A337348 - OEIS

%I #29 Sep 09 2020 09:22:36

%S 0,2,10,50,210,882,3570,14450,57970,232562,930930,3726450,14908530,

%T 59645042,238591090,954408050,3817675890,15270878322,61083688050,

%U 244335451250,977342504050,3909372812402,15637494045810,62549987368050,250199960657010,1000799887367282,4003199594208370

%N Numbers formed as the product of two numbers without consecutive equal binary digits and sharing no common bits between them.

%C The alternating, non-overlapping bits means that the divisors sum to 1 less than a power of 2.

%C They also resemble a zipper:

%C    10101010

%C    01010101.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,0,-20,16).

%F a(n) = A000975(n - 1) * A000975(n).

%F From _Colin Barker_, Aug 24 2020: (Start)

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

%F a(n) = 5*a(n-1) - 20*a(n-3) + 16*a(n-4) for n>4.

%F (End)

%F 18*a(n) = 4^(n+1) +(-2)^n +4 -9*2^n. - _R. J. Mathar_, Sep 09 2020

%e For n = 6, in binary form:

%e     101010

%e   x 010101

%e ----------

%e 1101110010 (882)

%t LinearRecurrence[{5, 0, -20, 16}, {0, 2, 10, 50}, 27] (* _Amiram Eldar_, Aug 24 2020 *)

%o (Python)

%o def a(n):

%o     x = y = ''

%o     for _ in range(n):

%o         x, y  = y + '1', x + '0'

%o     return int(x, 2) * int(y, 2)

%o (PARI) a(n) = (2 * 2^n \ 3) * (2 * 2^(n-1) \ 3) \\ _David A. Corneth_, Aug 24 2020

%o (PARI) concat(0, Vec(2*x^2 / ((1 - x)*(1 - 2*x)*(1 + 2*x)*(1 - 4*x)) + O(x^30))) \\ _Colin Barker_, Sep 04 2020

%Y Formed from the product of consecutive pairs of A000975.

%K nonn,base,easy

%O 1,2

%A _Matt Donahoe_, Aug 24 2020