A337348 Numbers formed as the product of two numbers without consecutive equal binary digits and sharing no common bits between them.

0, 2, 10, 50, 210, 882, 3570, 14450, 57970, 232562, 930930, 3726450, 14908530, 59645042, 238591090, 954408050, 3817675890, 15270878322, 61083688050, 244335451250, 977342504050, 3909372812402, 15637494045810, 62549987368050, 250199960657010, 1000799887367282, 4003199594208370

Offset

1,2

Comments

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

They also resemble a zipper:

10101010

01010101.

Links

Table of n, a(n) for n=1..27.

Index entries for linear recurrences with constant coefficients, signature (5,0,-20,16).

Formula

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

From Colin Barker, Aug 24 2020: (Start)

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

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

(End)

18*a(n) = 4^(n+1) +(-2)^n +4 -9*2^n. - R. J. Mathar, Sep 09 2020

Example

For n = 6, in binary form:
    101010
  x 010101
----------
1101110010 (882)

Mathematica

LinearRecurrence[{5, 0, -20, 16}, {0, 2, 10, 50}, 27] (* Amiram Eldar, Aug 24 2020 *)

Prog

(Python)
def a(n):
    x = y = ''
    for _ in range(n):
        x, y  = y + '1', x + '0'
    return int(x, 2) * int(y, 2)
(PARI) a(n) = (2 * 2^n \ 3) * (2 * 2^(n-1) \ 3) \\ David A. Corneth, Aug 24 2020
(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

Crossrefs

Formed from the product of consecutive pairs of A000975.

Sequence in context: A268108 A143147 A317111 * A218778 A320521 A371605

Adjacent sequences: A337345 A337346 A337347 * A337349 A337350 A337351

Keyword

nonn,base,easy

Author

Matt Donahoe, Aug 24 2020

Status

approved