OFFSET
0,5
COMMENTS
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(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
KEYWORD
AUTHOR
Paul Curtz, Jan 20 2009
EXTENSIONS
a(22) - a(57) from Nathaniel Johnston, Apr 13 2011
STATUS
approved