A062741 3 times pentagonal numbers: 3*n*(3*n-1)/2.

0, 3, 15, 36, 66, 105, 153, 210, 276, 351, 435, 528, 630, 741, 861, 990, 1128, 1275, 1431, 1596, 1770, 1953, 2145, 2346, 2556, 2775, 3003, 3240, 3486, 3741, 4005, 4278, 4560, 4851, 5151, 5460, 5778, 6105, 6441, 6786, 7140, 7503, 7875, 8256, 8646, 9045

Offset: 0

Author

Floor van Lamoen, Jul 21 2001

Keywords

• nonn
• easy

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

Examples

			The spiral begins:
            15
          16  14
        17   3  13
      18   4   2  12
    19   5   0   1  11
  20   6   7   8   9  10
		

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).

Crossrefs

Cf. A000326, A008585, A051682, A218470.

Cf. 3 times n-gonal numbers: A045943, A033428, A094159, A152773, A152751, A152759, A152767, A153783, A153448, A153875.

Programs

Magma

[Binomial(3*n,2): n in [0..50]]; // G. C. Greubel, Dec 26 2023

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 *)

PARI

a(n)=3*n*(3*n-1)/2 \\ Charles R Greathouse IV, Sep 24 2015

SageMath

[binomial(3*n,2) for n in range(51)] # G. C. Greubel, Dec 26 2023

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

a(n) = n*A008585(n) + Sum_{i=0..n-1} A008585(i) for n > 0. - Bruno Berselli, Dec 19 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

Extensions

Better definition and edited by Omar E. Pol, Dec 11 2008