

A294745


Lexicographically first sequence of distinct positive integers in which successive terms differ by distinct triangular numbers.


1



1, 2, 5, 11, 21, 6, 27, 55, 10, 46, 101, 23, 89, 180, 9, 114, 234, 3, 139, 292, 16, 206, 416, 38, 291, 591, 30, 355, 4, 410, 845, 25, 490, 986, 40, 568, 1163, 35, 665, 1331, 56, 759, 18, 798, 1659, 63, 966, 1956, 65, 1100, 19, 1195, 2420, 74, 1400, 22, 1453
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OFFSET

1,2


COMMENTS

a(1) = 1; for n > 1, a(n) is the smallest unused positive integer such that a(n)  a(n1) is a triangular number (A000217) that has not already been used as the absolute difference between two successive terms.
This sequence differs from Recamán's sequence (A005132) in its starting value (1 vs. 0) and the absolute distance between its successive terms a(n)  a(n1) (any unused triangular number vs. n).
Conjecture: this sequence is a permutation of the positive integers.
Terms that are less than all subsequent terms: a(1)=1, a(2)=2, a(18)=3, a(29)=4, a(157)=7, a(216)=8, a(254)=13, a(1220)=20; a(?)=33 (does not appear in the first 40000 terms).


LINKS

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000


EXAMPLE

a(1) = 1 since all terms are positive integers and the sequence is lexicographically first.
Writing the kth triangular number A000217(k) as T(k):
a(2) = 2 because 2 is the smallest unused positive number that differs from a(1)=1 by a triangular number: 2  1 = 1 = T(1);
a(3) = 5 because 5 is the smallest unused positive number that differs from a(2)=2 by a triangular number other than 1 (already used): 5  2 = 3 = T(2).
Similarly,
a(4) = 11: 11  5 = 6 = T(3);
a(5) = 21: 21  11 = 10 = T(4);
a(6) = 6:  6  21 = 15 = T(5);
a(7) = 27: 27  6 = 21 = T(6);
a(8) = 55: 55  27 = 28 = T(7);
a(9) = 10: 10  55 = 45 = T(9) (T(8) has not yet been used, but 55  45 = 10 gives a smaller unused number than 55  36 = 19).


PROG

(MAGMA)
a:=[1]; TUsed:=[]; for n in [2..57] do tBest:=0; k:=0; while true do k+:=1; T:=(k*(k+1)) div 2; if not (T in TUsed) then t:=a[n1]T; if t lt 1 then break; end if; if not (t in a) then tBest:=t; end if; end if; end while; if tBest eq 0 then k:=0; while true do k+:=1; T:=(k*(k+1)) div 2; if not (T in TUsed) then t:=a[n1]+T; if not (t in a) then tBest:=t; break; end if; end if; end while; end if; a[n]:=tBest; TUsed[#TUsed+1]:=Abs(a[n]a[n1]); end for; a;


CROSSREFS

Cf. A000217 (triangular numbers), A005132 (Recamán's sequence).
Sequence in context: A256310 A026390 A005575 * A050407 A113032 A100134
Adjacent sequences: A294742 A294743 A294744 * A294746 A294747 A294748


KEYWORD

nonn


AUTHOR

Jon E. Schoenfield, Dec 09 2017


STATUS

approved



