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A082109
Third row of number array A082105.
12
1, 13, 33, 61, 97, 141, 193, 253, 321, 397, 481, 573, 673, 781, 897, 1021, 1153, 1293, 1441, 1597, 1761, 1933, 2113, 2301, 2497, 2701, 2913, 3133, 3361, 3597, 3841, 4093, 4353, 4621, 4897, 5181, 5473, 5773, 6081, 6397, 6721, 7053, 7393, 7741, 8097, 8461
OFFSET
0,2
COMMENTS
Define b(n) = A000217(n), the triangular numbers. Using six consecutive terms to create the vertices of a triangle at points (b(n-2), b(n-1)), (b(n), b(n+1)), and (b(n+2), b(n+3)), one fourth the area of these triangles = a(n). - J. M. Bergot, Jul 30 2013
LINKS
Takao Komatsu, Ritika Goel, and Neha Gupta, The Frobenius number for the triple of the 2-step star numbers, arXiv:2409.14788 [math.CO], 2024. See p. 2.
FORMULA
a(n) = 4*n^2 + 8*n + 1.
a(n) = a(n-1) + 8*n + 4, with a(0)=1. - Vincenzo Librandi, Aug 08 2010
G.f.: (1 + 10*x - 3*x^2)/(1-x)^3. - Bruno Berselli, Apr 18 2011
E.g.f.: (1 + 12*x + 4*x^2)*exp(x). - G. C. Greubel, Dec 22 2022
From Amiram Eldar, Jan 18 2023: (Start)
Sum_{n>=0} 1/a(n) = 1/6 - cot(sqrt(3)*Pi/2)*sqrt(3)*Pi/12.
Sum_{n>=0} (-1)^n/a(n) = cosec(sqrt(3)*Pi/2)*sqrt(3)*Pi/12 - 1/6. (End)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {1, 13, 33}, 51] (* Vladimir Joseph Stephan Orlovsky, Oct 25 2008 *)
PROG
(PARI) a(n)=4*n^2+8*n+1 \\ Charles R Greathouse IV, Jun 17 2017
(Magma) [4*n^2+8*n+1: n in [0..60]]; // G. C. Greubel, Dec 22 2022
(SageMath) [4*n^2+8*n+1 for n in range(61)] # G. C. Greubel, Dec 22 2022
CROSSREFS
Column 2 of array A188646.
Sequence in context: A050659 A123161 A146052 * A024839 A146177 A146194
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 03 2003
STATUS
approved