A187272 a(n) = (n/4)*2^(n/2)*((1+sqrt(2))^2 + (-1)^n*(1-sqrt(2))^2).
0, 2, 6, 12, 24, 40, 72, 112, 192, 288, 480, 704, 1152, 1664, 2688, 3840, 6144, 8704, 13824, 19456, 30720, 43008, 67584, 94208, 147456, 204800, 319488, 442368, 688128, 950272, 1474560, 2031616, 3145728, 4325376, 6684672, 9175040, 14155776, 19398656, 29884416, 40894464, 62914560, 85983232
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- R. Kemp, On the number of words in the language {w in Sigma* | w = w^R }^2, Discrete Math., 40 (1982), 225-234. See Lemma 6 (p. 228).
- Index entries for linear recurrences with constant coefficients, signature (0,4,0,-4).
Programs
-
Magma
[Round((n/4)*2^(n/2)*((1+Sqrt(2))^2 + (-1)^n*(1-Sqrt(2))^2)): n in [0..30]]; // G. C. Greubel, Nov 28 2017
-
Maple
R:=(b,n)-> expand(simplify( (n/4)*b^(n/2)*((1+sqrt(b))^2+(-1)^n*(1-sqrt(b))^2) )); [seq(R(2,n),n=0..100)];
-
Mathematica
CoefficientList[Series[2 x (1 + x) (1 + 2 x) / (1 - 2 x^2)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 19 2013 *) -
PARI
x='x+O('x^30); concat([0], Vec(2*x*(1+x)*(1+2*x)/(1-2*x^2)^2)) \\ G. C. Greubel, Nov 28 2017 -
Python
def A187272(n): return (n<<(n+1>>1) if n&1 else 3*n<<(n-2>>1)) if n else 0 # Chai Wah Wu, Feb 18 2024
Formula
From Bruno Berselli, Mar 22 2011: (Start)
G.f.: 2*x*(1+x)*(1+2*x)/(1-2*x^2)^2.
a(n)/a(n-2) = 2*n/(n-2). (End)
a(2*n) = 3*n*2^n, a(2*n+1) = (2*n+1)*2^(n+1). - Andrew Howroyd, Mar 28 2016