[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
in T and not leading to a contradiction :
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
'6' is already in T):
Thus:
n : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
T = 1,3,6,2,7,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.
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