login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A023855 a(n) = 1*(n) + 2*(n-1) + 3*(n-2) + ... + (n+1-k)*k, where k = floor((n+1)/2). 15
1, 2, 7, 10, 22, 28, 50, 60, 95, 110, 161, 182, 252, 280, 372, 408, 525, 570, 715, 770, 946, 1012, 1222, 1300, 1547, 1638, 1925, 2030, 2360, 2480, 2856, 2992, 3417, 3570, 4047, 4218, 4750, 4940, 5530, 5740, 6391, 6622, 7337, 7590, 8372, 8648, 9500, 9800, 10725, 11050 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Given a rectangle of perimeter 2*n one can form rectangles having this perimeter for a number of different rectangles or squares depending on how large 2*n is.  The sequence lists the total areas of all such rectangles for each 2*n. - J. M. Bergot, Sep 14 2011

Conjecture: Antidiagonal sums of triangle A075462. - L. Edson Jeffery, Jan 20 2012

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1)

FORMULA

a(n) = (n+1)*(n+3)*(2*n+1)/24 if n is odd, or n*(n+1)*(n+2)/12 if n is even.

G.f.: x*(1+x+2*x^2)/((1-x)^4*(1+x)^3). - Ralf Stephan, Apr 28 2004

a(n) = Sum_{i=1..ceiling(n/2)} i*(n-i+1) = -ceiling(n/2)*(ceiling(n/2)+1)*(2*ceiling(n/2)-3n-2)/6. - Wesley Ivan Hurt, Sep 19 2013

a(n) = (4*n^3 + 15*n^2 + 14*n + 3 - 3*(n+1)^2*(-1)^n)/48. - Luce ETIENNE, Oct 22 2014

a(n) = (A000292(n) + (n mod 2)*(ceiling(n/2))^2)/2. - Luc Rousseau, Feb 25 2018

MAPLE

seq(-(1/3)*floor((k+1)/2)^3 + (k/2)*floor((k+1)/2)^2 + ((3*k+2)/6)*floor((k+1)/2), k=1..100); # Wesley Ivan Hurt, Sep 18 2013

MATHEMATICA

LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {1, 2, 7, 10, 22, 28, 50}, 50] (* Vincenzo Librandi, Jan 23 2012 *)

Table[-Ceiling[n/2] (Ceiling[n/2] + 1) (2 Ceiling[n/2] - 3 n - 2)/6, {n, 100}] (* Wesley Ivan Hurt, Sep 20 2013 *)

PROG

(PARI) a(n)=if(n%2, (n+1)*(n+3)*(2*n+1)/24, n*(n+1)*(n+2)/12)

(PARI) x='x+O('x^99); Vec(x*(1+x+2*x^2)/((1-x)^4*(1+x)^3)) \\ Altug Alkan, Mar 03 2018

(Haskell)

a023855 n = sum $ zipWith (*) [1 .. div (n+1) 2] [n, n-1 ..]

-- Reinhard Zumkeller, Jan 23 2012

CROSSREFS

Cf. A023856, A023857, A024305, A024854.

Sequence in context: A049830 A270879 A022302 * A191832 A066964 A066967

Adjacent sequences:  A023852 A023853 A023854 * A023856 A023857 A023858

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

EXTENSIONS

Formula, program, and slight revision by Charles R Greathouse IV, Feb 23 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 15 15:10 EST 2019. Contains 329999 sequences. (Running on oeis4.)