A093005 a(n) = n * ceiling(n/2).

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

Offset: 1

Author

Amarnath Murthy, Mar 29 2004

Keywords

• nonn
• easy

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 n-th 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(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

Numbers k such that floor(sqrt(2k)+1/2) | k. - Wesley Ivan Hurt, Dec 01 2020

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Torleiv Kløve, Linear recurring sequences in boolean rings, Math. Scand., 33 (1973), 5-12.
• Torleiv Kløve, Linear recurring sequences in boolean rings, Math. Scand., 33 (1973), 5-12. (Annotated scanned copy)
• Luc Rousseau, Illustration, a(n) viewed as a number of regions in an arrangement of lines / of circles.
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A000217, A000384, A001105, A006578, A008619 (a(n)/n), A181995.

Cf. A183207, A210977, A210978, A210981, A210982, A370980.

Programs

Magma

[n*(n+(n mod 2))/2: n in [1..70]]; // G. C. Greubel, Mar 14 2024

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 *)

PARI

a(n)=(n+1)\2*n \\ Charles R Greathouse IV, Jun 11 2015

Python

for n in range(1,55):
    print(n*((n+1)//2), end=",")   # Alex Ratushnyak, Apr 26 2012

SageMath

[n*(n +(n%2))/2 for n in range(1,71)] # G. C. Greubel, Mar 14 2024

Formula

a(n) = n*floor((n+1)/2).

a(n) = n*A008619(n).

a(2*n-1) = n*(2*n-1), a(2*n) = 2*n^2.

From Paul Barry, Jun 29 2006: (Start)

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)

a(n) = n * ceiling(n/2). - Wesley Ivan Hurt, Nov 14 2013

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

a(n) = A183207(n) - A370980(n) + 1, by Euler's formula. - Scott R. Shannon, Jul 07 2024

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