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A023855 a(n) = 1*(n) + 2*(n-1) + 3*(n-2) + ... + (n+1-k)*k, where k = floor((n+1)/2). 11
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..ceil(n/2)} i*(n-i+1) = -ceil(n/2)*(ceil(n/2)+1)*(2*ceil(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

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)

(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

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Last modified August 22 12:39 EDT 2017. Contains 290946 sequences.