

A093005


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


21



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.
Interleaves A000384 and A001105.  Paul Barry, Jun 29 2006
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
Similar to generalized hexagonal numbers A000217. Other members of this family are A210977, A006578, A210978, A181995, A210981, A210982.  Omar E. Pol, Aug 09 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

Harvey P. Dale, Table of n, a(n) for n = 1..1000
T. Kløve, Linear recurring sequences in boolean rings, Math. Scand., 33 (1973), 512.
T. Kløve, Linear recurring sequences in boolean rings, Math. Scand., 33 (1973), 512. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (1,2,2,1,1).
Luc Rousseau, Illustration, a(n) viewed as a number of regions in an arrangement of lines / of circles


FORMULA

a(2*n1) = n*(2*n1), a(2*n) = 2*n^2.
From Paul Barry, Jun 29 2006; (Start)
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)/4n*(1)^n/4. (End)
a(n) = n * ceiling(n/2).  Wesley Ivan Hurt, Nov 14 2013


MAPLE

A093005:=n>n*ceil(n/2); seq(A093005(n), n=1..100); # Wesley Ivan Hurt, Nov 14 2013


MATHEMATICA

a[n_Integer] := n*Floor[(n + 1)/2] (* Olivier Gérard, Jun 21 2007 *)
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):
. print n*((n+1)//2), # Alex Ratushnyak, Apr 26 2012
(PARI) a(n)=(n+1)\2*n \\ Charles R Greathouse IV, Jun 11 2015


CROSSREFS

Cf. A008619 (a(n)/n), A183207.
Sequence in context: A029933 A228366 A128913 * A049818 A066189 A278834
Adjacent sequences: A093002 A093003 A093004 * A093006 A093007 A093008


KEYWORD

nonn,easy


AUTHOR

Amarnath Murthy, Mar 29 2004


EXTENSIONS

Corrected and extended by Joshua Zucker, May 08 2006
New name from Alex Ratushnyak, Apr 26 2012
New name from Wesley Ivan Hurt, Nov 14 2013


STATUS

approved



