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A093353
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a(n) = (n + n mod 2)*(n + 1)/2.
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17
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2, 3, 8, 10, 18, 21, 32, 36, 50, 55, 72, 78, 98, 105, 128, 136, 162, 171, 200, 210, 242, 253, 288, 300, 338, 351, 392, 406, 450, 465, 512, 528, 578, 595, 648, 666, 722, 741, 800, 820, 882, 903, 968, 990, 1058, 1081, 1152, 1176, 1250, 1275, 1352, 1378, 1458
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OFFSET
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1,1
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COMMENTS
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a(n) is the sum of all parts in the integer partitions of n+1 into two parts, see example. - Wesley Ivan Hurt, Jan 26 2013
Also the independence number of the n X n torus grid graph. - Eric W. Weisstein, Sep 06 2017
Also the number of circles we can draw on vertices of an (n+1)-sided regular polygon (using only a compass). - Matej Veselovac, Jan 21 2020
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REFERENCES
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W. R. Hare, S. T. Hedetniemi, R. Laskar, and J. Pfaff, Complete coloring parameters of graphs, Proceedings of the sixteenth Southeastern international conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1985). Congr. Numer., Vol. 48 (1985), pp. 171-178. MR0830709 (87h:05088). See s_m on page 135. - N. J. A. Sloane, Apr 06 2012
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LINKS
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FORMULA
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a(2*n) = a(2*n-1) + n = A014105(n).
a(2*n+1) = a(2*n) + 3*n + 2 = A001105(n+1).
G.f.: x*(2+x+x^2)/((1-x)^3*(1+x)^2).
a(n) = (n+1)*(2*n+1-(-1)^n)/4. - Paul Barry, Mar 31 2008
a(n) = Sum_{i=1..floor((n+1)/2)} i + Sum_{i=ceiling((n+1)/2)..n} i. - Wesley Ivan Hurt, Jun 08 2013
Sum_{n>=1} 1/a(n) = Pi^2/12 + 2*(1-log(2)).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - 2*(1-log(2)). (End)
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EXAMPLE
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a(1) = 2 since 2 = (1+1) and the sum of the first and second parts in the partition is 2; a(2) = 3 since 3 = (1+2) and the sum of these parts is 3; a(3) = 8 since 4 = (1+3) = (2+2) and the sum of all the parts is 8. - Wesley Ivan Hurt, Jan 26 2013
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MAPLE
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a:=n->(n+1)*floor((n+1)/2); seq(a(k), k = 1..70);
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MATHEMATICA
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Table[(n+Mod[n, 2])*(n+1)/2, {n, 60}]
LinearRecurrence[{1, 2, -2, -1, 1}, {2, 3, 8, 10, 18}, 60]
Module[{nn = 60, ab}, ab = Transpose[ Partition[ Accumulate[ Range[nn]], 2]]; Flatten[ Transpose[ {ab[[1]] + Range[nn/2], ab[[2]]}]]]
(* End *)
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PROG
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(SageMath) [(n+1)*int((n+1)//2) for n in range(1, 71)] # G. C. Greubel, Mar 14 2024
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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