A339236 Irregular triangle of incomplete Leonardo numbers read by rows. T(n, k) = 2*(Sum_{j=0..k} binomial(n-j, j)) - 1, for n>=0 and 0<=k<=floor(n/2).

1, 1, 1, 3, 1, 5, 1, 7, 9, 1, 9, 15, 1, 11, 23, 25, 1, 13, 33, 41, 1, 15, 45, 65, 67, 1, 17, 59, 99, 109, 1, 19, 75, 145, 175, 177, 1, 21, 93, 205, 275, 287, 1, 23, 113, 281, 421, 463, 465, 1, 25, 135, 375, 627, 739, 753, 1, 27, 159, 489, 909, 1161, 1217, 1219

Offset

0,4

Links

Table of n, a(n) for n=0..63.

P. Catarino and A. Borges, A Note on Incomplete Leonardo Numbers, INTEGERS 20A (2020) A43.

Formula

T(n, floor(n/2)) = A001595(n).

Example

Triangle begins:
  1;
  1;
  1, 3;
  1, 5;
  1, 7, 9;
  1, 9, 15;
  1, 11, 23, 25;
  1, 13, 33, 41;
  1, 15, 45, 65, 67;
  1, 17, 59, 99, 109;
  ...

Mathematica

T[n_, k_] := 2 * Sum[Binomial[n - j, j], {j, 0, k}] - 1; Table[T[n, k], {n, 0, 14}, {k, 0, Floor[n/2]}] // Flatten (* Amiram Eldar, Nov 28 2020 *)

Prog

(PARI) T(n, k) = 2*sum(j=0, k, binomial(n-j, j)) -1;
row(n) = vector(n\2+1, k, k--; T(n, k));

Crossrefs

Cf. A001595 (Leonardo numbers: right diagonal).

Cf. A000012 (column 0), A005408 (column 1), A027688 (column 2).

Sequence in context: A197943 A318828 A318827 * A029669 A050329 A147005

Adjacent sequences: A339233 A339234 A339235 * A339237 A339238 A339239

Keyword

nonn,tabf

Author

Michel Marcus, Nov 28 2020

Status

approved