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

Author

Paul Barry, Mar 15 2003

Keywords

• nonn
• easy

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

Examples

			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.
		

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).

Crossrefs

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

Cf. A003154, A133694.

Programs

GAP

List([0..50],n->Binomial(3*n+1,2)); # Muniru A Asiru, Feb 28 2019

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 *)

PARI

a(n)=3*n*(3*n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

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)