A356036 Triangle read by rows, giving in the first column the powers of 3 (A000244) and in the next columns 4/3 times the previous row entry.

1, 3, 4, 9, 12, 16, 27, 36, 48, 64, 81, 108, 144, 192, 256, 243, 324, 432, 576, 768, 1024, 729, 972, 1296, 1728, 2304, 3072, 4096, 2187, 2916, 3888, 5184, 6912, 9216, 12288, 16384, 6561, 8748, 11664, 15552, 20736, 27648, 36864, 49152, 65536, 19683, 26244, 34992, 46656, 62208, 82944, 110592, 147456, 196608, 262144

Offset: 0

Author

Wolfdieter Lang, Aug 01 2022

Keywords

• nonn
• tabl
• easy

Comments

This is Boethius's triangle, with rows read as columns. See the link and reference.

Examples

			The triangle T begins:
n\k     0     1      2      3      4      5      6      7      8      9  ...
0:      1
1:      3     4
2:      9    12     16
3:     27    36     48     64
4:     81   108    144    192    256
5:    243   324    432    576    768   1024
6:    729   972   1296   1728   2304   3072   4096
7:   2187  2916   3888   5184   6912   9216  12288  16384
8:   6561  8748  11664  15552  20736  27648  36864  49152  65536
9:  19683 26244  34992  46656  62208  82944 110592 147456 196608 262144
...
		

References

• Thomas Sonar, 3000 Jahre Analysis, 2. Auflage, Springer Spektrum, 2016, p.94, Abb. 3.1.2 und Abb. 3.1.3.

Links

• Anicius Manlius Severinus Boethius, De Institutione Arithmetica, 1488, p. 55 of 104, top of left column.

Crossrefs

Columns: A000244, A003946, A257970, ...

Diagonals: A000302, A002001(n+1), A002063, A002063(n+3), A118265(n+4), ...

Row sums: A005061(n+1).

Programs

Mathematica

T[n_, k_] := 3^(n - k) * 4^k; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 05 2022 *)

Formula

T(n, k) = 3^(n-k)*4^k, for n >= 0, and k = 1, 2, ..., n.

G.f. of row polynomials R(n, y) = Sum_{k=0..n} T(n, k)*y^k: G(x, y) = 1/((1 - 3*x)*(1 - 4*x*y)).