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A036561 Nicomachus triangle read by rows, T(n, k) = 2^(n - k)*3^k, for 0 <= k <= n.

%I A036561 #188 Jan 21 2025 02:04:19
%S A036561 1,2,3,4,6,9,8,12,18,27,16,24,36,54,81,32,48,72,108,162,243,64,96,144,
%T A036561 216,324,486,729,128,192,288,432,648,972,1458,2187,256,384,576,864,
%U A036561 1296,1944,2916,4374,6561,512,768,1152,1728,2592,3888,5832,8748,13122,19683
%N A036561 Nicomachus triangle read by rows, T(n, k) = 2^(n - k)*3^k, for 0 <= k <= n.
%C A036561 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
%C A036561 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
%C A036561 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
%C A036561 The triangle sums, see A180662 for their definitions, link Nicomachus's 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
%C A036561 The diagonal sums Sum_{k} T(n - k, k) give A167762(n + 2). - _Michael Somos_, May 28 2012
%C A036561 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
%C A036561 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
%C A036561 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
%C A036561 A permutation of A003586. - _L. Edson Jeffery_, Sep 22 2014
%C A036561 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
%C A036561 Form of Zorach additive triangle (see A035312) where each number is sum of west and northwest numbers, with the additional condition that each number is GCD of the two numbers immediately below it. - _Michel Lagneau_, Dec 27 2018
%D A036561 Jay Kappraff, Beyond Measure, World Scientific, 2002, p. 148.
%D A036561 Flora R. Levin, The Manual of Harmonics of Nicomachus the Pythagorean, Phanes Press, 1994, p. 114.
%H A036561 Reinhard Zumkeller and Matthew House, <a href="/A036561/b036561.txt">Rows n = 0..300 of triangle, flattened</a> [Rows 0 through 120 were computed by Reinhard Zumkeller; rows 121 through 300 by Matthew House, Jul 09 2015]
%H A036561 Fred Daniel Kline, <a href="http://mathematica.stackexchange.com/questions/6067/">How do I convert this Nicomachus' Triangle to one with edges?</a>
%H A036561 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%H A036561 Pierre de la Ramée (Petrus Ramus), <a href="http://docnum.u-strasbg.fr/cdm/ref/collection/coll7/id/40857">P. Rami Arithmeticae (anno 1569) Liber 2, Cap. XVI "De inventione continue proportionalium" p.46 (leaf 0055)</a> describes this integer triangle in a layout close to the current OEIS 'tabl' layout.
%H A036561 Marko Riedel, <a href="/A036561/a036561.pdf">Proof of identity by Egorychev method</a>.
%H A036561 Thomas Scheuerle, <a href="/A036561/a036561.jpg">Version of this triangle from Boethius (480-524)</a>, Anicius Manlius Severinus Boethius, De institutione arithmetica, Medeltidshandskrift 1 (Mh 1), Lund University Library, early 10th century, page 4r.
%H A036561 Robert Sedgewick, <a href="https://sedgewick.io/wp-content/themes/sedgewick/papers/1996Shellsort.pdf">Analysis of shellsort and related algorithms</a>, Fourth European Symposium on Algorithms, Barcelona, September, 1996.
%F A036561 T(n,k) = A013620(n,k)/A007318(n,k). - _Reinhard Zumkeller_, May 14 2006
%F A036561 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
%F A036561 T(n,k) = 2^(k-1)*3^(n-1), n, k > 0 read by antidiagonals. - _Boris Putievskiy_, Jan 08 2013
%F A036561 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
%F A036561 G.f.: 1/((1-2x)(1-3yx)). - _Geoffrey Critzer_, Jun 23 2016
%F A036561 T(n,k) = (-1)^n * Sum_{q=0..n} (-1)^q * C(k+3*q, q) * C(n+2*q, n-q). - _Marko Riedel_, Jul 01 2024
%e A036561 The start of the sequence as a triangular array read by rows:
%e A036561    1
%e A036561    2   3
%e A036561    4   6   9
%e A036561    8  12  18  27
%e A036561   16  24  36  54  81
%e A036561   32  48  72 108 162 243
%e A036561   ...
%e A036561 The start of the sequence as a table T(n,k) n, k > 0:
%e A036561     1    2    4    8   16   32 ...
%e A036561     3    6   12   24   48   96 ...
%e A036561     9   18   36   72  144  288 ...
%e A036561    27   54  108  216  432  864 ...
%e A036561    81  162  324  648 1296 2592 ...
%e A036561   243  486  972 1944 3888 7776 ...
%e A036561   ...
%e A036561 - _Boris Putievskiy_, Jan 08 2013
%p A036561 A036561 := proc(n,k): 2^(n-k)*3^k end:
%p A036561 seq(seq(A036561(n,k),k=0..n),n=0..9);
%p A036561 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);
%p A036561 # _Johannes W. Meijer_, Sep 22 2010, Sep 10 2011
%t A036561 Flatten[Table[ 2^(i-j) 3^j, {i, 0, 12}, {j, 0, i} ]] (* Flatten added by _Harvey P. Dale_, Jun 07 2011 *)
%o A036561 (PARI) for(i=0,9,for(j=0,i,print1(3^j<<(i-j)", "))) \\ _Charles R Greathouse IV_, Dec 22 2011
%o A036561 (PARI) {T(n, k) = if( k<0 || k>n, 0, 2^(n - k) * 3^k)} /* _Michael Somos_, May 28 2012 */
%o A036561 (Haskell)
%o A036561 a036561 n k = a036561_tabf !! n !! k
%o A036561 a036561_row n = a036561_tabf !! n
%o A036561 a036561_tabf = iterate (\xs@(x:_) -> x * 2 : map (* 3) xs) [1]
%o A036561 -- _Reinhard Zumkeller_, Jun 08 2013
%o A036561 (Magma) /* As triangle: */ [[(2^(i-j)*3^j)/3: j in [1..i]]: i in [1..10]]; // _Vincenzo Librandi_, Oct 17 2014
%Y A036561 Cf. A001047 (row sums), A000400 (central terms), A013620, A007318.
%Y A036561 Cf. A003586, A000079, A000244, A007283, A025197, A005010, A003946, A005051.
%Y A036561 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
%Y A036561 Antidiagonal cumulative sum: A000392; square arrays cumulative sum: A160869. Antidiagonal products: 6^A000217; antidiagonal cumulative products: 6^A000292; square arrays products: 6^A005449; square array cumulative products: 6^A006002.
%K A036561 nonn,easy,tabl,nice
%O A036561 0,2
%A A036561 _N. J. A. Sloane_