A185877 - OEIS

A185877

Array T given by T(n,k) = k^2 +(2*n-3)*k -2*n +3, by antidiagonals.

1, 3, 1, 7, 5, 1, 13, 11, 7, 1, 21, 19, 15, 9, 1, 31, 29, 25, 19, 11, 1, 43, 41, 37, 31, 23, 13, 1, 57, 55, 51, 45, 37, 27, 15, 1, 73, 71, 67, 61, 53, 43, 31, 17, 1, 91, 89, 85, 79, 71, 61, 49, 35, 19, 1, 111, 109, 105, 99, 91, 81, 69, 55, 39, 21, 1, 133, 131, 127, 121, 113, 103, 91, 77, 61, 43, 23, 1, 157, 155, 151, 145, 137, 127, 115, 101, 85, 67, 47, 25, 1, 183, 181, 177, 171, 163, 153, 141, 127, 111, 93, 73, 51, 27, 1

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OFFSET

1,2

COMMENTS

A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ... (See A144112 for the definition of accumulation array).

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

FORMULA

T(n,k) = k^2 + (2*n-3)*k - 2*n + 3, k>=1, n>=1.

EXAMPLE

Northwest corner:

1, 3, 7, 13, 21

1, 5, 11, 19, 29

1, 7, 15, 25, 45

1, 9, 19, 31, 45

MATHEMATICA

(* This program generates A185877, its accumulation array A185878, and its weight array A185879. *)

f[n_, 0]:=0; f[0, k_]:=0;

f[n_, k_]:=k^2+(2n-3)k-2n+3;

TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]] (* A185877 *)

Table[f[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten

s[n_, k_]:=Sum[f[i, j], {i, 1, n}, {j, 1, k}]; (* accumulation array of {f(n, k)} *)

FullSimplify[s[n, k]] (* formula for A185878 *)

TableForm[Table[s[n, k], {n, 1, 10}, {k, 1, 15}]]

Table[s[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten

w[m_, n_]:=f[m, n]+f[m-1, n-1]-f[m, n-1]-f[m-1, n]/; Or[m>0, n>0];

TableForm[Table[w[n, k], {n, 1, 10}, {k, 1, 15}]] (* A185879 *)

Table[w[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten

CROSSREFS

Cf. A144112, A185878, A185879, A185880.

Row 1 to 3: A002061, A028387, A082111.

diag (1,5,...): A056108;

diag (3,11,...): A056106;

diag (7,19,...): A003215;

diag (13,29,...): A144391;

diag (1,7,...): A003215;

diag (1,9,...): A144390.

Sequence in context: A210198 A271258 A100584 * A135858 A193845 A372938

Adjacent sequences: A185874 A185875 A185876 * A185878 A185879 A185880

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 05 2011

STATUS

approved