/@(8Range[0,20])] (* _Harvey P. Dale_, Apr 22 2011 *)

%t Select[Range[100], MemberQ[{1, 5, 6, 7}, Mod[#, 8]] &] (* _Vincenzo Librandi_, May 30 2016 *)

%o (Magma) [n : n in [0..150] | n mod 8 in [1, 5, 6, 7]]; // _Wesley Ivan Hurt_, May 29 2016

%Y Cf. A047452, A047550.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_