OFFSET
0,2
COMMENTS
Write 0,1,2,3,4,... in a triangular spiral; then a(n) is the sequence found by reading from 0 in the vertical upward direction.
Number of edges in the join of two complete graphs of order 2n and n, K_2n * K_n - Roberto E. Martinez II, Jan 07 2002
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 0..10000
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = binomial(3*n, 2). - Zerinvary Lajos, Jan 02 2007
a(n) = (9*n^2 - 3*n)/2 = 3*n(3*n-1)/2 = A000326(n)*3. - Omar E. Pol, Dec 11 2008
a(n) = a(n-1) + 9*n - 6, with n > 0, a(0)=0. - Vincenzo Librandi, Aug 07 2010
G.f.: 3*x*(1+2*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(n) = A218470(9n+2). - Philippe Deléham, Mar 27 2013
From Amiram Eldar, Jan 10 2022: (Start)
Sum_{n>=1} 1/a(n) = log(3) - Pi/(3*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/(3*sqrt(3)) - 4*log(2)/3. (End)
E.g.f.: (3/2)*x*(2 + 3*x)*exp(x). - G. C. Greubel, Dec 26 2023
EXAMPLE
The spiral begins:
15
16 14
17 3 13
18 4 2 12
19 5 0 1 11
20 6 7 8 9 10
MAPLE
[seq(binomial(3*n, 2), n=0..45)]; # Zerinvary Lajos, Jan 02 2007
MATHEMATICA
3*PolygonalNumber[5, Range[0, 50]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 06 2019 *)
PROG
(PARI) a(n)=3*n*(3*n-1)/2 \\ Charles R Greathouse IV, Sep 24 2015
(Magma) [Binomial(3*n, 2): n in [0..50]]; // G. C. Greubel, Dec 26 2023
(SageMath) [binomial(3*n, 2) for n in range(51)] # G. C. Greubel, Dec 26 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Floor van Lamoen, Jul 21 2001
EXTENSIONS
Better definition and edited by Omar E. Pol, Dec 11 2008
STATUS
approved