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

0,2

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

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

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

Cf. A000326, A008585, A051682, A218470.

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

Sequence in context: A162441 A001803 A161738 * A185541 A176661 A117561

Adjacent sequences: A062738 A062739 A062740 * A062742 A062743 A062744

Keyword

nonn,easy

Author

Floor van Lamoen, Jul 21 2001

Extensions

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

Status

approved