A152994 Nine times hexagonal numbers: a(n) = 9*n*(2*n-1).

0, 9, 54, 135, 252, 405, 594, 819, 1080, 1377, 1710, 2079, 2484, 2925, 3402, 3915, 4464, 5049, 5670, 6327, 7020, 7749, 8514, 9315, 10152, 11025, 11934, 12879, 13860, 14877, 15930, 17019, 18144, 19305, 20502, 21735, 23004, 24309, 25650, 27027, 28440, 29889, 31374

Offset: 0

Author

Omar E. Pol, Dec 22 2008

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 0, in the direction 0, 9,..., in the square spiral whose vertices are the generalized 11-gonal numbers A195160. - Omar E. Pol, Sep 18 2011

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000384, A094159.

Similar sequences are listed in A316466.

Programs

GAP

List([0..40], n-> 9*n*(2*n-1)); # G. C. Greubel, Sep 01 2019

Magma

[9*n*(2*n-1): n in [0..40]]; // G. C. Greubel, Sep 01 2019

Maple

seq(9*n*(2*n-1), n=0..40); # G. C. Greubel, Sep 01 2019

Mathematica

Table[9*n*(2*n-1), {n,0,40}] (* G. C. Greubel, Sep 01 2019 *)
9*PolygonalNumber[6,Range[0,50]] (* Harvey P. Dale, Jul 24 2022 *)

PARI

a(n)=9*n*(2*n-1) \\ Charles R Greathouse IV, Jun 17 2017

Sage

[9*n*(2*n-1) for n in (0..40)] # G. C. Greubel, Sep 01 2019

Formula

a(n) = 18*n^2 - 9*n = A000384(n)*9 = A094159(n)*3.

a(n) = a(n-1) + 36*n - 27 for n>0, a(0)=0. - Vincenzo Librandi, Dec 15 2010

a(n) = Sum_{i = 2..10} P(i,n), where P(i,m) = m*((i-2)*m-(i-4))/2. - Bruno Berselli, Jul 04 2018

From G. C. Greubel, Sep 01 2019: (Start)

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

E.g.f.: 9*x*(1+2*x)*exp(x). (End)

From Amiram Eldar, Feb 27 2022: (Start)

Sum_{n>=1} 1/a(n) = 2*log(2)/9.

Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi - 2*log(2))/18. (End)