OFFSET
1,2
COMMENTS
To build T: always use the smallest integer not yet present in T and not leading to a contradiction.
All triangular numbers appear in their natural order.
Density of T: it appears that more than 50% of the terms are triangular.
Arie Groeneveld computed a million terms in less than 1.5 seconds using the language J.
REFERENCES
Eric Angelini, Postings to Sequence Fans Mailing List, Mar 02 2011 and Mar 03 2011.
EXAMPLE
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 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 contradiction":
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 triangular 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.
PROG
(J)
arr249558 =: monad define
n=. # (, 8+{:)^:(y>+/)^:_[ 6 15
ti=. (#~y>])+/\ 0 1 1 3 1 2, ; , &2 2&.>^:(<n) 1 2 2; 3$<1 2 2 2
tn=. (#ti){.+/\ 1+i.y
ni=. (#~y>])+/\ 3 1 3, ; , & 2 2&.>^:(<n) 3 2; 3$<3 2 2
nn=. (# ni){.+/\ 2 5 2 3 2 3 2, ; , &2 2&.>^:(<n)3 2 2; 3$<3 2 2 2
(tn, nn) /: ti, ni
)
Arie Groeneveld, Dec 02 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 01 2014
STATUS
approved