A321373 - OEIS

%I #20 Jul 27 2024 15:56:11

%S 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,

%T 13,22,31,44,41,2,6,9,16,27,42,65,84,87,2,7,10,19,32,53,86,127,172,

%U 169,2,8,11,22,37,64,107,170,257,340,343

%N 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.

%C Array:

%C 2, -1,  3,  1,  7,  9,  23,  41,  87, ... = (-1)^n*A140966(n)

%C 2,  0,  4,  4, 12, 20,  44,  84, 172, ... = abs(A084247(n+1))

%C 2,  1,  5,  7, 17, 31,  65, 127, 257, ... = A014551(n)

%C 2,  2,  6, 10, 22, 42,  86, 170, 342, ... = A078008(n+2) = A014113(n+1)

%C 2,  3,  7, 13, 27, 53, 107, 213, 427, ... = A048573(n)

%C 2,  4,  8, 16, 32, 64, 128, 256, 512, ... = A000079(n+1)

%C 2,  5,  9, 19, 37, 75, 149, 299, 597, ... = A062092(n)

%C 2,  6, 10, 22, 42, 86, 170, 342, 682, ... = A078008(n+3) = A014113(n+2).

%C T(n+1,k) = (-1)^k*A140966(k) + (n+1)*A001045(k).

%C Every row T(n+1,k) has the signature (1,2).

%C T(0,k) = 2, -2, 2, -2, ... = (-1)^n*2.

%C T(n+1,k) - T(0,k) = (n+1)*A001045(n).

%C 5*A001045(n) is not in the OEIS.

%e Triangle a(n):

%e   2;

%e   2, -1;

%e   2,  0,  3;

%e   2,  1,  4,  1;

%e   2,  2,  5,  4,  7;

%e   2,  3,  6,  7, 12,  9;

%e   2,  4,  7, 10, 17, 20, 23;

%e   etc.

%e Row sums: 2, 1, 5, 8, 20, 39, 83, 166, 338, 677, 1361, 2724, ... = b(n+2).

%e With b(0) = 2 and b(1) = 0, b(n) = b(n-1) + 2*b(n-2)  + n - 4, n > 1.

%e b(n) = A001045(n) - A097065(n-1).

%e b(n) = b(n-2) + A000225(n-2).

%t T[_, 0] = 2;

%t T[0, k_] := (2^k + 5(-1)^k)/3;

%t T[n_ /; n>0, k_ /; k>0] := T[n, k] = T[n-1, k] + (2^k + (-1)^(k+1))/3;

%t T[_, _] = 0;

%t Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 10 2018 *)

%Y Cf. A000079, A001045, A014113, A014551, A048573, A062092, A078008, A084247, A092297, A097073, A140360, A140966.

%Y Cf. A000225, A097065.

%K sign,tabl

%O 0,1

%A _Paul Curtz_, Nov 08 2018