A071930 -id:A071930 - cp's OEIS frontend

A003674 a(n) = 2^(n-1)*(2^n - (-1)^n).

0, 3, 6, 36, 120, 528, 2016, 8256, 32640, 131328, 523776, 2098176, 8386560, 33558528, 134209536, 536887296, 2147450880, 8590000128, 34359607296, 137439215616, 549755289600, 2199024304128, 8796090925056

Offset: 0

N. J. A. Sloane

M. Gardner, Riddles of the Sphinx, New Mathematical Library, M.A.A., 1987, p. 145. Math. Rev. 89i:00015.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A001045, A003683 (one-third), A062510, A071930.

[(4^n -(-2)^n)/2: n in [0..40]]; // G. C. Greubel, Feb 17 2023

Table[(4^n-(-2)^n)/2, {n,0,40}] (* G. C. Greubel, Feb 17 2023 *)

PARI
```
a(n)=if(n<0,0,2^(n-1)*(2^n-(-1)^n))
```

[(4^n-(-2)^n)/2 for n in range(41)] # G. C. Greubel, Feb 17 2023

G.f.: 3*x/((1+2*x)*(1-4*x)).
a(n) = 3*A003683(n).
Given the 2 X 2 matrix M = [1,3; 3,1], a(n) = term (1,2) in M^n, n>0. - Gary W. Adamson, Aug 06 2010
From G. C. Greubel, Feb 17 2023: (Start)
a(n) = 2*a(n-1) + 8*a(n-2).
a(n) = 3*2^(n-1)*A001045(n).
a(n) = 2^(n-1)*A062510(n).
a(n) = (1/2)*A071930(n+1).
E.g.f.: (1/2)*(exp(4*x) - exp(-2*x)). (End)

A377885 Cogrowth sequence of the 16-element quasihedral group SD16 = .

Original entry on oeis.org

1, 1, 1, 4, 28, 136, 544, 2080, 8128, 32512, 130816, 524800, 2099200, 8390656, 33550336, 134201344, 536854528, 2147516416, 8590065664, 34359869440, 137438691328, 549754765312, 2199022206976, 8796095119360, 35184380477440, 140737496743936, 562949936644096

Offset: 0

Sean A. Irvine, Nov 10 2024

Gives the even terms, all the odd terms are 0.

Also called QD16, Q8:C2. Gap identifier 16,8.

Index entries for linear recurrences with constant coefficients, signature (6, -12, 16).

Cf. A047849 (D4), A007582 (D8), A071930 (Q8), A377840 (C8 X C2), A377883 (M4(2)).

G.f.: (6*x^3-7*x^2+5*x-1) / ((4*x-1) * (4*x^2-2*x+1)).

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

Original entry on oeis.org

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

Indranil Ghosh, Jan 17 2017

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.

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

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)

Cf. A014550, A129868, A134808, A138148.

Partial sums of A071930.

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

def a(n):
    return (-2*(1+((-2)**n)-(2**(2*n+1))))/3

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

A377735 Cogrowth sequence of the 16-element group Q8 X C2 = .

Original entry on oeis.org

1, 1, 7, 103, 829, 7261, 66595, 598627, 5377849, 48426745, 435876607, 3922582687, 35303534581, 317733991381, 2859598948507, 25736384863003, 231627537879409, 2084647743751921, 18761829221081335, 168856464809568727, 1519708183900618669, 13677373637498037325

Offset: 0

Sean A. Irvine, Nov 10 2024

Gives the even terms, all the odd terms are 0.

Index entries for linear recurrences with constant coefficients, signature (8, 2, 72, -81).

Cf. A047854 (D4 X C2), A377840 (C8 X C2), A071930 (Q8).

G.f.: (27*x^3+3*x^2+7*x-1) / ((1-x) * (9*x-1) * (9*x^2+2*x+1)).

A377944 Cogrowth sequence of the 16-element dicyclic group Q16 = .

Original entry on oeis.org

1, 0, 1, 12, 28, 120, 544, 2016, 8128, 33024, 130816, 523776, 2099200, 8386560, 33550336, 134234112, 536854528, 2147450880, 8590065664, 34359607296, 137438691328, 549756862464, 2199022206976, 8796090925056, 35184380477440, 140737479966720, 562949936644096

Offset: 0

Sean A. Irvine, Nov 11 2024

Gives the even terms, all the odd terms are 0.

Also called Dic16, C8:C2. Gap identifier 16, 9.

Index entries for linear recurrences with constant coefficients, signature (2, 4, 16).

Cf. A071930 (Q8), A377656 (Dic12), A377735 (Q8 X C2), A377840 (C8 X C2), A007582 (D8), A377885 (SD16), A377883 (M4(2)).

G.f.: (6*x^3+3*x^2+2*x-1) / ((4*x-1) * (4*x^2+2*x+1)).

cp's OEIS Frontend

A003674 a(n) = 2^(n-1)*(2^n - (-1)^n).

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A377885 Cogrowth sequence of the 16-element quasihedral group SD16 = .

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

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A377735 Cogrowth sequence of the 16-element group Q8 X C2 = .

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A377944 Cogrowth sequence of the 16-element dicyclic group Q16 = .

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