A279260 - OEIS

%I #55 Feb 04 2026 08:37:34

%S 0,6,18,90,330,1386,5418,21930,87210,349866,1397418,5593770,22366890,

%T 89483946,357903018,1431677610,5726579370,22906579626,91625794218,

%U 366504225450,1466014804650,5864063412906,23456245263018,93824997829290,375299957762730,1501199898159786,6004799458421418

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

%C 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.

%C a(n) mod 6 = 0.

%H Indranil Ghosh, <a href="/A279260/b279260.txt">Table of n, a(n) for n = 0..1000</a>

%H Indranil Ghosh, <a href="/A279260/a279260.txt">Proof of 6|{(-2*(1+((-2)^n)-(2^(2*n+1))))/3}</a>

%H Brady Haran and Simon Pampena, <a href="https://www.youtube.com/watch?v=HPfAnX5blO0">Glitch Primes and Cyclops Numbers</a>, Numberphile video, (2015)

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-8).

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

%F From _Andrew Howroyd_, Oct 28 2025: (Start)

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

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

%F E.g.f.: 2*(2*exp(4*x) - exp(x) - exp(-2*x))/3. - _Stefano Spezia_, Oct 28 2025

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

%t A279260[n_] := -2*(1 + (-2)^n - 2^(2*n+1))/3; Array[A279260, 30, 0] (* or *)

%t LinearRecurrence[{3, 6, -8}, {0, 6, 18}, 30] (* _Paolo Xausa_, Feb 04 2026 *)

%o (Python)

%o def a(n):

%o     return (-2*(1+((-2)**n)-(2**(2*n+1))))/3

%o (PARI) a(n)=(-2*(1+((-2)^n)-(2^(2*n+1))))/3 \\ _Charles R Greathouse IV_, Jun 29 2018

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

%Y Partial sums of A071930.

%K nonn,base,easy

%O 0,2

%A _Indranil Ghosh_, Jan 17 2017