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

Author

Mohammad K. Azarian

Keywords

• nonn
• easy

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.
• Yaohui Zhu, Kaiming Sun, Zhengdong Luo, and Lingfeng Wang, Progressive Self-Learning for Domain Adaptation on Symbolic Regression of Integer Sequences, Proc. 39th AAAI Conf. Artif. Intel. (2025) Vol. 39, No. 1, 1692-1699. See p. 1698.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

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

Programs

GAP

List([0..50], n-> 8*Binomial(3*n,2)/3); # G. C. Greubel, Oct 09 2019

Magma

[8*Binomial(3*n,2)/3: n in [0..50]]; // G. C. Greubel, Oct 09 2019

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 *)

PARI

vector(51, n, 8*binomial(3*(n-1),2)/3 ) \\ G. C. Greubel, Jun 07 2017

Sage

[8*binomial(3*n,2)/3 for n in (0..50)] # G. C. Greubel, Oct 09 2019

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)

Extensions

More terms from Patrick De Geest