A276158 Triangle read by rows: T(n,k) = 6*k*(n + 1 - k) for 0 < k <= n; for k = 0, T(n,0) = n + 1.

1, 2, 6, 3, 12, 12, 4, 18, 24, 18, 5, 24, 36, 36, 24, 6, 30, 48, 54, 48, 30, 7, 36, 60, 72, 72, 60, 36, 8, 42, 72, 90, 96, 90, 72, 42, 9, 48, 84, 108, 120, 120, 108, 84, 48, 10, 54, 96, 126, 144, 150, 144, 126, 96, 54

Offset: 0

Author

Stefano Maruelli, Aug 22 2016

Keywords

• nonn
• tabl

Comments

The row sums of the triangle provide the positive terms of A000578.

Similar triangles can be generated by the formula P(n,k,m) = (Q(k+1,m)-Q(k,m))*(n+1-k), where Q(i,r) = i^r-(i-1)^r, 0 < k <= n, and P(n,0,m) = n+1. T(n,k) is the case m=3, that is T(n,k) = P(n,k,3).

T(9,k) for 0 <= k <= 9 provides the indegrees of the 10 non-leaf nodes of the network graph of the Kaprekar Process on 3 digits when the nodes are listed in numerical order. Namely, nodes 000, 099, 198, 297, 396, 495, 594, 693, 792, and 891 have indegrees 10, 54, 96, 126, 144, 150, 144, 126, 96, 54, respectively. Result derived empirically. See "Kaprekar Network Graph for 3 Digits". - Norman Whitehead, May 16 2022

Examples

			Triangle starts:
----------------------------------------------
n \ k |  0   1    2    3    4    5    6    7
----------------------------------------------
0     |  1;
1     |  2,  6;
2     |  3, 12,  12;
3     |  4, 18,  24,  18;
4     |  5, 24,  36,  36,  24;
5     |  6, 30,  48,  54,  48,  30;
6     |  7, 36,  60,  72,  72,  60,  36;
7     |  8, 42,  72,  90,  96,  90,  72,  42;
...
		

Links

• Norman Whitehead, Kaprekar Network Graph for 3 Digits
• Norman Whitehead, Kaprekar Network Graph Node Count Verification

Crossrefs

Cf. A000578, A008458, A276189.

Programs

Magma

[IsZero(k) select n+1 else 6*k*(n+1-k): k in [0..n], n in [0..10]]; // Bruno Berselli, Aug 31 2016

Magma

/* As triangle (see the second comment): */ m:=3; Q:=func; P:=func; [[P(n, k, m): k in [0..n]]: n in [0..10]]; // Bruno Berselli, Aug 31 2016

Maple

T:= (n, k) -> `if`(k=0, n+1, 6*k*(n+1-k)):
seq(seq(T(n, k), k=0..n), n=0..30); # Robert Israel, Aug 31 2016

Mathematica

Table[If[k == 0, n + 1, 6 k (n + 1 - k)], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 25 2016 *)

PARI

T(n, k) = if (k==0, n+1, 6*k*(n+1-k));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Aug 25 2016

Formula

Sum_{k=0..n} T(n,k) = T(n,0)^3 = A000578(n+1).

G.f. as triangle: (1+4*x*y + x^2*y^2)/((1-x)^2*(1-x*y)^2). - Robert Israel, Aug 31 2016

T(n,n-h) = (h+1)*A008458(n-h) for 0 <= h <= n. Therefore, the main diagonal of the triangle is A008458. - Bruno Berselli, Aug 31 2016

Extensions

Corrected and rewritten by Bruno Berselli, Sep 01 2016