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Numbers that are congruent to {1, 2, 4, 6} mod 8.
1

%I #33 Sep 08 2022 08:44:57

%S 1,2,4,6,9,10,12,14,17,18,20,22,25,26,28,30,33,34,36,38,41,42,44,46,

%T 49,50,52,54,57,58,60,62,65,66,68,70,73,74,76,78,81,82,84,86,89,90,92,

%U 94,97,98,100,102,105,106,108,110,113,114,116,118,121,122,124,126,129,130

%N Numbers that are congruent to {1, 2, 4, 6} mod 8.

%H G. C. Greubel, <a href="/A047411/b047411.txt">Table of n, a(n) for n = 1..10000</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 _R. J. Mathar_, Mar 10 2008: (Start)

%F a(n) = a(n-4)+8.

%F O.g.f.: 2/(-1+x)^2+1/(2(x^2+1))+7/(4(-1+x))+1/(4(x+1)). (End)

%F a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5. - _R. J. Mathar_, Feb 11 2010

%F From _Wesley Ivan Hurt_, May 22 2016:

%F a(n) = (8n-7-i^(2n)+i^(1-n)-i^(1+n))/4 where i=sqrt(-1).

%F a(2n+2) = A016825(n) n>0, a(2n-1) = A047461(n). (End)

%F E.g.f.: (4 + sin(x) + (4*x - 3)*sinh(x) + 4*(x - 1)*cosh(x))/2. - _Ilya Gutkovskiy_, May 23 2016

%F a(n) = (8*n-7-cos(n*Pi)+2*sin(n*Pi/2))/4. - _Wesley Ivan Hurt_, Oct 05 2017

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 - sqrt(2)*log(3-2*sqrt(2))/16. - _Amiram Eldar_, Dec 23 2021

%p A047411 := proc(n) if n <= 4 then op(n,[1,2,4,6]); else procname(n-4)+8; end if; end proc: seq(A047411(n), n=1..99); # _R. J. Mathar_, Feb 11 2010

%t Table[(8n-7-I^(2n)+I^(1-n)-I^(1+n))/4, {n, 80}] (* _Wesley Ivan Hurt_, May 22 2016 *)

%t LinearRecurrence[{1,0,0,1,-1}, {1,2,4,6,9}, 50] (* _G. C. Greubel_, May 23 2016 *)

%o (Magma) [n : n in [0..150] | n mod 8 in [1, 2, 4, 6]]; // _Wesley Ivan Hurt_, May 22 2016

%Y Cf. A016825, A047461.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_

%E Extended by _R. J. Mathar_, Feb 11 2010