A079297 Triangle read by rows: the k-th column is an arithmetic progression with difference 2k-1 and the top entry is the hexagonal number k*(2*k-1) (A000384).

1, 2, 6, 3, 9, 15, 4, 12, 20, 28, 5, 15, 25, 35, 45, 6, 18, 30, 42, 54, 66, 7, 21, 35, 49, 63, 77, 91, 8, 24, 40, 56, 72, 88, 104, 120, 9, 27, 45, 63, 81, 99, 117, 135, 153, 10, 30, 50, 70, 90, 110, 130, 150, 170, 190, 11, 33, 55, 77, 99, 121, 143, 165, 187, 209, 231, 12

Offset

1,2

Comments

The n-th row consists of the odd multiples of n from n*1 to n*(2n-1).

References

R. Honsberger, Ingenuity in Mathematics, Random House, 1970, p. 88.

Links

Table of n, a(n) for n=1..67.

Formula

n-th row adds to n^3.

T(n, k) = n*(2*k - 1) for 1 <= k <= n.

Sum_{k=1..n} (-1)^(n-k) * T(n,k) = n^2. - Saurabh Kumar, Feb 05 2026

G.f.: x*(1 - y - 3*x*y - 3*x*y^2 + 6*x^2*y^2)/((1 - x)^2*(1 - x*y)^3). - Stefano Spezia, Feb 08 2026

Example

Triangle begins:
  1
  2 6
  3 9 15
  4 12 20 28
  5 15 25 35 45
  6 18 30 42 54 66

Mathematica

Flatten[Table[n(2k-1), {n, 1, 12}, {k, 1, n}]]

Crossrefs

Cf. A000578 (row sums), A000290.

Sequence in context: A308541 A136190 A378822 * A276941 A143219 A109465

Adjacent sequences: A079294 A079295 A079296 * A079298 A079299 A079300

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Mar 04 2003

Extensions

More terms from Dean Hickerson, Apr 06 2003

Status

approved