A245524 a(n) = n^2 - floor(n/2)*(-1)^n.
1, 3, 10, 14, 27, 33, 52, 60, 85, 95, 126, 138, 175, 189, 232, 248, 297, 315, 370, 390, 451, 473, 540, 564, 637, 663, 742, 770, 855, 885, 976, 1008, 1105, 1139, 1242, 1278, 1387, 1425, 1540, 1580, 1701, 1743, 1870, 1914, 2047, 2093, 2232, 2280, 2425, 2475
Offset: 1
Examples
a(3) = 10; The partitions of 2*3 = 6 into two parts are: (5,1), (4,2), (3,3). Since 3 is odd, we sum the even numbers from the smallest parts together with the odd numbers from the largest parts to get: (2) + (3+5) = 10. a(4) = 14; The partitions of 4*2 = 8 into two parts are: (7,1), (6,2), (5,3), (4,4). Since 4 is even, we sum the odd numbers from the smallest parts together with the even numbers from the largest parts to get: (1+3) + (4+6) = 14.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- Index entries for sequences related to partitions
- Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
Programs
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Magma
[n^2-Floor(n/2)*(-1)^n : n in [1..50]];
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Maple
A245524:=n->n^2-floor(n/2)*(-1)^n: seq(A245524(n), n=1..50);
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Mathematica
Table[n^2 - Floor[n/2] (-1)^n, {n, 50}] CoefficientList[Series[(1 + 2 x + 5 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Jul 25 2014 *) -
SageMath
[n^2-(-1)^n*(n//2) for n in range(1,71)] # G. C. Greubel, Mar 14 2024
Formula
a(n) = (4*n^2 - 1 - (2*n - 1)*(-1)^n) / 4.
G.f.: x*(1+2*x+5*x^2)/((1+x)^2*(1-x)^3). - Robert Israel, Jul 25 2014
a(n) = (2*n-1)*(n-floor(n/2)). - Wesley Ivan Hurt, Jan 10 2017
a(n) = n^2 + Sum_{k=1..n-1} (-1)^k*k for n>1. Example: for n=5, a(5) = 5^2 + (4 - 3 + 2 - 1) = 27. - Bruno Berselli, May 23 2018
E.g.f.: (1/2)*(x*(2*x+3)*cosh(x) + (2*x^2+x-1)*sinh(x)). - G. C. Greubel, Mar 14 2024
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Wesley Ivan Hurt, Jun 20 2024
Comments