A096222 - OEIS

%I #13 Jul 19 2020 11:51:33

%S 1,3,9,30,100,360,1296,4896,18496,71808,278784,1098240,4326400,

%T 17172480,68161536,271589376,1082146816,4320165888,17247043584,

%U 68920934400,275415040000,1101122764800,4402342526976,17605073043456,70403108110336

%N Number of different rectangles when a piece of paper is folded n times in alternate directions.

%C Similar to A000537, which counts all possible rectangles in an n X n array of squares. In this sequence we count the rectangles in an a X b array of squares, where a=2^floor(n/2) and b=2^ceiling(n/2). Note that a(n) is the product of two triangular numbers.

%F a(n) = a b (a+1) (b+1)/4, where a=2^floor(n/2) and b=2^ceiling(n/2).

%F a(n) (mod 10^k) is cyclic. For (mod 10) the cycle is 0, 0, 0, 6, 6, 6, 8, 4. - _Robert G. Wilson v_, Jul 31 2004

%F Conjectures from _Colin Barker_, Feb 04 2020: (Start)

%F G.f.: (1 - 3*x - 9*x^2 + 24*x^3) / ((1 - 2*x)*(1 - 4*x)*(1 - 8*x^2)).

%F a(n) = 6*a(n-1) - 48*a(n-3) + 64*a(n-4) for n>3.

%F (End)

%e a(1) = 3: fold a 1 X 2 rectangle down the middle; there are 3 rectangles, the one on the left, the one on the right and the one we started with. a(2) = 9 : fold a 2 X 2 rectangle along the X and Y axes; there 4 rectangles of size 1 X 1, 4 of size 1 X 2 or 2 X 1 and 1 of size 2 X 2.

%t Table[a=2^Floor[n/2]; b=2^Ceiling[n/2]; Sum[i*j, {i, a}, {j, b}], {n, 20}]

%Y Cf. A000537.

%K nonn

%O 0,2

%A Bill Liebeskind (billlieb(AT)hotmail.com), Jul 29 2004

%E Edited by _T. D. Noe_, Jul 30 2004

%E More terms from _Robert G. Wilson v_, Jul 31 2004