cp's OEIS Frontend

A355564 Triangle read by rows: T(n,k) = n*(1+2*k) - k*(1+k), n >= 1, 0 <= k <= n-1.

%I A355564 #12 Aug 30 2022 14:27:39
%S A355564 1,2,4,3,7,9,4,10,14,16,5,13,19,23,25,6,16,24,30,34,36,7,19,29,37,43,
%T A355564 47,49,8,22,34,44,52,58,62,64,9,25,39,51,61,69,75,79,81,10,28,44,58,
%U A355564 70,80,88,94,98,100,11,31,49,65,79,91,101,109,115,119,121
%N A355564 Triangle read by rows: T(n,k) = n*(1+2*k) - k*(1+k), n >= 1, 0 <= k <= n-1.
%C A355564 T(n,k) is the number of potentially nonzero elements in a square, n X n band matrix of bandwidth k, i.e., the number of matrix elements (i,j) for which |i-j| <= k.
%C A355564 T(n,0) = n, as a zero-bandwidth matrix is diagonal, and T(n,n-1) = n^2, as the band encompasses the entire matrix.
%e A355564 Triangle starts:
%e A355564   1;
%e A355564   2,  4;
%e A355564   3,  7,  9;
%e A355564   4, 10, 14, 16;
%e A355564   5, 13, 19, 23, 25;
%e A355564   6, 16, 24, 30, 34, 36;
%e A355564   7, 19, 29, 37, 43, 47, 49;
%e A355564   ...
%e A355564 Example: For n = 6 and k = 2, we have a band matrix of the form
%e A355564     [. . . 0 0 0]
%e A355564     [. . . . 0 0]
%e A355564     [. . . . . 0]
%e A355564     [0 . . . . .],
%e A355564     [0 0 . . . .]
%e A355564     [0 0 0 . . .]
%e A355564 where dots represent the entries which may have nonzero values. The number of such entries is T(6,2) = 24.
%t A355564 Flatten[Table[n (1 + 2 k) - k (1 + k), {n, 1, 10}, {k, 0, n - 1}]]
%Y A355564 The sequence is complementary to A095832, i.e.: T(n,k) = n^2 - A095832(n,k).
%Y A355564 Differences within a row T(n,k+1) - T(n,k) give A212012.
%K A355564 nonn,easy,tabl
%O A355564 1,2
%A A355564 _Lucas B. Vieira_, Jul 07 2022