OFFSET
0,2
COMMENTS
Also minimal 3-covers of a labeled n-set that cover 3 points of that set uniquely (if offset is 3). Cf. A057524 for unlabeled case. - Vladeta Jovovic, Sep 02 2000
Let M=[1,0,0,i;0,1,i,0;0,i,1,0;i,0,0,1], i=sqrt(-1). Then 1/det(I-xM) = 1/(1-4x)^4. - Paul Barry, Apr 27 2005
With a different offset, number of n-permutations (n=4) of 5 objects u, v, w, z, x with repetition allowed, containing exactly three u's. Example: a(1)=16 because we have uuuv, uuvu, uvuu, vuuu, uuuw, uuwu, uwuu, wuuu, uuuz, uuzu, uzuu, zuuu, uuux, uuxu, uxuu and xuuu. - Zerinvary Lajos, May 19 2008
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (16,-96,256,-256).
FORMULA
a(n) = binomial(n+3, 3)*4^n.
G.f.: 1/(1-4*x)^4.
a(n) = Sum_{a+b+c+d+e+f+g+h=n} f(a)*f(b)*f(c)*f(d)*f(e)*f(f)*f(g)*f(h) with f(n)=A000984(n). - Philippe Deléham, Jan 22 2004
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=0} 1/a(n) = 108*log(4/3) - 30.
Sum_{n>=0} (-1)^n/a(n) = 300*log(5/4) - 66. (End)
E.g.f.: exp(4*x)*(3 + 36*x + 72*x^2 + 32*x^3)/3. - Stefano Spezia, Jan 01 2023
MAPLE
seq(seq(binomial(i, j)*4^(i-3), j =i-3), i=3..33); # Zerinvary Lajos, Dec 03 2007
seq(binomial(n+3, 3)*4^n, n=0..30); # Zerinvary Lajos, May 19 2008
MATHEMATICA
Table[4^n*Binomial[n+3, 3], {n, 0, 30}] (* G. C. Greubel, Jul 20 2019 *)
PROG
(Sage) [lucas_number2(n, 4, 0)*binomial(n, 3)/2^6 for n in range(3, 33)] # Zerinvary Lajos, Mar 11 2009
(Magma) [4^n*Binomial(n+3, 3): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011
(PARI) Vec(1/(1-4*x)^4+O(x^30)) \\ Charles R Greathouse IV, Oct 03 2016
(GAP) List([0..30], n-> 4^n*Binomial(n+3, 3) ) # G. C. Greubel, Jul 20 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
STATUS
approved