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
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
Programs
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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
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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
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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 *)
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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(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
Comments