OFFSET
0,4
COMMENTS
Number of rectangles that can be formed from the vertices of an n-dimensional cube. E.g., a(3)=12 because the three-dimensional cube has six faces plus six rectangles passing through the center of the cube. Cf. A064436: each rectangle on the cube provides an opportunity for a function not to be a linear threshold function, by alternating in value around the rectangle. - Matthew Cook, Jan 26 2004
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..150
Index entries for linear recurrences with constant coefficients, signature (12,-44,48).
FORMULA
a(n) = (2^n)*Stirling2(n+3, 3), n >= 0, with Stirling2(n, m) = A008277(n, m).
G.f.: x^2/((1-2*x)*(1-4*x)*(1-6*x)).
E.g.f.: (exp(2*x) - 8*exp(4*x) + 9*exp(6*x))/2!.
a(n) = ((6^n - 2^n)/4 - (4^n - 2^n)/2)/2. - Zerinvary Lajos, Jun 05 2009
From Taras Goy, May 31 2026: (Start)
a(n) = 12*a(n-1) - 44*a(n-2) + 48*a(n-3), where a(0)=0, a(1)=0, a(2)=1.
a(n) = 10*a(n-1) - 24*a(n-2) + 2^(n-2), where a(0)=0, a(1)=0. (End)
MAPLE
[seq(9/2*6^n-4*4^n+1/2*2^n, n=0..20)]; # Detlef Pauly (dettodet(AT)yahoo.de), Dec 04 2001
MATHEMATICA
CoefficientList[Series[x^2/((1 - 2 x) (1 - 4 x) (1 - 6 x)), {x, 0, 20}], x] (* Michael De Vlieger, Jan 31 2018 *)
PROG
(SageMath) [((6^n - 2^n)/4-(4^n - 2^n)/2)/2 for n in range(0, 21)] # Zerinvary Lajos, Jun 05 2009
(Magma) [6^n/8 - 4^(n-1) + 2^(n-3): n in [0..25]]; // Vincenzo Librandi, Apr 26 2011
(PARI) a(n)=6^n/8-4^(n-1)+2^(n-3) \\ Charles R Greathouse IV, May 17 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved