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A062741 3 times pentagonal numbers: 3*n*(3*n-1)/2. 28
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 (list; graph; refs; listen; history; text; internal format)
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
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).
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. 3 times n-gonal numbers: A045943, A033428, A094159, A152773, A152751, A152759, A152767, A153783, A153448, A153875.
Sequence in context: A162441 A001803 A161738 * A185541 A176661 A117561
KEYWORD
nonn,easy
AUTHOR
Floor van Lamoen, Jul 21 2001
EXTENSIONS
Better definition and edited by Omar E. Pol, Dec 11 2008
STATUS
approved

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Last modified April 16 01:01 EDT 2024. Contains 371696 sequences. (Running on oeis4.)