A118413 Triangle read by rows: T(n,k) = (2*n-1)*2^(k-1), 0

1, 3, 6, 5, 10, 20, 7, 14, 28, 56, 9, 18, 36, 72, 144, 11, 22, 44, 88, 176, 352, 13, 26, 52, 104, 208, 416, 832, 15, 30, 60, 120, 240, 480, 960, 1920, 17, 34, 68, 136, 272, 544, 1088, 2176, 4352, 19, 38, 76, 152, 304, 608, 1216, 2432, 4864, 9728, 21, 42, 84, 168

Offset: 1

Author

Reinhard Zumkeller, Apr 27 2006

Keywords

• nonn
• tabl
• easy

Comments

Central terms give A118415; row sums give A118414;

T(n,1) = A005408(n-1);

T(n,2) = A016825(n-1) for n>1;

T(n,3) = A017113(n-1) for n>2;

T(n,4) = A051062(n-1) for n>3;

T(n,n-2) = A052951(n-1) for n>2;

T(n,n) = A014480(n-1) = A118416(n,n);

A001511(T(n,k)) = A002260(n,k);

A003602(T(n,k)) = A002024(n,k).

G.f.: x*y*(1 + x + 2*x*y - 6*x^2*y)/((1 - x)^2*(1 - 2*x*y)^2). - Stefano Spezia, Dec 22 2024

Examples

			   1
   3   6
   5  10  20
   7  14  28  56
   9  18  36  72 144
  11  22  44  88 176 352
  13  26  52 104 208 416  832
  15  30  60 120 240 480  960 1920
  17  34  68 136 272 544 1088 2176 4352
  19  38  76 152 304 608 1216 2432 4864 9728
  ...
		

Crossrefs

Cf. A001511, A002024, A002260, A003602, A005408, A014480, A016825, A017113, A051062, A052951, A117303, A118414, A118415, A118416, A117303.

Programs

Mathematica

Select[Flatten[Table[(2n-1)2^(k-1),{n,20},{k,0,n}]],IntegerQ] (* Harvey P. Dale, Jan 17 2024 *)

Python

from math import isqrt, comb
def A118413(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    return ((a<<1)-1)<Chai Wah Wu, Jun 20 2025