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A321373
Array T(n,k) read by antidiagonals where the first row is (-1)^k*A140966(k) and each subsequent row is obtained by adding A001045(k) to the preceding one.
1
2, 2, -1, 2, 0, 3, 2, 1, 4, 1, 2, 2, 5, 4, 7, 2, 3, 6, 7, 12, 9, 2, 4, 7, 10, 17, 20, 23, 2, 5, 8, 13, 22, 31, 44, 41, 2, 6, 9, 16, 27, 42, 65, 84, 87, 2, 7, 10, 19, 32, 53, 86, 127, 172, 169, 2, 8, 11, 22, 37, 64, 107, 170, 257, 340, 343
OFFSET
0,1
COMMENTS
Array:
2, -1, 3, 1, 7, 9, 23, 41, 87, ... = (-1)^n*A140966(n)
2, 0, 4, 4, 12, 20, 44, 84, 172, ... = abs(A084247(n+1))
2, 1, 5, 7, 17, 31, 65, 127, 257, ... = A014551(n)
2, 2, 6, 10, 22, 42, 86, 170, 342, ... = A078008(n+2) = A014113(n+1)
2, 3, 7, 13, 27, 53, 107, 213, 427, ... = A048573(n)
2, 4, 8, 16, 32, 64, 128, 256, 512, ... = A000079(n+1)
2, 5, 9, 19, 37, 75, 149, 299, 597, ... = A062092(n)
2, 6, 10, 22, 42, 86, 170, 342, 682, ... = A078008(n+3) = A014113(n+2).
T(n+1,k) = (-1)^k*A140966(k) + (n+1)*A001045(k).
Every row T(n+1,k) has the signature (1,2).
T(0,k) = 2, -2, 2, -2, ... = (-1)^n*2.
T(n+1,k) - T(0,k) = (n+1)*A001045(n).
5*A001045(n) is not in the OEIS.
EXAMPLE
Triangle a(n):
2;
2, -1;
2, 0, 3;
2, 1, 4, 1;
2, 2, 5, 4, 7;
2, 3, 6, 7, 12, 9;
2, 4, 7, 10, 17, 20, 23;
etc.
Row sums: 2, 1, 5, 8, 20, 39, 83, 166, 338, 677, 1361, 2724, ... = b(n+2).
With b(0) = 2 and b(1) = 0, b(n) = b(n-1) + 2*b(n-2) + n - 4, n > 1.
b(n) = A001045(n) - A097065(n-1).
b(n) = b(n-2) + A000225(n-2).
MATHEMATICA
T[_, 0] = 2;
T[0, k_] := (2^k + 5(-1)^k)/3;
T[n_ /; n>0, k_ /; k>0] := T[n, k] = T[n-1, k] + (2^k + (-1)^(k+1))/3;
T[_, _] = 0;
Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 10 2018 *)
KEYWORD
sign,tabl
AUTHOR
Paul Curtz, Nov 08 2018
STATUS
approved