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A036561 Triangle of numbers in which i-th row is {2^(i-j)*3^j, 0<=j<=i}; i >= 0. 31
1, 2, 3, 4, 6, 9, 8, 12, 18, 27, 16, 24, 36, 54, 81, 32, 48, 72, 108, 162, 243, 64, 96, 144, 216, 324, 486, 729, 128, 192, 288, 432, 648, 972, 1458, 2187, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The triangle pertaining to this sequence has the property that every row, every column and every diagonal contains a nontrivial geometric progression. More interestingly every line joining any two elements contains a nontrivial geometric progression. - Amarnath Murthy, Jan 02 2002

Kappraff states (pp. 148-149): "I shall refer to this as Nicomachus' table since an identical table of numbers appeared in the Arithmetic of Nicomachus of Gerasa (circa 150 A.D.)" The table was rediscovered during the Italian Renaissance by Leon Battista Alberti, who incorporated the numbers in dimensions of his buildings and in a system of musical proportions. Kappraff states "Therefore a room could exhibit a 4:6 or 6:9 ratio but not 4:9. This ensured that ratios of these lengths would embody musical ratios". - Gary W. Adamson, Aug 18 2003

After Nichomachus and Alberti several Renaissance authors described this table. See for instance Pierre de la Ramée in 1569 (facsimile of a page of his Arithmetic Treatise in Latin in the links section). - Olivier Gérard, Jul 04 2013

Row sums give A001047; central terms give A000400; T(n,k)=A013620(n,k)/A007318(n,k). - Reinhard Zumkeller, May 14 2006

The triangle sums, see A180662 for their definitions, link Nicomachus' table with eleven different sequences, see the crossrefs. It is remarkable that these eleven sequences can be described with simple elegant formulas. The mirror of this triangle is A175840. - Johannes W. Meijer, Sep 22 2010

The diagonal sums Sum_{k} T(n - k, k) give A167762(n + 2). - Michael Somos, May 28 2012

Where d(n) is the divisor count function, then d(T(i,j)) = A003991, the rows of which sum to the tetrahedral numbers A000292(n+1). For example, the sum of the divisors of row 4 of this triangle (i = 4), gives d(16) + d(24) + d(36) + d(54) + d(81) = 5 + 8 + 9 + 8 + 5 = 35 = A000292(5). In fact, where p and q are distinct primes, the aforementioned relationship to the divisor function and tetrahedral numbers can be extended to any triangle of numbers in which the i-th row is of form {p^(i-j)*q^j, 0<=j<=i}; i >= 0 (e.g., A003593, A003595). - Raphie Frank, Nov 18 2012, corrected Dec 07 2012

T(n,k) = 2^(k-1)*3^(n-1), n, k >0 read by antidiagonals. - Boris Putievskiy, Jan 08 2013

Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then 2*x and 3*x are in S, and duplicates are deleted as they occur; see A232559. - Clark Kimberling, Nov 28 2013

Partial sums of rows produce Stirling numbers of the 2nd kind: A000392(n+2) = Sum_{m=1..(n^2+n)/2} a(m). - Fred Daniel Kline, Sep 22 2014

A permutation of A003586. - L. Edson Jeffery, Sep 22 2014

Form a word of length i by choosing a (possibly empty) word on alphabet {0,1} then concatenating a word of length j on alphabet {2,3,4}. T(i,j) is the number of such words. - Geoffrey Critzer, Jun 23 2016

REFERENCES

Jay Kappraff, Beyond Measure, World Scientific, 2002, p. 148.

Flora R. Levin, The Manual of Harmonics of Nicomachus the Pythagorean, Phanes Press, 1994, p. 114.

LINKS

Reinhard Zumkeller and Matthew House, Rows n = 0..300 of triangle, flattened [Rows 0 through 120 were computed by Reinhard Zumkeller; rows 121 through 300 by Matthew House, Jul 09 2015]

Fred Daniel Kline, How do I convert this Nicomachus' Triangle to one with edges?

