A160869 -id:A160869 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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
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
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
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
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

			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

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.

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.
Marko Riedel, Proof of identity by Egorychev method.
Thomas Scheuerle, Version of this triangle from Boethius (480-524), Anicius Manlius Severinus Boethius, De institutione arithmetica, Medeltidshandskrift 1 (Mh 1), Lund University Library, early 10th century, page 4r.
Robert Sedgewick, Analysis of shellsort and related algorithms, Fourth European Symposium on Algorithms, Barcelona, September, 1996.

Cf. A001047 (row sums), A000400 (central terms), A013620, A007318.
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
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.

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

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

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

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

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

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

T(n,k) = A013620(n,k)/A007318(n,k). - Reinhard Zumkeller, May 14 2006
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
T(n,k) = 2^(k-1)*3^(n-1), n, k > 0 read by antidiagonals. - Boris Putievskiy, Jan 08 2013
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
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

A160870 Array read by antidiagonals: T(n,k) is the number of sublattices of index n in generic k-dimensional lattice (n >= 1, k >= 1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 7, 1, 1, 7, 13, 15, 1, 1, 6, 35, 40, 31, 1, 1, 12, 31, 155, 121, 63, 1, 1, 8, 91, 156, 651, 364, 127, 1, 1, 15, 57, 600, 781, 2667, 1093, 255, 1, 1, 13, 155, 400, 3751, 3906, 10795, 3280, 511, 1, 1, 18, 130, 1395, 2801, 22932, 19531, 43435, 9841, 1023, 1

Offset: 1

N. J. A. Sloane, Nov 19 2009

			Array begins:
  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
  1,3,7,15,31,63,127,255,511,1023,2047,4095,8191,16383,32767,65535,...
  1,4,13,40,121,364,1093,3280,9841,29524,88573,265720,797161,2391484,...
  1,7,35,155,651,2667,10795,43435,174251,698027,2794155,11180715,...
  1,6,31,156,781,3906,19531,97656,488281,2441406,12207031,61035156,...
  ...

Günter Scheja, Uwe Storch, Lehrbuch der Algebra, Teil 2. BG Teubner, Stuttgart, 1988. [§63, Aufg. 13]

Álvar Ibeas, First 100 antidiagonals, flattened
Michael Baake, Solution of the coincidence problem in dimensions d≤4, arXiv:math/0605222 [math.MG], 2006. [Appx. A]
B. Gruber, Alternative formulas for the number of sublattices, Acta Cryst. A53 (1997) 807-808.
J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
Yi Ming Zou, Gaussian binomials and the number of sublattices, arXiv:math/0610684 [math.CO], 2006.
Yi Ming Zou, Gaussian binomials and the number of sublattices, Acta Cryst. A62 (2006) 409-410.

Columns: A000203, A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038997.
Rows: A000012, A000225, A003462, A006095, A003463, A160869, A023000, A006096.
Transposed array: A128119.

T[, 1] = 1; T[1, ] = 1; T[n_, k_] := T[n, k] = DivisorSum[n, (n/#)^(k-1) *T[#, k-1]&]; Table[T[n-k+1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 04 2015 *)

T(n,k)={ if ( (n==1) || (k==1), 1, sumdiv(n,d, d*T(d, k-1)) ) }

T(n,1) = 1; T(1,k) = 1; T(n, k) = Sum_{d|n} d*T(d, k-1).
From Álvar Ibeas, Oct 31 2015: (Start)
T(n,k) = Sum_{d|n} (n/d)^(k-1) * T(d, k-1).
T(Product(p^e), k) = Product(Gaussian_poly[e+k-1, e]_p). (End)

A263950 Array read by antidiagonals: T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 7, 1, 1, 6, 13, 15, 1, 1, 6, 28, 40, 31, 1, 1, 12, 31, 120, 121, 63, 1, 1, 8, 91, 156, 496, 364, 127, 1, 1, 12, 57, 600, 781, 2016, 1093, 255, 1, 1, 12, 112, 400, 3751, 3906, 8128, 3280, 511, 1, 1, 18, 117, 960, 2801, 22932, 19531, 32640

Offset: 1

Álvar Ibeas, Oct 30 2015

All the enumerated lattices have full rank k, since the quotient group is finite.
For m>=1, T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_nm x (C_m)^(k-1); and also, (C_nm)^(k-1) x C_m.
Also, number of subgroups of (C_n)^k isomorphic to C_n (and also, to (C_n)^{k-1}), cf. [Butler, Lemma 1.4.1].
T(n,k) is the sum of the divisors d of n^(k-1) such that n^(k-1)/d is k-free. Namely, the coefficient in n^(-(k-1)*s) of the Dirichlet series zeta(s) * zeta(s-1) / zeta(ks).
Also, number of isomorphism classes of connected (C_n)-fold coverings of a connected graph with circuit rank k.
Columns are multiplicative functions.

