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A337348 Numbers formed as the product of two numbers without consecutive equal binary digits and sharing no common bits between them. 0
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 (list; graph; refs; listen; history; text; internal format)
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 A180266

Adjacent sequences:  A337345 A337346 A337347 * A337349 A337350 A337351

KEYWORD

nonn,base,easy

AUTHOR

Matt Donahoe, Aug 24 2020

STATUS

approved

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Last modified May 10 21:08 EDT 2021. Contains 343780 sequences. (Running on oeis4.)