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%I #70 Nov 23 2023 15:18:08
%S 0,6,21,45,78,120,171,231,300,378,465,561,666,780,903,1035,1176,1326,
%T 1485,1653,1830,2016,2211,2415,2628,2850,3081,3321,3570,3828,4095,
%U 4371,4656,4950,5253,5565,5886,6216,6555,6903,7260,7626,8001,8385,8778,9180
%N Staggered diagonal of triangular spiral in A051682.
%C Staggered diagonal of triangular spiral in A051682, between (0,4,17) spoke and (0,7,23) spoke.
%C Binomial transform of (0, 6, 9, 0, 0, 0, ...).
%C If Y is a fixed 3-subset of a (3n+1)-set X then a(n) is the number of (3n-1)-subsets of X intersecting Y. - _Milan Janjic_, Oct 28 2007
%C Partial sums give A085788. - _Leo Tavares_, Nov 23 2023
%H Muniru A Asiru, <a href="/A081266/b081266.txt">Table of n, a(n) for n = 0..10000</a>
%H Tomislav Došlić and Luka Podrug, <a href="https://arxiv.org/abs/2304.12121">Sweet division problems: from chocolate bars to honeycomb strips and back</a>, arXiv:2304.12121 [math.CO], 2023.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Milan Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv:1301.4550 [math.CO], 2013.
%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 Leo Tavares, <a href="/A081266/a081266.jpg">Star illustration</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 6*C(n,1) + 9*C(n,2).
%F a(n) = 3*n*(3*n+1)/2.
%F G.f.: (6*x+3*x^2)/(1-x)^3.
%F a(n) = A000217(3*n); a(2*n) = A144314(n). - _Reinhard Zumkeller_, Sep 17 2008
%F a(n) = 3*A005449(n). - _R. J. Mathar_, Mar 27 2009
%F a(n) = 9*n+a(n-1)-3 for n>0, a(0)=0. - _Vincenzo Librandi_, Aug 08 2010
%F a(n) = A218470(9n+5). - _Philippe Deléham_, Mar 27 2013
%F a(n) = Sum_{k=0..3n} (-1)^(n+k)*k^2. - _Bruno Berselli_, Aug 29 2013
%F E.g.f.: 3*exp(x)*x*(4 + 3*x)/2. - _Stefano Spezia_, Jun 06 2021
%F From _Amiram Eldar_, Aug 11 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 2 - Pi/(3*sqrt(3)) - log(3).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/(3*sqrt(3)) + 4*log(2)/3 - 2. (End)
%F From _Leo Tavares_, Nov 23 2023: (Start)
%F a(n) = 3*A000217(n) + 3*A000290(n).
%F a(n) = A003154(n+1) - A133694(n+1). (End)
%e a(1)=9*1+0-3=6, a(2)=9*2+6-3=21, a(3)=9*3+21-3=45.
%e For n=3, a(3) = -0^2+1^2-2^2+3^2-4^2+5^2-6^2+7^2-8^2+9^2 = 45.
%p seq(binomial(3*n+1,2), n=0..45); # _Zerinvary Lajos_, Jan 21 2007
%t LinearRecurrence[{3,-3,1},{0,6,21},50] (* _Harvey P. Dale_, Aug 29 2015 *)
%o (PARI) a(n)=3*n*(3*n+1)/2 \\ _Charles R Greathouse IV_, Jun 17 2017
%o (GAP) List([0..50],n->Binomial(3*n+1,2)); # _Muniru A Asiru_, Feb 28 2019
%Y Cf. A000217, A000290, A005449, A014105, A022266, A033585, A062725, A144312, A144314, A218470.
%Y Cf. A003154, A133694.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Mar 15 2003
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