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Pierre de la Ramée (Petrus Ramus), P. Rami Arithmeticae (anno 1569) Liber 2, Cap. XVI "De inventione continue proportionalium" p.46 (leaf 0055) describes this integer triangle in a layout close to the current OEIS 'tabl' layout.

Robert Sedgewick, Analysis of shellsort and related algorithms, Fourth European Symposium on Algorithms, Barcelona, September, 1996.

FORMULA

T(n,k) = T(n,k-1) + T(n-1,k-1) for n>=1 and 1<=k<=n with T(n,0) = 2^n for n>=0. - Johannes W. Meijer, Sep 22 2010

a(n) = 2^(A004736(n)-1)*3^(A002260(n)-1), n > 0, or a(n) = 2^(j-1)*3^(i-1) n > 0, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor[(-1+sqrt(8*n-7))/2]. - Boris Putievskiy, Jan 08 2013

G.f.: 1/((1-2x)(1-3yx)). - Geoffrey Critzer, Jun 23 2016

EXAMPLE

The start of the sequence as a triangular array read by rows:

   1

   2   3

   4   6   9

   8  12  18  27

  16  24  36  54  81

  32  48  72 108 162 243

  ...

The start of the sequence as a table T(n,k) n, k > 0:

    1    2    4    8   16   32 ...

    3    6   12   24   48   96 ...

    9   18   36   72  144  288 ...

   27   54  108  216  432  864 ...

   81  162  324  648 1296 2592 ...

  243  486  972 1944 3888 7776 ...

  ...

- Boris Putievskiy, Jan 08 2013

MAPLE

A036561 := proc(n, k): 2^(n-k)*3^k end:

seq(seq(A036561(n, k), k=0..n), n=0..9);

T := proc(n, k) option remember: if k=0 then 2^n elif k>=1 then procname(n, k-1) + procname(n-1, k-1) fi: end: seq(seq(T(n, k), k=0..n), n=0..9);

# Johannes W. Meijer, Sep 22 2010, Sep 10 2011

MATHEMATICA

Flatten[Table[ 2^(i-j) 3^j, {i, 0, 12}, {j, 0, i} ]] (* Flatten added by Harvey P. Dale, Jun 07 2011 *)

PROG

(PARI) for(i=0, 9, for(j=0, i, print1(3^j<<(i-j)", "))) \\ Charles R Greathouse IV, Dec 22 2011

(PARI) {T(n, k) = if( k<0 || k>n, 0, 2^(n - k) * 3^k)} /* Michael Somos, May 28 2012 */

(Haskell)

a036561 n k = a036561_tabf !! n !! k

a036561_row n = a036561_tabf !! n

a036561_tabf = iterate (\xs@(x:_) -> x * 2 : map (* 3) xs) [1]

-- Reinhard Zumkeller, Jun 08 2013

(MAGMA) /* As triangle: */ [[(2^(i-j)*3^j)/3: j in [1..i]]: i in [1..10]]; // Vincenzo Librandi, Oct 17 2014

CROSSREFS

Cf. A003586, A000079, A000244, A007283, A025197, A005010, A003946, A005051.

Triangle sums (see the comments): A001047 (Row1); A015441 (Row2); A005061 (Kn1, Kn4); A016133 (Kn2, Kn3); A016153 (Fi1, Fi2); A016140 (Ca1, Ca4); A180844 (Ca2, Ca3); A180845 (Gi1, Gi4); A180846 (Gi2, Gi3); A180847 (Ze1, Ze4); A016185 (Ze2, Ze3). - Johannes W. Meijer, Sep 22 2010, Sep 10 2011

Sequence in context: A285321 A253561 A119919 * A082976 A242407 A047419

Adjacent sequences:  A036558 A036559 A036560 * A036562 A036563 A036564

KEYWORD

nonn,easy,tabl,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 17 16:34 EDT 2018. Contains 316285 sequences. (Running on oeis4.)