A129234 Triangle read by rows: T(n,k) = n/k + k - 1 if n mod k = 0; otherwise T(n,k)=0 (1 <= k <= n).

1, 2, 2, 3, 0, 3, 4, 3, 0, 4, 5, 0, 0, 0, 5, 6, 4, 4, 0, 0, 6, 7, 0, 0, 0, 0, 0, 7, 8, 5, 0, 5, 0, 0, 0, 8, 9, 0, 5, 0, 0, 0, 0, 0, 9, 10, 6, 0, 0, 6, 0, 0, 0, 0, 10, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 12, 7, 6, 6, 0, 7, 0, 0, 0, 0, 0, 12, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 14, 8, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 14

Offset

1,2

Comments

Row sums = A129235: (1, 4, 6, 11, 10, 20, 14, ...). Moebius transform of A129234 = A129236. Inverse Moebius transform of A129234 = A129237.

Links

Table of n, a(n) for n=1..105.

Formula

G.f. = G(t,z) = Sum_{k>=1} t^k*z^k*(k-(k-1)*z^k)/(1-z^k)^2. - Emeric Deutsch, Apr 17 2007

Example

First few rows of the triangle:
  1;
  2, 2;
  3, 0, 3;
  4, 3, 0, 4;
  5, 0, 0, 0, 5;
  6, 4, 4, 0, 0, 6;
  7, 0, 0, 0, 0, 0, 7;
  ...

Maple

T:=proc(n, k) if n mod k = 0 then n/k+k-1 else 0 fi end: for n from 1 to 16 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Apr 17 2007

Mathematica

T[n_, k_]:=If[Mod[n, k]==0, n/k+k-1, 0]; Table[T[n, k], {n, 14}, {k, n}]//Flatten (* James C. McMahon, Jan 17 2025 *)

Prog

(PARI) row(n) = vector(n, k, if (!(n%k), n/k + k - 1, 0)); \\ Michel Marcus, Jan 17 2025

Crossrefs

Cf. A129235, A129236, A129237.

Sequence in context: A103516 A233558 A319929 * A213081 A127446 A046157

Adjacent sequences: A129231 A129232 A129233 * A129235 A129236 A129237

Keyword

nonn,tabl

Author

Gary W. Adamson, Apr 05 2007

Extensions

Edited by Emeric Deutsch, Apr 17 2007

Status

approved