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 *)
LinearRecurrence[{4, -4}, {1, 2, 5, 11}, 40] (* Harvey P. Dale, Sep 27 2024 *)
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
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Jun 07 2007
STATUS
approved