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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.

%I A321373 #20 Jul 27 2024 15:56:11
%S A321373 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 A321373 13,22,31,44,41,2,6,9,16,27,42,65,84,87,2,7,10,19,32,53,86,127,172,
%U A321373 169,2,8,11,22,37,64,107,170,257,340,343
%N 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.
%C A321373 Array:
%C A321373 2, -1,  3,  1,  7,  9,  23,  41,  87, ... = (-1)^n*A140966(n)
%C A321373 2,  0,  4,  4, 12, 20,  44,  84, 172, ... = abs(A084247(n+1))
%C A321373 2,  1,  5,  7, 17, 31,  65, 127, 257, ... = A014551(n)
%C A321373 2,  2,  6, 10, 22, 42,  86, 170, 342, ... = A078008(n+2) = A014113(n+1)
%C A321373 2,  3,  7, 13, 27, 53, 107, 213, 427, ... = A048573(n)
%C A321373 2,  4,  8, 16, 32, 64, 128, 256, 512, ... = A000079(n+1)
%C A321373 2,  5,  9, 19, 37, 75, 149, 299, 597, ... = A062092(n)
%C A321373 2,  6, 10, 22, 42, 86, 170, 342, 682, ... = A078008(n+3) = A014113(n+2).
%C A321373 T(n+1,k) = (-1)^k*A140966(k) + (n+1)*A001045(k).
%C A321373 Every row T(n+1,k) has the signature (1,2).
%C A321373 T(0,k) = 2, -2, 2, -2, ... = (-1)^n*2.
%C A321373 T(n+1,k) - T(0,k) = (n+1)*A001045(n).
%C A321373 5*A001045(n) is not in the OEIS.
%e A321373 Triangle a(n):
%e A321373   2;
%e A321373   2, -1;
%e A321373   2,  0,  3;
%e A321373   2,  1,  4,  1;
%e A321373   2,  2,  5,  4,  7;
%e A321373   2,  3,  6,  7, 12,  9;
%e A321373   2,  4,  7, 10, 17, 20, 23;
%e A321373   etc.
%e A321373 Row sums: 2, 1, 5, 8, 20, 39, 83, 166, 338, 677, 1361, 2724, ... = b(n+2).
%e A321373 With b(0) = 2 and b(1) = 0, b(n) = b(n-1) + 2*b(n-2)  + n - 4, n > 1.
%e A321373 b(n) = A001045(n) - A097065(n-1).
%e A321373 b(n) = b(n-2) + A000225(n-2).
%t A321373 T[_, 0] = 2;
%t A321373 T[0, k_] := (2^k + 5(-1)^k)/3;
%t A321373 T[n_ /; n>0, k_ /; k>0] := T[n, k] = T[n-1, k] + (2^k + (-1)^(k+1))/3;
%t A321373 T[_, _] = 0;
%t A321373 Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 10 2018 *)
%Y A321373 Cf. A000079, A001045, A014113, A014551, A048573, A062092, A078008, A084247, A092297, A097073, A140360, A140966.
%Y A321373 Cf. A000225, A097065.
%K A321373 sign,tabl
%O A321373 0,1
%A A321373 _Paul Curtz_, Nov 08 2018