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A093005
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a(n) = n * ceiling(n/2).
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29
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1, 2, 6, 8, 15, 18, 28, 32, 45, 50, 66, 72, 91, 98, 120, 128, 153, 162, 190, 200, 231, 242, 276, 288, 325, 338, 378, 392, 435, 450, 496, 512, 561, 578, 630, 648, 703, 722, 780, 800, 861, 882, 946, 968, 1035, 1058, 1128, 1152, 1225, 1250, 1326, 1352, 1431, 1458
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OFFSET
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1,2
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COMMENTS
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Old name was: The lone multiple of n among the next n numbers.
Another old name: a(n) = n*floor((n+1)/2).
Consider the triangle
1
2 3
4 5 6
7 8 9 10
11 12 13 14 15
16 17 18 19 20 21
22 23 24 25 26 27 28
... Then sequence contains the multiple of n in the n-th row.
Termwise products of the natural numbers and the natural numbers repeated.
Number of pairs (x,y) having the same parity, with x in {0,...,n} and y in {0,...,2n}. - Clark Kimberling, Jul 02 2012
For even n, a(n) gives the sum of all the parts in the partitions of n into exactly two parts. For odd n>1, a(n) gives n plus the sum of all the parts in the partitions of n into exactly two parts. - Wesley Ivan Hurt, Nov 14 2013
Number of regions of the plane that do not contain the origin, in the arrangement of lines with polar equations rho = 1/cos(theta-k*2*Pi/n), k=0..n-1; or, by inversion, number of bounded regions in the arrangement of circles with radius 1 and centers the n-th roots of unity. - Luc Rousseau, Feb 08 2019
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LINKS
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FORMULA
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a(n) = n*floor((n+1)/2).
a(2*n-1) = n*(2*n-1), a(2*n) = 2*n^2.
G.f.: x*(1+x+2*x^2)/((1+x)^2*(1-x)^3).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = n*((2*n+1) - (-1)^n)/4. (End)
E.g.f.: (1/2)*x*( (x+2)*cosh(x) + (x+1)*sinh(x) ). - G. C. Greubel, Mar 14 2024
Sum_{n>=1} 1/a(n) = Pi^2/12 + 2*log(2). - Amiram Eldar, Mar 15 2024
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MAPLE
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MATHEMATICA
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Table[n*Ceiling[n/2], {n, 60}] (* or *) LinearRecurrence[{1, 2, -2, -1, 1}, {1, 2, 6, 8, 15}, 60] (* Harvey P. Dale, May 08 2014 *)
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PROG
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(Python)
for n in range(1, 55):
(Magma) [n*(n+(n mod 2))/2: n in [1..70]]; // G. C. Greubel, Mar 14 2024
(SageMath) [n*(n +(n%2))/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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EXTENSIONS
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STATUS
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approved
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