			There are 7 = A160870(4,2) lattices of volume 4 in Z^2. Among them, only one (<(2,0), (0,2)>) gives the quotient group C_2 x C_2, whereas the rest give C_4. Hence, T(4,2) = 6 and T(1,2) = 1.
Array begins:
      k=1    k=2    k=3    k=4    k=5    k=6
n=1     1      1      1      1      1      1
n=2     1      3      7     15     31     63
n=3     1      4     13     40    121    364
n=4     1      6     28    120    496   2016
n=5     1      6     31    156    781   3906
n=6     1     12     91    600   3751  22932

Lynne M. Butler, Subgroup lattices and symmetric functions. Mem. Amer. Math. Soc., Vol. 112, No. 539, 1994.

Álvar Ibeas, First 100 antidiagonals, flattened
Jin Ho Kwak, Jang-Ho Chun, and Jaeun Lee, Enumeration of regular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11(2), 1998, pp. 273-285. [Page 277]
Jin Ho Kwak and Jaeun Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. [Examples 2 and 3]

Rows (1-11): A000012, A000225, A003462, A006516, A003463, A160869, A023000, A016152, A016142(n-1), A046915(n-1), A016123(n-1).
Columns (1-17): A000012, A001615, A160889, A160891, A160893, A160895, A160897, A160908, A160953, A160957, A160960, A160972, A161010, A161025, A161139, A161167, A161213.
Main diagonal gives A344210.
Cf. A000203, A008683, A160870.

f[p_, e_, k_] := p^((k - 1)*(e - 1))*(p^k - 1)/(p - 1); T[n_, 1] = T[1, k_] = 1; T[n_, k_] := Times @@ (f[First[#], Last[#], k] & /@ FactorInteger[n]); Table[T[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 08 2022 *)

T(n,k) = J_k(n) / J_1(n) = (Sum_{d|n} mu(n/d) * d^k) / phi(n).
T(n,k) = n^(k-1) * Product_{p|n, p prime} (p^k - 1) / ((p - 1) * p^(k-1)).
Dirichlet g.f. of k-th column: zeta(s-k+1) * Product_{p prime} (1 + p^(-s) + p^(1-s) + ... + p^(k-2-s)).
If n is squarefree, T(n,k) = A160870(n,k) = A000203(n^(k-1)).
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{i=1..n} T(i, k) ~ c * n^k, where c = (1/k) * Product_{p prime} (1 + (p^(k-1)-1)/((p-1)*p^k)).
Sum_{i>=1} 1/T(i, k) = zeta(k-1)*zeta(k) * Product_{p prime} (1 - 2/p^k + 1/p^(2*k-1)), for k > 2. (End)
T(n,k) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^k). - Ridouane Oudra, Apr 03 2025

A059387 Jordan function J_n(6) (see A059379).

Original entry on oeis.org

0, 2, 24, 182, 1200, 7502, 45864, 277622, 1672800, 10057502, 60406104, 362617862, 2176246800, 13059091502, 78359364744, 470170602902, 2821066795200, 16926530173502, 101559568985784, 609358577224742, 3656154952230000

Offset: 0

N. J. A. Sloane, Jan 29 2001

a(n) = A000225(n) * A024023(n) = (2^n - 1) * (3^n - 1) . a(n) is the number of n-tuples of elements e_1,e_2,...,e_n in the cyclic group C_6 such that the subgroup generated by e_1,e_2,...,e_n is C_6. - Sharon Sela (sharonsela(AT)hotmail.com), Jun 02 2002

Szalay proves that this sequence contains no squares except for 0. He & Liu prove that this sequence contains no higher powers aside from 2. - Charles R Greathouse IV, Jan 10 2025

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Bo He and Chang Liu, The diophantine equation (2^k-1)*(3^k-1)=x^n, arXiv preprint (2025). arXiv:2501.04050 [math.NT].
L. Szalay, On the Diophantine equation (2n - 1)(3n - 1) = x^2, Publicationes Mathematicae Debrecen, 57(1-2) (2000), pp. 1-9.

Index entries for linear recurrences with constant coefficients, signature (12,-47,72,-36).

Cf. A000225, A024023.

