Search: a077814 -id:a077814
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A078112
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Coefficients a(n) in the unique expansion sin(1) = Sum[a(n)/n!, n>=1], where a(n) satisfies 0<=a(n)<n.
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+10
2
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0, 1, 2, 0, 0, 5, 6, 0, 0, 9, 10, 0, 0, 13, 14, 0, 0, 17, 18, 0, 0, 21, 22, 0, 0, 25, 26, 0, 0, 29, 30, 0, 0, 33, 34, 0, 0, 37, 38, 0, 0, 41, 42, 0, 0, 45, 46, 0, 0, 49, 50, 0, 0, 53, 54, 0, 0, 57, 58, 0, 0, 61, 62, 0, 0, 65, 66, 0, 0, 69, 70, 0, 0, 73, 74, 0, 0, 77, 78, 0, 0, 81, 82, 0, 0, 85
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OFFSET
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1,3
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LINKS
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FORMULA
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a(n) = floor(n!*sin(1)) - n*floor((n-1)!*sin(1)). a(n)=0 if n==0 or 1 (mod 4); a(n)=n-1 if n==2 or 3 (mod 4). - Benoit Cloitre, Dec 07 2002
a(n) = 2*a(n-1)-3*a(n-2)+4*a(n-3)-3*a(n-4)+2*a(n-5)-a(n-6) for n>6.
G.f.: x^2*(1-x^2+2*x^3) / ((1-x)^2*(1+x^2)^2). (End)
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EXAMPLE
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sum(i=1,10,a(i)/i!)=0.84147073..., sin(1)=0.841470984...
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PROG
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(PARI) concat(0, Vec(x^2*(1-x^2+2*x^3)/((1-x)^2*(1+x^2)^2) + O(x^100))) \\ Colin Barker, Feb 15 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A087620
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#{0<=k<=n: k*n is divisible by 4}.
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+10
2
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1, 1, 2, 1, 5, 2, 4, 2, 9, 3, 6, 3, 13, 4, 8, 4, 17, 5, 10, 5, 21, 6, 12, 6, 25, 7, 14, 7, 29, 8, 16, 8, 33, 9, 18, 9, 37, 10, 20, 10, 41, 11, 22, 11, 45, 12, 24, 12, 49, 13, 26, 13, 53, 14, 28, 14, 57, 15, 30, 15, 61, 16, 32, 16, 65, 17, 34, 17, 69, 18, 36, 18, 73, 19, 38, 19, 77, 20
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OFFSET
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0,3
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COMMENTS
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With the similar remainder 1, 2 and 3 sequences provides a four-fold partition of A000027.
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} if (k*n mod 4 = 0, 1, 0).
a(n) = (6+4*n+i^n*(-i+n)+(-i)^n*(i+n)+2*(-1)^n*(1+n))/8 where i=sqrt(-1).
a(n) = 2*a(n-4)-a(n-8) for n>7.
G.f.: (3*x^4+x^3+2*x^2+x+1) / ((x-1)^2*(x+1)^2*(x^2+1)^2).
(End)
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MATHEMATICA
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CoefficientList[Series[(3 x^4 + x^3 + 2 x^2 + x + 1)/((x - 1)^2 (x + 1)^2 (x^2 + 1)^2), {x, 0, 80}], x] (* Vincenzo Librandi, May 03 2015 *)
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PROG
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(PARI) Vec((3*x^4+x^3+2*x^2+x+1)/((x-1)^2*(x+1)^2*(x^2+1)^2) + O(x^100)) \\ Colin Barker, May 03 2015
(Magma) I:=[1, 1, 2, 1, 5, 2, 4, 2]; [n le 8 select I[n] else 2*Self(n-4)-Self(n-8): n in [1..80]]; // Vincenzo Librandi, May 03 2015
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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A131728
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a(4n) = n, a(4n+1) = 2n+1, a(4n+2) = n+1, a(4n+3) = 0.
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+10
0
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0, 1, 1, 0, 1, 3, 2, 0, 2, 5, 3, 0, 3, 7, 4, 0, 4, 9, 5, 0, 5, 11, 6, 0, 6, 13, 7, 0, 7, 15, 8, 0, 8, 17, 9, 0, 9, 19, 10, 0, 10, 21, 11, 0, 11, 23, 12, 0, 12, 25, 13, 0, 13, 27, 14, 0, 14, 29, 15, 0, 15, 31, 16, 0, 16, 33, 17, 0, 17, 35, 18, 0, 18, 37, 19, 0, 19, 39, 20, 0, 20, 41, 21, 0, 21
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OFFSET
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0,6
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LINKS
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FORMULA
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MATHEMATICA
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LinearRecurrence[{2, -3, 4, -3, 2, -1}, {0, 1, 1, 0, 1, 3}, 90] (* Harvey P. Dale, Aug 15 2013 *)
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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