A081266 Staggered diagonal of triangular spiral in A051682.

0, 6, 21, 45, 78, 120, 171, 231, 300, 378, 465, 561, 666, 780, 903, 1035, 1176, 1326, 1485, 1653, 1830, 2016, 2211, 2415, 2628, 2850, 3081, 3321, 3570, 3828, 4095, 4371, 4656, 4950, 5253, 5565, 5886, 6216, 6555, 6903, 7260, 7626, 8001, 8385, 8778, 9180

Offset

0,2

Comments

Staggered diagonal of triangular spiral in A051682, between (0,4,17) spoke and (0,7,23) spoke.

Binomial transform of (0, 6, 9, 0, 0, 0, ...).

If Y is a fixed 3-subset of a (3n+1)-set X then a(n) is the number of (3n-1)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007

Partial sums give A085788. - Leo Tavares, Nov 23 2023

Links

Muniru A Asiru, Table of n, a(n) for n = 0..10000

Tomislav Došlić and Luka Podrug, Sweet division problems: from chocolate bars to honeycomb strips and back, arXiv:2304.12121 [math.CO], 2023.

Milan Janjic, Two Enumerative Functions

Milan Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.

Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).

Leo Tavares, Star illustration

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

Formula

a(n) = 6*C(n,1) + 9*C(n,2).

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

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

a(n) = A000217(3*n); a(2*n) = A144314(n). - Reinhard Zumkeller, Sep 17 2008

a(n) = 3*A005449(n). - R. J. Mathar, Mar 27 2009

a(n) = 9*n+a(n-1)-3 for n>0, a(0)=0. - Vincenzo Librandi, Aug 08 2010

a(n) = A218470(9n+5). - Philippe Deléham, Mar 27 2013

a(n) = Sum_{k=0..3n} (-1)^(n+k)*k^2. - Bruno Berselli, Aug 29 2013

E.g.f.: 3*exp(x)*x*(4 + 3*x)/2. - Stefano Spezia, Jun 06 2021

From Amiram Eldar, Aug 11 2022: (Start)

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

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

From Leo Tavares, Nov 23 2023: (Start)

a(n) = 3*A000217(n) + 3*A000290(n).

a(n) = A003154(n+1) - A133694(n+1). (End)

Example

a(1)=9*1+0-3=6, a(2)=9*2+6-3=21, a(3)=9*3+21-3=45.
For n=3, a(3) = -0^2+1^2-2^2+3^2-4^2+5^2-6^2+7^2-8^2+9^2 = 45.

Maple

seq(binomial(3*n+1, 2), n=0..45); # Zerinvary Lajos, Jan 21 2007

Mathematica

LinearRecurrence[{3, -3, 1}, {0, 6, 21}, 50] (* Harvey P. Dale, Aug 29 2015 *)

Prog

(PARI) a(n)=3*n*(3*n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017
(GAP) List([0..50], n->Binomial(3*n+1, 2)); # Muniru A Asiru, Feb 28 2019

Crossrefs

Cf. A000217, A000290, A005449, A014105, A022266, A033585, A062725, A144312, A144314, A218470.

Cf. A003154, A133694.

Sequence in context: A180857 A119868 A175729 * A087863 A212656 A051941

Adjacent sequences: A081263 A081264 A081265 * A081267 A081268 A081269

Keyword

nonn,easy

Author

Paul Barry, Mar 15 2003

Status

approved