A047558 Numbers that are congruent to {1, 3, 6, 7} mod 8.
1, 3, 6, 7, 9, 11, 14, 15, 17, 19, 22, 23, 25, 27, 30, 31, 33, 35, 38, 39, 41, 43, 46, 47, 49, 51, 54, 55, 57, 59, 62, 63, 65, 67, 70, 71, 73, 75, 78, 79, 81, 83, 86, 87, 89, 91, 94, 95, 97, 99, 102, 103, 105, 107, 110, 111, 113, 115, 118, 119, 121, 123, 126
Offset: 1
Links
- Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Programs
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Magma
[n : n in [0..150] | n mod 8 in [1, 3, 6, 7]]; // Wesley Ivan Hurt, May 29 2016
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Maple
A047558:=n->(8*n-3-I^(2*n)-I^(1-n)+I^(1+n))/4: seq(A047558(n), n=1..100); # Wesley Ivan Hurt, May 29 2016
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Mathematica
Select[Range[150], MemberQ[{1,3,6,7}, Mod[#,8]]&] (* Harvey P. Dale, Jul 31 2014 *)
Formula
From Wesley Ivan Hurt, May 29 2016: (Start)
G.f.: x*(1+2*x+3*x^2+x^3+x^4) / ((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-3-i^(2*n)-i^(1-n)+i^(1+n))/4 where i=sqrt(-1).
E.g.f.: (2 - sin(x) + (4*x - 1)*sinh(x) + (4*x - 2)*cosh(x))/2. - Ilya Gutkovskiy, May 30 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (2+sqrt(2))*Pi/16 + sqrt(2)*log(2+sqrt(2))/8 - (2+sqrt(2))*log(2)/16. - Amiram Eldar, Dec 24 2021