A155118 Array T(n,k) read by antidiagonals: the k-th term of the n-th iterated differences of A140429.

0, 1, 1, 1, 2, 3, 3, 4, 6, 9, 5, 8, 12, 18, 27, 11, 16, 24, 36, 54, 81, 21, 32, 48, 72, 108, 162, 243, 43, 64, 96, 144, 216, 324, 486, 729, 85, 128, 192, 288, 432, 648, 972, 1458, 2187, 171, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 341, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683

Offset: 0

Author

Paul Curtz, Jan 20 2009

Keywords

• nonn
• tabl
• easy

Comments

Deleting column k=0 and reading by antidiagonals yields A036561.

Deleting column k=0 and reading the antidiagonals downwards yields A175840.

Examples

			The array starts in row n=0 with columns k>=0 as:
   0   1    3    9    27    81    243    729    2187  ... A140429;
   1   2    6   18    54   162    486   1458    4374  ... A025192;
   1   4   12   36   108   324    972   2916    8748  ... A003946;
   3   8   24   72   216   648   1944   5832   17496  ... A080923;
   5  16   48  144   432  1296   3888  11664   34992  ... A257970;
  11  32   96  288   864  2592   7776  23328   69984  ...
  21  64  192  576  1728  5184  15552  46656  139968  ...
Antidiagonal triangle begins as:
   0;
   1,   1;
   1,   2,   3;
   3,   4,   6,   9;
   5,   8,  12,  18,  27;
  11,  16,  24,  36,  54,  81;
  21,  32,  48,  72, 108, 162, 243;
  43,  64,  96, 144, 216, 324, 486, 729;
  85, 128, 192, 288, 432, 648, 972, 1458, 2187; - _G. C. Greubel_, Mar 25 2021
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A001045, A004054, A046936.

Programs

Magma

t:= func< n,k | k eq 0 select (2^(n-k) -(-1)^(n-k))/3 else 2^(n-k)*3^(k-1) >;
[t(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 25 2021

Maple

T:=proc(n,k)if(k>0)then return 2^n*3^(k-1):else return (2^n - (-1)^n)/3:fi:end:
for d from 0 to 8 do for m from 0 to d do print(T(d-m,m)):od:od: # Nathaniel Johnston, Apr 13 2011

Mathematica

t[n_, k_]:= If[k==0, (2^(n-k) -(-1)^(n-k))/3, 2^(n-k)*3^(k-1)];
Table[t[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 25 2021 *)

Sage

def A155118(n,k): return (2^(n-k) -(-1)^(n-k))/3 if k==0 else 2^(n-k)*3^(k-1)
flatten([[A155118(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 25 2021

Formula

For the square array:

T(n,k) = 2^n*3^(k-1), k>0.

T(n,k) = T(n-1,k+1) - T(n-1,k), n>0.

Rows:

T(0,k) = A140429(k) = A000244(k-1).

T(1,k) = A025192(k).

T(2,k) = A003946(k).

T(3,k) = A080923(k+1).

T(4,k) = A257970(k+3).

Columns:

T(n,0) = A001045(n) (Jacobsthal numbers J_{n}).

T(n,1) = A000079(n).

T(n,2) = A007283(n).

T(n,3) = A005010(n).

T(n,4) = A175806(n).

T(0,k) - T(k+1,0) = 4*A094705(k-2).

From G. C. Greubel, Mar 25 2021: (Start)

For the antidiagonal triangle:

t(n, k) = T(n-k, k).

t(n, k) = (2^(n-k) - (-1)^(n-k))/3 (J_{n-k}) if k = 0 else 2^(n-k)*3^(k-1).

Sum_{k=0..n} t(n, k) = 3^n - J_{n+1}, where J_{n} = A001045(n).

Sum_{k=0..n} t(n, k) = A004054(n-1) for n >= 1. (End)

Extensions

a(22) - a(57) from Nathaniel Johnston, Apr 13 2011