cp's OEIS Frontend

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

Method A = 'frequency' followed by 'digit'-indication.

A001155, A001140, A001141, A001143, A001145, A001151 and A001154 are all identical apart from the last digit of each term (the seed). This is because digits other than 1, 2 and 3 never arise elsewhere in the terms (other than at the end of each of them) of look-and-say sequences of this type (as is mentioned by Carmine Suriano in A006751). - Chayim Lowen, Jul 16 2015

a(n+1) - a(n) is divisible by 10^5 for n > 5. - Altug Alkan, Dec 04 2015

Second bisection is A134752.

It appears that if s(n) is a first order rational sequence of the form s(1)=5, s(n)= (2*s(n-1)+1)/(s(n-1)+2),n>1, then s(n)= a(n)/(a(n)-4), n>1. - Gary Detlefs, Nov 16 2010

Mahler exhibits this sequence with n>=1 as a proof that there exists an infinite number of x coprime to 3, such that x belongs to A125293 and x^2 belongs to A005836. - Michel Marcus, Nov 12 2012

Table starts

.....3.......6........12.........24...........48.............96

.....9......24........72........216..........648...........1944

....27......96.......432.......1944.........8856..........40392

....81.....384......2592......17496.......121176.........842616

...243....1536.....15552.....157464......1658232.......17587584

...729....6144.....93312....1417176.....22692312......367125912

..2187...24576....559872...12754584....310536504.....7663517136

..6561...98304...3359232..114791256...4249585944...159971190624

.19683..393216..20155392.1033121304..58154132088..3339300422232

.59049.1572864.120932352.9298091736.795819434328.69705848287656

Sum of consecutive pairs of elements of A025192.

The alternating sign sequence with g.f. (1-x^2)/(1+3x) gives the diagonal sums of A110168. - Paul Barry, Jul 14 2005

Let M = an infinite lower triangular matrix with the odd integers (1,3,5,...) in every column, with the leftmost column shifted up one row. Then A080923 = lim_{n->inf} M^n. - Gary W. Adamson, Feb 18 2010

a(n+1), n >= 0, with o.g.f. ((1-x^2)/(1-3*x)-1)/x = (3-x)/(1-3*x) provides the coefficients in the formal power series for tan(3*x)/tan(x) = (3-z)/(1-3*z) = Sum_{n>=0} a(n+1)*z^n, with z = tan(x)^2. Convergence holds for 0 <= z < 1/3, i.e., |x| < Pi/6, approximately 0.5235987758. For the numerator and denominator of this o.g.f. see A034867 and A034839, respectively. - Wolfdieter Lang, Jan 18 2013

T(n,0) = A000244(n); T(n,n) = A000400(n) = A100851(n,n);

T(n,1) = A008776(n) for n>0;

T(n,2) = A003946(n+1) for n>1;

T(n,3) = A005051(n+1) for n>2;

T(n,n-1) = A081341(n+1) for n>0;

row sums give A016137.

Also length of successive strings generated by an alternating Kolakoski (2,4) rule starting at 4 (i.e. string begins with 2 if previous string ends with 4 and vice et versa) : 4-->2222-->44224422-->444422224422444422224422-->... and length of strings are 1,4,8,24,72,... - Benoit Cloitre, Oct 15 2005

Also number of words of length n over alphabet {1,2,3} with no fixed points (a fixed point is value i in position i). - Margaret Archibald, Jun 23 2020

Also maximum leaf number of the (n+3)-Apollonian network for n >= 0. - Eric W. Weisstein, Jan 17 2018