

A093005


a(n) = n * ceiling(n/2).


27



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

1,2


COMMENTS

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 nth 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(thetak*2*Pi/n), k=0..n1; or, by inversion, number of bounded regions in the arrangement of circles with radius 1 and centers the nth roots of unity.  Luc Rousseau, Feb 08 2019


LINKS



FORMULA

a(n) = n*floor((n+1)/2).
a(2*n1) = n*(2*n1), a(2*n) = 2*n^2.
G.f.: x*(1+x+2*x^2)/((1+x)^2*(1x)^3).
a(n) = a(n1) + 2*a(n2)  2*a(n3)  a(n4) + a(n5).
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


MAPLE



MATHEMATICA

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 *)


PROG

(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


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



