A047576 Numbers that are congruent to {1, 5, 6, 7} mod 8.
1, 5, 6, 7, 9, 13, 14, 15, 17, 21, 22, 23, 25, 29, 30, 31, 33, 37, 38, 39, 41, 45, 46, 47, 49, 53, 54, 55, 57, 61, 62, 63, 65, 69, 70, 71, 73, 77, 78, 79, 81, 85, 86, 87, 89, 93, 94, 95, 97, 101, 102, 103, 105, 109, 110, 111, 113, 117, 118, 119, 121, 125
Offset: 1
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- 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, 5, 6, 7]]; // Wesley Ivan Hurt, May 29 2016
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Maple
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
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Mathematica
Flatten[#+{1,5,6,7}&/@(8Range[0,20])] (* Harvey P. Dale, Apr 22 2011 *) Select[Range[100], MemberQ[{1, 5, 6, 7}, Mod[#, 8]] &] (* Vincenzo Librandi, May 30 2016 *)
Formula
From Wesley Ivan Hurt, May 29 2016: (Start)
G.f.: x*(1+4*x+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-1+i^(2*n)-(2+i)*i^(-n)-(2-i)*i^n)/4 where i=sqrt(-1).
E.g.f.: (2 - sin(x) - 2*cos(x) - sinh(x) + 4*x*exp(x))/2. - Ilya Gutkovskiy, May 30 2016
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