A111297 First differences of A109975.
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
Offset: 0
Examples
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; ...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
- Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
- Index entries for linear recurrences with constant coefficients, signature (4,-4).
Programs
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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
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Maple
1,2, seq((n+8)*2^(n-3), n = 2..30); # G. C. Greubel, Sep 27 2022
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Mathematica
CoefficientList[Series[(1-2x+x^2-x^3)/(1-2x)^2, {x,0,40}], x] (* Vincenzo Librandi, Jun 27 2012 *) LinearRecurrence[{4,-4},{1,2,5,11},40] (* Harvey P. Dale, Sep 27 2024 *) -
PARI
a=[1,2,5,11]; for(i=1,99,a=concat(a,4*a[#a]-4*a[#a-1])); a \\ Charles R Greathouse IV, Jun 01 2011
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SageMath
[(n+8)*2^(n-3) - int(n==1)/4 for n in range(40)] # G. C. Greubel, Sep 27 2022
Formula
Equals binomial transform of [1, 1, 2, 1, 3, 1, 4, 1, 5, ...] - Gary W. Adamson, Apr 25 2008
From Paul Barry, Mar 18 2009: (Start)
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
From Amiram Eldar, Jan 13 2021: (Start)
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)
E.g.f.: (1/4)*((4+x)*exp(2*x) - x). - G. C. Greubel, Sep 27 2022
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