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%I #71 Dec 26 2023 11:23:12
%S 0,3,15,36,66,105,153,210,276,351,435,528,630,741,861,990,1128,1275,
%T 1431,1596,1770,1953,2145,2346,2556,2775,3003,3240,3486,3741,4005,
%U 4278,4560,4851,5151,5460,5778,6105,6441,6786,7140,7503,7875,8256,8646,9045
%N 3 times pentagonal numbers: 3*n*(3*n-1)/2.
%C 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.
%C 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
%H Nathaniel Johnston, <a href="/A062741/b062741.txt">Table of n, a(n) for n = 0..10000</a>
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1805.10680">A generating polynomial for the pretzel knot</a>, arXiv:1805.10680 [math.CO], 2018.
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3470205">The groupoid of the Triangular Numbers and the generation of related integer sequences</a>, Politecnico di Torino, Italy (2019).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = binomial(3*n, 2). - _Zerinvary Lajos_, Jan 02 2007
%F a(n) = (9*n^2 - 3*n)/2 = 3*n(3*n-1)/2 = A000326(n)*3. - _Omar E. Pol_, Dec 11 2008
%F a(n) = a(n-1) + 9*n - 6, with n > 0, a(0)=0. - _Vincenzo Librandi_, Aug 07 2010
%F G.f.: 3*x*(1+2*x)/(1-x)^3. - _Bruno Berselli_, Jan 21 2011
%F a(n) = A218470(9n+2). - _Philippe Deléham_, Mar 27 2013
%F a(n) = n*A008585(n) + Sum_{i=0..n-1} A008585(i) for n > 0. - _Bruno Berselli_, Dec 19 2013
%F From _Amiram Eldar_, Jan 10 2022: (Start)
%F Sum_{n>=1} 1/a(n) = log(3) - Pi/(3*sqrt(3)).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/(3*sqrt(3)) - 4*log(2)/3. (End)
%F E.g.f.: (3/2)*x*(2 + 3*x)*exp(x). - _G. C. Greubel_, Dec 26 2023
%e The spiral begins:
%e 15
%e 16 14
%e 17 3 13
%e 18 4 2 12
%e 19 5 0 1 11
%e 20 6 7 8 9 10
%p [seq(binomial(3*n,2),n=0..45)]; # _Zerinvary Lajos_, Jan 02 2007
%t 3*PolygonalNumber[5,Range[0,50]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Mar 06 2019 *)
%o (PARI) a(n)=3*n*(3*n-1)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [Binomial(3*n,2): n in [0..50]]; // _G. C. Greubel_, Dec 26 2023
%o (SageMath) [binomial(3*n,2) for n in range(51)] # _G. C. Greubel_, Dec 26 2023
%Y Cf. A000326, A008585, A051682, A218470.
%Y Cf. 3 times n-gonal numbers: A045943, A033428, A094159, A152773, A152751, A152759, A152767, A153783, A153448, A153875.
%K nonn,easy
%O 0,2
%A _Floor van Lamoen_, Jul 21 2001
%E Better definition and edited by _Omar E. Pol_, Dec 11 2008
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