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

0,2

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

Links

Table of n, a(n) for n=0..54.

Norman Whitehead, Kaprekar Network Graph for 3 Digits

Norman Whitehead, Kaprekar Network Graph Node Count Verification

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

Example

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;
...

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 *)

Prog

(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
(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<i, r | i^r-(i-1)^r>; P:=func<n, k, m | IsZero(k) select n+1 else (Q(k+1, m)-Q(k, m))*(n+1-k)>; [[P(n, k, m): k in [0..n]]: n in [0..10]]; // Bruno Berselli, Aug 31 2016

Crossrefs

Cf. A000578, A008458, A276189.

Sequence in context: A304755 A243618 A063929 * A092393 A352793 A207901

Adjacent sequences: A276155 A276156 A276157 * A276159 A276160 A276161

Keyword

nonn,tabl

Author

Stefano Maruelli, Aug 22 2016

Extensions

Corrected and rewritten by Bruno Berselli, Sep 01 2016

Status

approved