

A014642


Even octagonal numbers: a(n) = 4*n*(3*n1).


15



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
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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 cubeframes (see illustrations in A000567), which separate and appear according to formula along the axes on the zerocentered and onecentered hexagonal number spirals, as well as the axes of the zerocentered and onecentered 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(n1) + 24*n  16 (with a(0)=0).  Vincenzo Librandi, Nov 20 2010
G.f.: x*(8+16*x)/(13*x+3*x^2x^3).  Colin Barker, Jan 06 2012
a(n) = 3*a(n1)  3*a(n2) + a(n3).  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*(n1), 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



