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A249558 a(a(n)) is a triangular number. 0

%I

%S 1,3,6,2,7,10,15,9,21,28,12,36,14,45,55,17,66,19,78,22,91,105,24,120,

%T 26,136,29,153,171,31,190,33,210,35,231,253,38,276,40,300,42,325,44,

%U 351,378,47,406,49,435,51,465,53,496,56,528,561

%N a(a(n)) is a triangular number.

%C To build T: always use the smallest integer not yet present in T and not leading to a contradiction.

%C All triangular numbers appear in their natural order.

%C Density of T: it appears that more than 50% of the terms are triangular.

%C _Arie Groeneveld_ computed a million terms in less than 1.5 seconds using the language J.

%D Eric Angelini, Postings to Sequence Fans Mailing List, Mar 02 2011 and Mar 03 2011.

%e T = 1,... meaning that the first term of T is a triangular number (true)

%e 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

%e Then:

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

%e T = 1,3,

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

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

%e T = 1,3,t,

%e Thus:

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

%e T = 1,3,6,

%e Thus:

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

%e T = 1,3,6,* * t

%e Now we need to replace the first star with "the smallest integer not yet present in T and not leading to a contradiction":

%e Thus:

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

%e T = 1,3,6,2,* t

%e 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):

%e Thus:

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

%e T = 1,3,6,2,7,t

%e and:

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

%e T = 1,3,6,2,7,t t

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

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

%e T = 1,3,6,2,7,10,15,

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

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

%e T = 1,3,6,2,7,10,15, t t

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

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

%e T = 1,3,6,2,7,10,15,9, t t

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

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

%e T = 1,3,6,2,7,10,15,9,t, t t

%e Now two more triangular numbers:

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

%e T = 1,3,6,2,7,10,15,9,21,28, t

%e ... etc.

%o (J)

%o arr249558 =: monad define

%o n=. # (,8+{:)^:(y>+/)^:_[ 6 15

%o ti=. (#~y>])+/\ 0 1 1 3 1 2, ;,&2 2&.>^:(<n) 1 2 2;3$<1 2 2 2

%o tn=. (#ti){.+/\ 1+i.y

%o ni=. (#~y>])+/\ 3 1 3, ;,& 2 2&.>^:(<n) 3 2;3$<3 2 2

%o nn=. (# ni){.+/\ 2 5 2 3 2 3 2, ;,&2 2&.>^:(<n)3 2 2;3$<3 2 2 2

%o (tn,nn) /: ti,ni

%o )

%o _Arie Groeneveld_, Dec 02 2014

%Y Cf. A000217.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Nov 01 2014

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Last modified May 12 20:52 EDT 2021. Contains 343829 sequences. (Running on oeis4.)