A014642 Even octagonal numbers: a(n) = 4*n*(3*n-1).

0, 8, 40, 96, 176, 280, 408, 560, 736, 936, 1160, 1408, 1680, 1976, 2296, 2640, 3008, 3400, 3816, 4256, 4720, 5208, 5720, 6256, 6816, 7400, 8008, 8640, 9296, 9976, 10680, 11408, 12160, 12936, 13736, 14560, 15408, 16280, 17176, 18096, 19040, 20008, 21000, 22016

Offset

0,2

Comments

8 times pentagonal numbers. - Omar E. Pol, Dec 11 2008

Sequence found by reading the line from 0, in the direction 0, 8, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012

The sequence forms the even nesting cube-frames (see illustrations in A000567), which separate and appear according to formula along the axes on the zero-centered and one-centered hexagonal number spirals, as well as the axes of the zero-centered and one-centered square number spirals. See illustrations in links. - John Elias, Jul 20 2022

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

John Elias, Octagonal Nesting Cubes on the Hexagonal Number Spiral Octagonal Nesting Cubes on the Square Number Spiral

Craig Knecht, Number of positions the remaining tiles can occupy in a 4*n length polyiamond bilayer when one tile is missing.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = A000326(n)*8. - Omar E. Pol, Dec 11 2008

a(n) = A049450(n)*4 = A033579(n)*2. - Omar E. Pol, Dec 13 2008

a(n) = a(n-1) + 24*n - 16 (with a(0)=0). - Vincenzo Librandi, Nov 20 2010

G.f.: x*(8+16*x)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 06 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - G. C. Greubel, Jun 07 2017

E.g.f.: 4*x*(2 + 3*x)*exp(x). - G. C. Greubel, Oct 09 2019

From Amiram Eldar, Mar 24 2021: (Start)

Sum_{n>=1} 1/a(n) = 3*log(3)/8 - Pi/(8*sqrt(3)).

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2 - Pi/(4*sqrt(3)). (End)

Maple

seq(8*binomial(3*n, 2)/3, n=0..50); # G. C. Greubel, Oct 09 2019

Mathematica

LinearRecurrence[{3, -3, 1}, {0, 8, 40}, 50] (* G. C. Greubel, Jun 07 2017 *)
PolygonalNumber[8, Range[0, 90, 2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 19 2020 *)

Prog

(PARI) vector(51, n, 8*binomial(3*(n-1), 2)/3 ) \\ G. C. Greubel, Jun 07 2017
(Magma) [8*Binomial(3*n, 2)/3: n in [0..50]]; // G. C. Greubel, Oct 09 2019
(Sage) [8*binomial(3*n, 2)/3 for n in (0..50)] # G. C. Greubel, Oct 09 2019
(GAP) List([0..50], n-> 8*Binomial(3*n, 2)/3); # G. C. Greubel, Oct 09 2019

Crossrefs

Cf. A000567, A000326, A001082, A014641, A014793, A014794, A033579, A049450.

Sequence in context: A226904 A305075 A069083 * A211631 A279273 A143943

Adjacent sequences: A014639 A014640 A014641 * A014643 A014644 A014645

Keyword

nonn,easy

Author

Mohammad K. Azarian

Extensions

More terms from Patrick De Geest

Status

approved