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A047507 Numbers that are congruent to {0, 4, 6, 7} mod 8.

%I A047507 #21 Dec 15 2023 19:17:11
%S A047507 0,4,6,7,8,12,14,15,16,20,22,23,24,28,30,31,32,36,38,39,40,44,46,47,
%T A047507 48,52,54,55,56,60,62,63,64,68,70,71,72,76,78,79,80,84,86,87,88,92,94,
%U A047507 95,96,100,102,103,104,108,110,111,112,116,118,119,120,124
%N A047507 Numbers that are congruent to {0, 4, 6, 7} mod 8.
%H A047507 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A047507 G.f.: x^2*(4+2*x+x^2+x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - _R. J. Mathar_, Nov 06 2015
%F A047507 From _Wesley Ivan Hurt_, May 27 2016: (Start)
%F A047507 a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
%F A047507 a(n) = (8*n-3+i^(2*n)-(1+2*i)*i^(-n)-(1-2*i)*i^n)/4 where i=sqrt(-1).
%F A047507 a(2k) = A047535(k), a(2k-1) = A047451(k). (End)
%F A047507 E.g.f.: (2 - 2*sin(x) - cos(x) + (4*x - 2)*sinh(x) + (4*x - 1)*cosh(x))/2. - _Ilya Gutkovskiy_, May 27 2016
%F A047507 Sum_{n>=2} (-1)^n/a(n) = (6-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8 - sqrt(2)*Pi/16. - _Amiram Eldar_, Dec 23 2021
%F A047507 a(n) = -A003485(-n) = a(n+4) - 8 for all n in Z. - _Michael Somos_, Dec 12 2023
%e A047507 G.f. = 4*x^2 + 6*x^3 + 7*x^4 + 8*x^5 + 12*x^6 + 14*x^7 + 15*x^8 + 16*x^9 + ... - _Michael Somos_, Dec 12 2023
%p A047507 A047507:=n->(8*n-3+I^(2*n)-(1+2*I)*I^(-n)-(1-2*I)*I^n)/4: seq(A047507(n), n=1..100); # _Wesley Ivan Hurt_, May 27 2016
%t A047507 Table[(8n-3+I^(2n)-(1+2*I)*I^(-n)-(1-2*I)*I^n)/4, {n, 80}] (* _Wesley Ivan Hurt_, May 27 2016 *)
%t A047507 a[ n_] := 2*n - Max[0, 2 - Mod[1-n, 4]]; (* _Michael Somos_, Dec 12 2023 *)
%o A047507 (Magma) [n : n in [0..150] | n mod 8 in [0, 4, 6, 7]]; // _Wesley Ivan Hurt_, May 27 2016
%o A047507 (PARI) {a(n) = 2*n - max(0, 2 - (1-n)%4)}; /* _Michael Somos_, Dec 12 2023 */
%Y A047507 Cf. A003485, A047451, A047535.
%K A047507 nonn,easy
%O A047507 1,2
%A A047507 _N. J. A. Sloane_