# [seqfan] a(a(n)) is a triangular number

Eric Angelini Eric.Angelini at kntv.be
Wed Mar 2 17:16:33 CET 2011

```... and to build T: always use the smallest integer not yet present

T = 1,3,6,2,7,10,15,9,21,28,12,36,14,45,55,17,66,19,78,22,91,105,24,...

All triangular numbers appear in their natural order
Density of T: more than 50% of the terms are triangular

Could someone please compute a hundred terms or so?
And find the exact density of T around the infinity?
;-)
Best,
É.

---

Modus operandi:

T = 1,
... meaning that the first term of T is a triangular number (true)
The next term can't be 2 as '2' would mean that the second term
of T is a triangular term -- which is false, 2 is not a triangular
number -- see http://oeis.org/A000217

Then:

n : 1 2 3 4 5 6  7  8 9  10 11 12 13 14 15 16 17 18 19 20
T = 1,3,

We have to put a triangular number 't' in third position:

n : 1 2 3 4 5 6  7  8 9  10 11 12 13 14 15 16 17 18 19 20
T = 1,3,t,

Thus:

n : 1 2 3 4 5 6  7  8 9  10 11 12 13 14 15 16 17 18 19 20
T = 1,3,6,

Thus:

n : 1 2 3 4 5 6  7  8 9  10 11 12 13 14 15 16 17 18 19 20
T = 1,3,6,* * t

Now we need to replace the first star with "the smallest
integer not yet present in T and not leading to a contra-
diction":
Thus:

n : 1 2 3 4 5 6  7  8 9  10 11 12 13 14 15 16 17 18 19 20
T = 1,3,6,2,* t

Next star is replaced by "the smallest etc." which can't
be 4, 5 or 6 -- thus 7 (the 4th term of T is not a trian-
gular number, the 5th neither -- as it would be '5' -- and
Thus:

n : 1 2 3 4 5 6  7  8 9  10 11 12 13 14 15 16 17 18 19 20
T = 1,3,6,2,7,t

and:

n : 1 2 3 4 5 6  7  8 9  10 11 12 13 14 15 16 17 18 19 20
T = 1,3,6,2,7,t  t

We replace the next two 't' with two triangular numbers:

n : 1 2 3 4 5 6  7  8 9  10 11 12 13 14 15 16 17 18 19 20
T = 1,3,6,2,7,10,15,

and mark accordingly the 10th and 15th term of T with 't':

n : 1 2 3 4 5 6  7  8 9  10 11 12 13 14 15 16 17 18 19 20
T = 1,3,6,2,7,10,15,     t              t

The next 'hole' we have to fill in T is the 8th; we put '9':

n : 1 2 3 4 5 6  7  8 9  10 11 12 13 14 15 16 17 18 19 20
T = 1,3,6,2,7,10,15,9,   t              t

and we add accordingly a 't' in 9th position:

n : 1 2 3 4 5 6  7  8 9  10 11 12 13 14 15 16 17 18 19 20
T = 1,3,6,2,7,10,15,9,t, t              t

Now two more triangular numbers:

n : 1 2 3 4 5 6  7  8 9  10 11 12 13 14 15 16 17 18 19 20
T = 1,3,6,2,7,10,15,9,21,28,            t

... etc.

```