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A155118 Array T(n,k) read by antidiagonals: the k-th term of the n-th iterated differences of A140429. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

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

LINKS

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

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)

EXAMPLE

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

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 *)

PROG

(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

(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

CROSSREFS

Cf. A001045, A004054, A046936.

Sequence in context: A218947 A017818 A228362 * A091275 A046936 A187067

Adjacent sequences:  A155115 A155116 A155117 * A155119 A155120 A155121

KEYWORD

nonn,tabl,easy

AUTHOR

Paul Curtz, Jan 20 2009

EXTENSIONS

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

STATUS

approved

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Last modified April 18 09:06 EDT 2021. Contains 343087 sequences. (Running on oeis4.)