[(2^n-1)*(3^n-1): n in [0..30]]; // Vincenzo Librandi, Jun 05 2011

A059387:=n->(2^n-1)*(3^n-1); seq(A059387(n), n=0..50); # Wesley Ivan Hurt, Nov 14 2013

Table[(2^n-1)*(3^n-1),{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Apr 28 2010 *)

for(n=0,30, print1((2^n-1)*(3^n-1), ", ")) \\ G. C. Greubel, Jan 29 2018

G.f.: -2*x*(6*x^2-1) / ((x-1)*(2*x-1)*(3*x-1)*(6*x-1)). - Colin Barker, Dec 06 2012
a(n-1) = (limit of (Sum_{k>=0} (1/(6*k + 1)^s - 1/(6*k + 2)^s - 2/(6*k + 3)^s - 1/(6*k + 4)^s + 1/(6*k + 5)^s + 2/(6*k + 6)^s) as s -> n))/zeta(n)*6^(n - 1). - Mats Granvik, Nov 14 2013
a(n) = 2*A160869(n). - R. J. Mathar, Nov 23 2018

A128119 Square array T(n,m) read by antidiagonals: number of sublattices of index m in generic n-dimensional lattice.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 4, 1, 1, 15, 13, 7, 1, 1, 31, 40, 35, 6, 1, 1, 63, 121, 155, 31, 12, 1, 1, 127, 364, 651, 156, 91, 8, 1, 1, 255, 1093, 2667, 781, 600, 57, 15, 1, 1, 511, 3280, 10795, 3906, 3751, 400, 155, 13, 1, 1, 1023, 9841, 43435, 19531, 22932, 2801, 1395, 130, 18, 1

Offset: 1

Ralf Stephan, May 09 2007

Differs from sum of divisors of m^(n-1) in 4th column!

			Array starts:
1,1,1,1,1,1,1,1,1,
1,3,4,7,6,12,8,15,13,
1,7,13,35,31,91,57,155,130,
1,15,40,155,156,600,400,1395,1210,
1,31,121,651,781,3751,2801,11811,11011,
1,63,364,2667,3906,22932,19608,97155,99463,
1,127,1093,10795,19531,138811,137257,788035,896260,
1,255,3280,43435,97656,836400,960800,6347715,8069620,

Günter Scheja, Uwe Storch, Lehrbuch der Algebra, Teil 2. BG Teubner, Stuttgart, 1988. [§63, Aufg. 13]

Álvar Ibeas, First 100 antidiagonals, flattened
Michael Baake, Solution of the coincidence problem in dimensions d≤4, arXiv:math/0605222 [math.MG], 2006. [Appx. A]
B. Gruber, Alternative formulas for the number of sublattices, Acta Cryst. A53 (1997) 807-808.
Yi Ming Zou, Gaussian binomials and the number of sublattices, arXiv:math/0610684 [math.CO], 2006.

Rows include A000203, A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038997, A038998, A038999.
Columns include A000225, A003462, A006095, A003463, A160869, A023000, A006096, A006100, A046915.
Transposed array is A160870.

T[n_, m_] := If[m == 1, 1, Product[{p, e} = pe; (p^(e+j)-1)/(p^j-1), {pe, FactorInteger[m]}, {j, 1, n-1}]];
Table[T[n-m+1, m], {n, 1, 11}, {m, 1, n}] // Flatten (* Jean-François Alcover, Dec 10 2018 *)

T(n,m)=local(k,v);v=factor(m);k=matsize(v)[1];prod(i=1,k,prod(j=1,n-1,(v[i,1]^(v[i,2]+j)-1)/(v[i,1]^j-1)))

Dirichlet g.f. of n-th row: Product_{i=0..n-1} zeta(s-i).
If m is squarefree, T(n,m) = A000203(m^(n-1)). - Álvar Ibeas, Jan 17 2015
T(n, Product(p^e)) = Product(Gaussian_poly[e+n-1, e]p). - _Álvar Ibeas, Oct 31 2015

Edited by Charles R Greathouse IV, Oct 28 2009

cp's OEIS Frontend

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

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A160870 Array read by antidiagonals: T(n,k) is the number of sublattices of index n in generic k-dimensional lattice (n >= 1, k >= 1).

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A263950 Array read by antidiagonals: T(n,k) is the number of lattices L in Z^k such that the quotient group Z^k / L is C_n.

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A059387 Jordan function J_n(6) (see A059379).

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A128119 Square array T(n,m) read by antidiagonals: number of sublattices of index m in generic n-dimensional lattice.

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