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 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(1) = 1; for n > 1, a(n) is the smallest unused positive integer such that |a(n) - a(n-1)| 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(n-1)| (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 k-th 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[n-1]-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[n-1]+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[n-1]); 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

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Last modified August 17 16:59 EDT 2018. Contains 313816 sequences. (Running on oeis4.)