A279260 Numbers which are cyclops palindromic in their binary reflected Gray code representation.

0, 6, 18, 90, 330, 1386, 5418, 21930, 87210, 349866, 1397418, 5593770, 22366890, 89483946, 357903018, 1431677610, 5726579370, 22906579626, 91625794218, 366504225450, 1466014804650, 5864063412906, 23456245263018, 93824997829290, 375299957762730, 1501199898159786, 6004799458421418

Offset

0,2

Comments

Cyclops palindromic numbers in base 2 are numbers with middle bit 0, having equal number of 1's on both side of the 0. There is a single 0 bit in the middle and the total number of bits is odd. The middle '0' represents the eye of a cyclops.

a(n) mod 6 = 0.

Links

Indranil Ghosh, Table of n, a(n) for n = 0..1000

Indranil Ghosh, Proof of 6|{(-2*(1+((-2)^n)-(2^(2*n+1))))/3}

Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video, (2015)

Index entries for linear recurrences with constant coefficients, signature (3,6,-8).

Formula

a(n) = (-2*(1+((-2)^n)-(2^(2*n+1))))/3.

From Andrew Howroyd, Oct 28 2025: (Start)

a(n) = 6*A084175(n).

G.f.: 6*x/((1 - x)*(1 + 2*x)*(1 - 4*x)). (End)

E.g.f.: 2*(2*exp(4*x) - exp(x) - exp(-2*x))/3. - Stefano Spezia, Oct 28 2025

Example

90 is in the sequence because the binary reflected Gray code representation of 90 is '1110111' which is a cyclops palindromic binary number.

Mathematica

A279260[n_] := -2*(1 + (-2)^n - 2^(2*n+1))/3; Array[A279260, 30, 0] (* or *)
LinearRecurrence[{3, 6, -8}, {0, 6, 18}, 30] (* Paolo Xausa, Feb 04 2026 *)

Prog

(Python)
def a(n):
    return (-2*(1+((-2)**n)-(2**(2*n+1))))/3
(PARI) a(n)=(-2*(1+((-2)^n)-(2^(2*n+1))))/3 \\ Charles R Greathouse IV, Jun 29 2018

Crossrefs

Cf. A014550, A084175, A129868, A134808, A138148.

Partial sums of A071930.

Sequence in context: A239420 A219590 A260664 * A379178 A371987 A294471

Adjacent sequences: A279257 A279258 A279259 * A279261 A279262 A279263

Keyword

nonn,base,easy

Author

Indranil Ghosh, Jan 17 2017

Status

approved