A214604 Odd numbers by transposing the right half of A176271, triangle read by rows: T(n,k) = A176271(n - 1 + k, n), 1 <= k <= n.

1, 5, 9, 11, 17, 25, 19, 27, 37, 49, 29, 39, 51, 65, 81, 41, 53, 67, 83, 101, 121, 55, 69, 85, 103, 123, 145, 169, 71, 87, 105, 125, 147, 171, 197, 225, 89, 107, 127, 149, 173, 199, 227, 257, 289, 109, 129, 151, 175, 201, 229, 259, 291, 325, 361, 131, 153, 177, 203, 231, 261, 293, 327, 363, 401, 441

Offset

1,2

Links

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

Formula

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

Sum_{k=1..n} T(n, k) = A214659(n).

T(2*n-1, n) = A214660(n) (main diagonal).

T(n, 1) = A028387(n-1).

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

T(n, k) = A214661(n,k) + 2*A025581(n,k).

T(n, k) = 2*A000290(A094727(n,k)) - A214661(n,k).

Example

.     Take the first n elements of the n-th diagonal (northeast to
.     southwest) 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:              _   5                  5  9
. 3:            _   9  11               11 17 25
. 4:         __  __  17  19             19 27 37 49
. 5:       __  __  25  27  29           29 39 51 65 ..
. 6:     __  __  __  37  39  41         41 53 67 .. .. ..
. 7:   __  __  __  49  51  53  55       55 69 .. .. .. .. ..
. 8: __  __  __  __  65  67  69  71     71 .. .. .. .. .. .. .. .

Mathematica

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

Prog

(Haskell)
import Data.List (transpose)
a214604 n k = a214604_tabl !! (n-1) !! (k-1)
a214604_row n = a214604_tabl !! (n-1)
a214604_tabl = zipWith take [1..] $ transpose a176271_tabl
(Magma) [(n+k)^2-n-3*k+1: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 10 2024
(SageMath) flatten([[(n+k)^2-n-3*k+1 for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Mar 10 2024

Crossrefs

Cf. A000290, A016754, A028387, A025581, A094727, A176271.

Cf. A214659 (row sums), A214660 (main diagonal), A214661.

Sequence in context: A314601 A314602 A314603 * A101284 A362156 A102182

Adjacent sequences: A214601 A214602 A214603 * A214605 A214606 A214607

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Jul 25 2012

Status

approved