login
T(n, k) = n*k - 1.
2

%I #31 Jul 28 2023 10:38:35

%S 0,1,3,2,5,8,3,7,11,15,4,9,14,19,24,5,11,17,23,29,35,6,13,20,27,34,41,

%T 48,7,15,23,31,39,47,55,63,8,17,26,35,44,53,62,71,80,9,19,29,39,49,59,

%U 69,79,89,99,10,21,32,43,54,65,76,87,98,109,120,11,23,35,47,59,71,83,95,107,119,131,143

%N T(n, k) = n*k - 1.

%C Previous name was: Triangle T(n,k), 0 < k <= n, T(n,1) = n - 1, T(n,k) = T(n,k-1) + n; read by rows.

%C This simple triangle arose analyzing f(x) = x/(n + e^(c/x)), for n <> 0. f(x) converges towards a rational number for large values of x, if x is rational. T(n+1,k)/(n+1)^2 equals the fractional portion of f(x) if x is large and restricted to the positive integers, c = 1 and n>=1, whereby the value of the fractional portion changes on a cycle with period n+1 (as k goes from 1 to n+1) for each n in the denominator of f(x). Other, somewhat similar triangles (or repeating fractional patterns) arise with other rational values of n or c, or other rational increments of x (even if a large irrational initial value of x is used).

%C Let S(n) = row sums = Sum(k>=1, T(n,k)), then:

%C S(n) = A077414(n); S(n)/(n+2) = A000217(n); S(n)/n = A000096(n);

%C Let Sq(n) = sum of squares of row elements = Sum(k>=1, T(n,k)^2), then:

%C Sq(n)/n^2 - 1/n = A058373(n)

%C Let D(n) = diagonal sums = Sum(k>=1, T(n-k+1, k)) then:

%C D(2n) = A131423(n); D(2n-1) = 2/3*n^3 + 1/2*n^2 - 7/6*n;

%C D(2n) - D(2n-1) = A000217(n); D(2n+1) - D(2n) = A115067(n);

%C D(2n+2) - D(2n)= A056220(n+1); D(2n+1) - D(2n -1) = A014106(n).

%C Equals A144204 with the first column of negative ones removed. - _Georg Fischer_, Jul 26 2023

%H Boris Putievskiy, <a href="/A223544/b223544.txt">Rows n = 1..140 of triangle, flattened</a>

%F Also note: T(n+1,k) = T(n,k)+ k, and T(n,n) = n^2 - 1.

%F a(n) = A075362(n)-1; a(n)=i(t+1)-1, where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Jul 24 2013

%F T(n, k) = n*k - 1. - _Georg Fischer_, Jul 26 2023

%e Triangle begins as:

%e 0;

%e 1, 3;

%e 2, 5, 8;

%e 3, 7, 11, 15;

%e 4, 9 14, 19, 24;

%e 5, 11, 17, 23, 29, 35;

%e 6, 13, 20, 27, 34, 41, 48;

%e 7, 15, 23, 31, 39, 47, 55, 63;

%e 8, 17, 26, 35, 44, 53, 62, 71, 80;

%Y Cf. A075362, A144204.

%K nonn,tabl

%O 1,3

%A _Richard R. Forberg_, Jul 19 2013

%E Simpler name from _Georg Fischer_, Jul 26 2023