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A032438 a(n) = n^2 - floor((n+1)/2)^2. 4
0, 0, 3, 5, 12, 16, 27, 33, 48, 56, 75, 85, 108, 120, 147, 161, 192, 208, 243, 261, 300, 320, 363, 385, 432, 456, 507, 533, 588, 616, 675, 705, 768, 800, 867, 901, 972, 1008, 1083, 1121, 1200, 1240, 1323, 1365, 1452, 1496, 1587, 1633, 1728, 1776, 1875, 1925 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The answer to a question from Mike and Laurie Crain (2crains(AT)concentric.net): how many even numbers are there in an n X n multiplication table starting at 1 X 1?
a(n+1) is the number of pairs (x,y) with x and y in {0,...,n}, x and y of the same parity, and x+y >= n. - Clark Kimberling, Jul 02 2012
From J. M. Bergot, Aug 08 2013: (Start)
Define a triangle to have T(1,1)=0 and T(n,c) = n^2 - c^2. The difference of the sum of the terms in antidiagonal(n+1) and those in antidiagonal(n)=a(n).
Column 0 is vertical and T(n,n)=0. The first few rows are 0; 3,0; 8,5,0; 15,12,7,0; 24,21,16,9,0; 35,32,27,20,11,0; the first few antidiagonals are 0; 3; 8,0; 15,5; 24,12,0; 35,21,7; 48,32,16,0; the first few sum of terms in the antidiagonals are 0, 3, 8, 20, 36, 63, 96, 144, 200, 275, 360, 468, 588, 735, 896, 1088, 1296, 1539. (End)
Sum of the largest parts in the partitions of 2n into two distinct odd parts. For example, a(5) = 16; the partitions of 2(5) = 10 into two distinct odd parts are (9,1) and (7,3). The sum of the largest parts is then 9+7 = 16. - Wesley Ivan Hurt, Nov 27 2017
LINKS
FORMULA
a(n) = n^2 - A008794(n+1).
G.f.: x^2*(x^2 + 2*x + 3)/(1-x^2)^2/(1-x). - Ralf Stephan, Jun 10 2003
a(n) = (1/8)*(2*n*(3*n-1)+(2*n+1)*(-1)^n-1). a(-n-1) = A014255(n). - Bruno Berselli, Sep 27 2011
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n > 4. - Harvey P. Dale, Nov 24 2011
E.g.f.: (x*(1 + 3*x)*cosh(x) + (3*x^2 + 3*x - 1)*sinh(x))/4. - Stefano Spezia, Aug 01 2022
MAPLE
A032438:=n->n^2-floor((n+1)/2)^2; seq(A032438(n), n=0..100) # Wesley Ivan Hurt, Nov 25 2013
MATHEMATICA
Table[n^2-Floor[((n+1)/2)]^2, {n, 0, 50}] (* or *) LinearRecurrence[ {1, 2, -2, -1, 1}, {0, 0, 3, 5, 12}, 51]
PROG
(Magma) [n^2-Floor( (n+1)/2 )^2 : n in [0..60]]; // Vincenzo Librandi, Sep 27 2011
(PARI) a(n)=n^2 - ((n+1)\2)^2 \\ Charles R Greathouse IV, Feb 19 2017
CROSSREFS
First differences are in A059029, partial sums in A143785.
Sequence in context: A269928 A057587 A213036 * A025083 A203150 A237351
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved

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Last modified February 26 11:46 EST 2024. Contains 370352 sequences. (Running on oeis4.)