A152420 Irregular triangle read by rows: T(n,k) = n*(n-2) - (k-n)*(k-n-2), with 0 <= k <= 2*n.

0, -4, -1, 0, -8, -3, 0, 1, 0, -12, -5, 0, 3, 4, 3, 0, -16, -7, 0, 5, 8, 9, 8, 5, 0, -20, -9, 0, 7, 12, 15, 16, 15, 12, 7, 0, -24, -11, 0, 9, 16, 21, 24, 25, 24, 21, 16, 9, 0, -28, -13, 0, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 0, -32, -15, 0, 13, 24, 33, 40, 45, 48, 49, 48, 45, 40, 33, 24, 13, 0

Offset

0,2

Comments

The row sums are: {0, -5, -10, -7, 12, 55, 130, 245, 408, 627, 910, ...}.

Links

G. C. Greubel, Rows n = 0..50 of triangle, flattened

Formula

T(n, k) = 4*( n*(n-1) - k*(k-1) ) = 4*( (n-k)*(n+k-1) ) with n and k ranging over half-integer steps.

T(n, k) = n*(n-2) - (k-n)*(k-n-2), with 0 <= k <= 2*n, n >= 0. - G. C. Greubel, Dec 04 2019

Example

Irregular triangle begins as:
    0;
   -4,  -1, 0;
   -8,  -3, 0,  1,  0;
  -12,  -5, 0,  3,  4,  3,  0;
  -16,  -7, 0,  5,  8,  9,  8,  5,  0;
  -20,  -9, 0,  7, 12, 15, 16, 15, 12,  7,  0;
  -24, -11, 0,  9, 16, 21, 24, 25, 24, 21, 16,  9,  0;
  -28, -13, 0, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 0;
  -32, -15, 0, 13, 24, 33, 40, 45, 48, 49, 48, 45, 40, 33, 24, 13, 0;

Maple

seq(seq( n*(n-2) - (k-n)*(k-n-2), k=0..2*n), n=0..10); # G. C. Greubel, Dec 04 2019

Mathematica

Table[4*(n*(n-1) - k*(k-1)), {n, 0, 5, 1/2}, {k, -n, n, 1/2}]//FlattenTable[n*(n-2) - (k-n)*(k-n-2), {n, 0, 5}, {k, 0, 2*n}]//Flatten (* G. C. Greubel, Dec 04 2019 *)

Prog

(PARI) T(n, k) = n*(n-2) - (k-n)*(k-n-2); \\ G. C. Greubel, Dec 04 2019
(Magma) [n*(n-2) - (k-n)*(k-n-2): k in [0..2*n], n in [0..10]]; // G. C. Greubel, Dec 04 2019
(Sage) [[n*(n-2) - (k-n)*(k-n-2) for k in (0..2*n)] for n in (0..10)] # G. C. Greubel, Dec 04 2019
(GAP) Flat(List([0..10], n-> List([0..2*n], k-> n*(n-2) - (k-n)*(k-n-2) ))); # G. C. Greubel, Dec 04 2019

Crossrefs

Cf. A152439.

Sequence in context: A245099 A176219 A231350 * A050465 A134575 A095831

Adjacent sequences: A152417 A152418 A152419 * A152421 A152422 A152423

Keyword

sign,tabf

Author

Roger L. Bagula and Gary W. Adamson, Dec 03 2008

Extensions

Keyword tabf by Michel Marcus, Apr 08 2013

New name from G. C. Greubel, Dec 04 2019

Status

approved