A100345 Triangle read by rows: T(n,k) = n*(n+k), 0 <= k <= n.

0, 1, 2, 4, 6, 8, 9, 12, 15, 18, 16, 20, 24, 28, 32, 25, 30, 35, 40, 45, 50, 36, 42, 48, 54, 60, 66, 72, 49, 56, 63, 70, 77, 84, 91, 98, 64, 72, 80, 88, 96, 104, 112, 120, 128, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190

Offset: 0

Author

Reinhard Zumkeller, Nov 18 2004

Keywords

• nonn
• tabl
• easy

Comments

Distinct members (except 0) are in A071562. Numbers occurring at least twice are in A175040. - Franklin T. Adams-Watters, Apr 04 2010

Examples

			Triangle begins:
   0
   1   2
   4   6   8
   9  12  15  18
  16  20  24  28  32
  25  30  35  40  45  50
  36  42  48  54  60  66  72
  49  56  63  70  77  84  91  98
  64  72  80  88  96 104 112 120 128
		

Crossrefs

Cf. A000290, A000384, A001105, A002378, A005563, A014107, A028347, A028552, A028557, A028560, A028563, A028566, A028569, A046092, A054000, A071355, A071562, A085789, A098603, A175040.

Programs

Mathematica

Table[n(n+k),{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Aug 16 2018 *)

PARI

row(n) = vector(n+1, k, n*(n+k-1)); \\ Amiram Eldar, May 09 2025

Formula

T(n,0) = A000290(n).

T(n,1) = A002378(n) for n > 0.

T(n,2) = A005563(n) for n > 1.

T(n,3) = A028552(n) for n > 2.

T(n,4) = A028347(n+2) for n > 3.

T(n,5) = A028557(n) for n > 4.

T(n,6) = A028560(n) for n > 5.

T(n,7) = A028563(n) for n > 6.

T(n,8) = A028566(n) for n > 7.

T(n,9) = A028569(n) for n > 8.

T(n,10) = A098603(n) for n > 9.

T(n,n-5) = A071355(n-4) for n > 4.

T(n,n-4) = A054000(n-1) for n > 3.

T(n,n-3) = A014107(n) for n > 2.

T(n,n-2) = A046092(n-1) for n > 1.

T(n,n-1) = A000384(n) for n > 0.

T(n,n) = A001105(n).

Row sums give A085789 for n > 0.

G.f.: x*(1 + 2*y + 6*x^3*y^2 - 3*x^2*y*(1 + 2*y) + x*(1 - 3*y + 2*y^2))/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Jul 03 2025