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

0,2

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).

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

Example

    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; ...

Maple

1, 2, seq((n+8)*2^(n-3), n = 2..30); # G. C. Greubel, Sep 27 2022

Mathematica

CoefficientList[Series[(1-2x+x^2-x^3)/(1-2x)^2, {x, 0, 40}], x]  (* Vincenzo Librandi, Jun 27 2012 *)

Prog

(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
(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

Crossrefs

Cf. A109975, A158416.

Sequence in context: A286945 A371797 A350326 * A077864 A052980 A190512

Adjacent sequences: A111294 A111295 A111296 * A111298 A111299 A111300

Keyword

nonn,easy

Author

Paul Curtz, Jun 07 2007

Status

approved