A175840 Mirror image of Nicomachus' table: T(n,k) = 3^(n-k)*2^k for n>=0 and 0<=k<=n.

1, 3, 2, 9, 6, 4, 27, 18, 12, 8, 81, 54, 36, 24, 16, 243, 162, 108, 72, 48, 32, 729, 486, 324, 216, 144, 96, 64, 2187, 1458, 972, 648, 432, 288, 192, 128, 6561, 4374, 2916, 1944, 1296, 864, 576, 384, 256, 19683, 13122, 8748, 5832, 3888, 2592, 1728, 1152, 768, 512

Offset: 0

Author

Johannes W. Meijer, Sep 21 2010, Jul 13 2011, Jun 03 2012

Keywords

• easy
• nonn
• tabl

Comments

Lenstra calls these numbers the harmonic numbers of Philippe de Vitry (1291-1361). De Vitry wanted to find pairs of harmonic numbers that differ by one. Levi ben Gerson, also known as Gersonides, proved in 1342 that there are only four pairs with this property of the form 2^n*3^m. See also Peterson’s story ‘Medieval Harmony’.

This triangle is the mirror image of Nicomachus' table A036561. The triangle sums, see the crossrefs, mirror those of A036561. See A180662 for the definitions of these sums.

Examples

			1;
3, 2;
9, 6, 4;
27, 18, 12, 8;
81, 54, 36, 24, 16;
243, 162, 108, 72, 48, 32;
		

Links

• Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
• J. O'Connor and E.F. Robertson, Nicomachus of Gerasa, The MacTutor History of Mathematics archive, 2010.
• Jay Kappraff, The Arithmetic of Nicomachus of Gerasa and its Applications to Systems of Proportion, Nexus Network Journal, vol. 2, no. 4 (October 2000).
• Hendrik Lenstra, Aeternitatem Cogita, Nieuw Archief voor Wiskunde, 5/2, maart 2001, pp. 23-28.
• Ivars Peterson, Medieval Harmony, Math Trek, Mathematical Association of America, 1998.

Crossrefs

Triangle sums: A001047 (Row1), A015441 (Row2), A016133 (Kn1 & Kn4), A005061 (Kn2 & Kn3), A016153 (Fi1& Fi2), A180844 (Ca1 & Ca4), A016140 (Ca2, Ca3), A180846 (Gi1 & Gi4), A180845 (Gi2 & Gi3), A016185 (Ze1 & Ze4), A180847 (Ze2 & Ze3).

Cf. A000079, A000244, A000400, A003586.

Programs

Haskell

a175840 n k = a175840_tabf !! n !! k
a175840_row n = a175840_tabf !! n
a175840_tabf = iterate (\xs@(x:_) -> x * 3 : map (* 2) xs) [1]
-- Reinhard Zumkeller, Jun 08 2013

Maple

A175840 := proc(n,k): 3^(n-k)*2^k end: seq(seq(A175840(n,k),k=0..n),n=0..9);

Mathematica

Flatten[Table[3^(n-k) 2^k,{n,0,10},{k,0,n}]] (* Harvey P. Dale, May 08 2013 *)

Formula

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

T(n,n-k) = T(n,n-k+1) + T(n-1,n-k) for n>=1 and 1<=k<=n with T(n,n) = 2^n for n>=0.