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A111297
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First differences of A109975.
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12
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1, 2, 5, 11, 24, 52, 112, 240, 512, 1088, 2304, 4864, 10240, 21504, 45056, 94208, 196608, 409600, 851968, 1769472, 3670016, 7602176, 15728640, 32505856, 67108864, 138412032, 285212672, 587202560, 1207959552, 2483027968, 5100273664
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OFFSET
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0,2
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LINKS
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FORMULA
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Equals binomial transform of [1, 1, 2, 1, 3, 1, 4, 1, 5, ...] - Gary W. Adamson, Apr 25 2008
G.f.: (1-2*x+x^2-x^3)/(1-2*x)^2.
a(n) = Sum_{k=0..n} C(n,k)*Sum_{j=0..floor(k/2)} C(j+1,k-j).
a(n) = Sum_{k=0..n} C(n,k)*A158416(k). (End)
a(n) = Sum_{k=0..n-2} (k+5)*binomial(n-2,k) for n >= 2. - Philippe Deléham, Apr 20 2009
a(n) = 2*a(n-1) + 2^(n-3) for n > 2, a(0) = 1, a(1) = 2, a(2) = 5. - Philippe Deléham, Mar 02 2012
G.f.: Q(0), where Q(k) = 1 + (k+1)*x/(1 - x - x*(1-x)/(x + (k+1)*(1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 24 2013
a(n) = (n+8) * 2^(n-3), for n >= 2.
Sum_{n>=0} 1/a(n) = 2048*log(2) - 893149/630.
Sum_{n>=0} (-1)^n/a(n) = 523549/630 - 2048*log(3/2). (End)
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EXAMPLE
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11 = 2 * 5 + 1;
24 = 2 * 11 + 2;
52 = 2 * 24 + 4;
112 = 2 * 52 + 8;
240 = 2 * 112 + 16;
512 = 2 * 240 + 32;
1088 = 2 * 512 + 64;
2304 = 2 * 1088 + 128; ...
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MAPLE
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1, 2, seq((n+8)*2^(n-3), n = 2..30); # G. C. Greubel, Sep 27 2022
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MATHEMATICA
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CoefficientList[Series[(1-2x+x^2-x^3)/(1-2x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 27 2012 *)
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PROG
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(Magma) I:=[1, 2, 5, 11]; [n le 4 select I[n] else 4*Self(n-1)-4*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jun 27 2012
(SageMath) [(n+8)*2^(n-3) - int(n==1)/4 for n in range(40)] # G. C. Greubel, Sep 27 2022
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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STATUS
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approved
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