A245524 -id:A245524 - cp's OEIS frontend

A128623 Triangle read by rows: A128621 * A000012 as infinite lower triangular matrices.

1, 2, 2, 6, 3, 3, 8, 8, 4, 4, 15, 10, 10, 5, 5, 18, 18, 12, 12, 6, 6, 28, 21, 21, 14, 14, 7, 7, 32, 32, 24, 24, 16, 16, 8, 8, 45, 36, 36, 27, 27, 18, 18, 9, 9, 50, 50, 40, 40, 30, 30, 20, 20, 10, 10, 66, 55, 55, 44, 44, 33, 33, 22, 22, 11, 11, 72, 72, 60, 60, 48, 48, 36, 36, 24, 24, 12, 12, 91, 78, 78, 65, 65, 52, 52, 39, 39, 26, 26, 13, 13

Offset: 1

Gary W. Adamson, Mar 14 2007

			First few rows of the triangle are:
   1;
   2,  2;
   6,  3,  3;
   8,  8,  4,  4;
  15, 10, 10,  5,  5;
  18, 18, 12, 12,  6, 6;
  28, 21, 21, 14, 14, 7, 7;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000027, A093005, A093353, A115514, A128621, A245524.

Cf. A128624 (row sums).

[n*Floor((n-k+2)/2): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 13 2024

Table[n*Floor[(n-k+2)/2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 13 2024 *)

flatten([[n*((n-k+2)//2) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 13 2024

Sum_{k=1..n} T(n, k) = A128624(n) (row sums).
T(n,k) = n*(1+floor((n-k)/2)), 1 <= k <= n. - R. J. Mathar, Jun 27 2012
From G. C. Greubel, Mar 13 2024: (Start)
T(n, k) = n*A115514(n, k).
T(n, k) = Sum_{j=k..n} A128621(n, j).
T(n, 1) = A093005(n).
T(n, 2) = A093353(n-1), n >= 2.
T(n, n) = A000027(n).
T(2*n-1, n) = A245524(n).
Sum_{k=1..n} (-1)^k*T(n, k) = (1/2)*(1-(-1)^n)*A000384(floor((n+1)/2)). (End)

a(41) = 27 inserted and more terms from Georg Fischer, Jun 05 2023

A245534 a(n) = n^2 + floor(n/2)*(-1)^n.

Original entry on oeis.org

1, 5, 8, 18, 23, 39, 46, 68, 77, 105, 116, 150, 163, 203, 218, 264, 281, 333, 352, 410, 431, 495, 518, 588, 613, 689, 716, 798, 827, 915, 946, 1040, 1073, 1173, 1208, 1314, 1351, 1463, 1502, 1620, 1661, 1785, 1828, 1958, 2003, 2139, 2186, 2328, 2377, 2525

Offset: 1

Wesley Ivan Hurt, Jul 25 2014

Consider the partitions of 2n into two parts: When n is odd, a(n) gives the total sum of the odd numbers from the smallest parts and the even numbers from the largest parts of these partitions. When n is even, a(n) gives the total sum of the even numbers from the smallest parts and the odd numbers from the largest parts (see example).

			a(3) = 8; The partitions of 2*3 = 6 into two parts are: (5,1), (4,2), (3,3). Since 3 is odd, we sum the odd numbers from the smallest parts together with the even numbers from the largest parts to get: (1+3) + (4) = 8.
a(4) = 18; 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 even numbers from the smallest parts together with the odd numbers from the largest parts to get: (2+4) + (5+7) = 18.

Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A001057, A000290. See A245524 for a very similar sequence.

Magma
```
[n^2+Floor(n/2)*(-1)^n: n in [1..50]];
```

A245534:=n->n^2+floor(n/2)*(-1)^n: seq(A245534(n), n=1..50);

Mathematica
```
Table[n^2 + Floor[n/2] (-1)^n, {n, 50}]
```

a(n) = n^2 + (n\2)*(-1)^n; \\ Michel Marcus, Aug 06 2014

G.f.: x*(1 + 4*x + x^2 + 2*x^3)/((1 + x)^2*(1 - x)^3).
a(n) = (4*n^2 + 1 + (2*n - 1)*(-1)^n)/4.
a(n) = A000290(n) + A001057(n-1) for n > 0.
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) = 23. - Bruno Berselli, May 23 2018

cp's OEIS Frontend

A128623 Triangle read by rows: A128621 * A000012 as infinite lower triangular matrices.

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