A214661 Odd numbers obtained by transposing the left half of A176271 into rows of a triangle: T(n,k) = A176271(n - 1 + k, k), 1 <= k <= n.

1, 3, 9, 7, 15, 25, 13, 23, 35, 49, 21, 33, 47, 63, 81, 31, 45, 61, 79, 99, 121, 43, 59, 77, 97, 119, 143, 169, 57, 75, 95, 117, 141, 167, 195, 225, 73, 93, 115, 139, 165, 193, 223, 255, 289, 91, 113, 137, 163, 191, 221, 253, 287, 323, 361, 111, 135, 161, 189, 219, 251, 285, 321, 359, 399, 441

Offset

1,2

Links

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

Formula

T(n, k) = (n+k)^2 - 3*n - k + 1.

T(n,k) = A176271(n+k-1, k).

T(n, k) = A214604(n,k) - 2*A025581(n,k).

T(n, k) = 2*A000290(A094727(n,k)) - A214604(n,k).

T(2*n-1, n) = A214675() (main diagonal).

T(n,1) = A002061(n).

T(n,n) = A016754(n-1).

Sum_{k=1..n} T(n, k) = A051673(n) (row sums).

Example

.     Take the first n elements of the n-th diagonal (northwest to
.     southeast) of the triangle on the left side
.     and write this as n-th row on the triangle of the right side.
. 1:                1                    1
. 2:              3   _                  3  9
. 3:            7   9  __                7 15 25
. 4:         13  15  __  __             13 23 35 49
. 5:       21  23  25  __  __           21 33 47 63 ..
. 6:     31  33  35  __  __  __         31 45 61 .. .. ..
. 7:   43  45  47  49  __  __  __       43 59 .. .. .. .. ..
. 8: 57  59  61  63  __  __  __  __     57 .. .. .. .. .. .. .. .

Mathematica

Table[(n+k)^2-3*n-k+1, {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Mar 10 2024 *)

Prog

(Haskell)
import Data.List (transpose)
a214661 n k = a214661_tabl !! (n-1) !! (k-1)
a214661_row n = a214661_tabl !! (n-1)
a214661_tabl = zipWith take [1..] $ transpose $ map reverse a176271_tabl
(Magma) [(n+k)^2-3*n-k+1: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 10 2024
(SageMath) flatten([[(n+k)^2-3*n-k+1 for k in range(1, n+1)] for n in range(1, 16)]) // G. C. Greubel, Mar 10 2024

Crossrefs

Cf. A000290, A002061, A016754, A025581, A094727, A176271, A214604.

Cf. A051673 (row sums), A214675 (main diagonal).

Sequence in context: A267363 A146179 A294734 * A178414 A220654 A302158

Adjacent sequences: A214658 A214659 A214660 * A214662 A214663 A214664

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Jul 25 2012

Status

approved