A087447 a(0) = a(1) = 1; for n > 1, a(n) = (n+2)*2^(n-2).

1, 1, 4, 10, 24, 56, 128, 288, 640, 1408, 3072, 6656, 14336, 30720, 65536, 139264, 294912, 622592, 1310720, 2752512, 5767168, 12058624, 25165824, 52428800, 109051904, 226492416, 469762048, 973078528, 2013265920, 4160749568

Offset: 0

Author

Paul Barry, Sep 05 2003

Keywords

• nonn
• easy

Comments

Binomial transform of A005408 (with interpolated zeros). Binomial transform is A087448. a(n+2) = 2*A045623(n+1); a(n+1) = A001792(n) + (0^n - (-2)^n)/2. The sequence 1,4,10,... given by 2^n*(n+3)/2 - 0^n/2 is the binomial transform of 1,3,3,5,5,...

Equals real part of binomial transform of [1, 2*i, 3, 4*i, 5, 6*i, ...]; i=sqrt(-1). - Gary W. Adamson, Sep 21 2008

An elephant sequence, see A175655. For the central square 24 A[5] vectors, with decimal values between 27 and 432, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A057711 (without the leading 0). - Johannes W. Meijer, Aug 15 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..3000
• Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
• Index entries for linear recurrences with constant coefficients, signature (4,-4).

Crossrefs

Essentially same as A079859.

Cf. A001792, A005408, A045623, A057711, A087448, A175655.

Programs

Mathematica

Join[{1, 1}, Table[(n + 2) 2^(n - 2), {n, 2, 30}]]  (* Harvey P. Dale, Feb 22 2011 *)

Python

def A087447(n): return n+2<1 else 1 # Chai Wah Wu, Oct 03 2024

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*(2k+1). - Paul Barry, Nov 29 2004

From Colin Barker, Mar 23 2012: (Start)

G.f.: (1-x)*(1-2*x+2*x^2)/(1-2*x)^2.

a(n) = 4*a(n-1) - 4*a(n-2) for n > 3. (End)

E.g.f.: (1 - x + exp(2*x)*(1 + x))/2. - Stefano Spezia, Jan 31 2023

Extensions

Definition corrected (by a factor of 2) by R. J. Mathar, Feb 21 2009