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A081266
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Staggered diagonal of triangular spiral in A051682.
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18
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0, 6, 21, 45, 78, 120, 171, 231, 300, 378, 465, 561, 666, 780, 903, 1035, 1176, 1326, 1485, 1653, 1830, 2016, 2211, 2415, 2628, 2850, 3081, 3321, 3570, 3828, 4095, 4371, 4656, 4950, 5253, 5565, 5886, 6216, 6555, 6903, 7260, 7626, 8001, 8385, 8778, 9180
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OFFSET
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0,2
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COMMENTS
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Staggered diagonal of triangular spiral in A051682, between (0,4,17) spoke and (0,7,23) spoke.
Binomial transform of (0, 6, 9, 0, 0, 0, ...).
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
a(n) = A000217(3*n); a(2*n) = A144314(n). - Reinhard Zumkeller, Sep 17 2008
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LINKS
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Muniru A Asiru, Table of n, a(n) for n = 0..10000
Milan Janjic, Two Enumerative Functions
M. Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = 6*C(n,1) + 9*C(n,2).
a(n) = 3*n*(3*n+1)/2.
G.f.: (6*x+3*x^2)/(1-x)^3.
a(n) = 3*A005449(n). - R. J. Mathar, Mar 27 2009
a(n) = 9*n+a(n-1)-3 for n>0, a(0)=0. - Vincenzo Librandi, Aug 08 2010
a(n) = A218470(9n+5). - Philippe Deléham, Mar 27 2013
a(n) = Sum_{k=0..3n} (-1)^(n+k)*k^2. - Bruno Berselli, Aug 29 2013
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EXAMPLE
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a(1)=9*1+0-3=6, a(2)=9*2+6-3=21, a(3)=9*3+21-3=45.
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.
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MAPLE
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seq(binomial(3*n+1, 2), n=0..45); # Zerinvary Lajos, Jan 21 2007
a:=n->sum(j, j=0..n): seq(a(3*n), n=0..45); # Zerinvary Lajos, Apr 30 2007
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MATHEMATICA
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s=0; lst={s}; Do[s+=n++ +6; AppendTo[lst, s], {n, 0, 7!, 9}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
LinearRecurrence[{3, -3, 1}, {0, 6, 21}, 50] (* Harvey P. Dale, Aug 29 2015 *)
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PROG
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(PARI) a(n)=3*n*(3*n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017
(GAP) List([0..50], n->Binomial(3*n+1, 2)); # Muniru A Asiru, Feb 28 2019
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CROSSREFS
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Cf. A000290, A014105, A022266, A033585, A062725, A144312.
Sequence in context: A180857 A119868 A175729 * A087863 A212656 A051941
Adjacent sequences: A081263 A081264 A081265 * A081267 A081268 A081269
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KEYWORD
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nonn,easy
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AUTHOR
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Paul Barry, Mar 15 2003
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STATUS
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approved
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