| Displaying 1-3 of 3 results found.
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1
0, 1, 4, 2, 1715, 31, 3, 5, 16, 14, 12, 10, 8, 6, 476, 101, 99, 97, 23, 303, 7, 9, 11, 13, 15, 17, 52, 50, 48, 46, 44, 42, 40, 38, 36, 34, 32, 30, 28, 683, 681, 22, 20, 18, 24, 77, 699, 697, 703, 695, 693, 691, 689, 687, 1723, 1725, 1721, 267, 269, 2686, 67, 263, 19, 21
COMMENTS
Conjecture that a(n) is never -1.
Lexicographically earliest injective nonnegative sequence a(n) satisfying |a(n+1) - a(n)| = n for all n.
+10
6
0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 64, 40, 65, 39, 66, 38, 67, 37, 68, 36, 69, 35, 70, 34, 71, 33, 72, 32, 73, 31, 74, 30, 75, 29, 76, 28, 77, 27, 78, 26, 79, 133, 188, 132, 189, 131, 190, 130, 191, 129, 192, 128, 193, 127, 194
COMMENTS
The map n -> a(n) is an injective map to the nonnegative integers, i.e., no two terms are identical.
Appears not to contain numbers from the following sets (grouped intentionally): {4, 5}, {14, 15, 16, 17, 18, 19}, {44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61}, etc. The numbers of terms in these groups appears to be A008776. - Paul Raff, Mar 15 2010 [This is correct: by the formula below, a(2*3^k+1...2*3^(k+1)) take all the values in the range [3^(k+1)-1, 5*3^k-2] U [7*3^k-1, 3^(k+2)-2], so the numbers not appearing are those in the range [5*3^k-1, 7*3^k-2] for some k. - Jianing Song, Oct 07 2022] It appears that the starting points of the gaps (4, 14, 44, 134, 404, 1214, ...) are given by A181655(2n) = A198643(n-1), and thus the ending points (5, 19, 61, ...) by A181655(2n) + A048473(n-1). The first differences have signs (grouped intentionally): +++, -, +++, -+-+-+-+- (5 times "-"), +++, -+...+- (17 times "-"), +++, ... where the number of minus signs is again given by A048473 = A008776-1. (End) A correspondent, Dennis Reichard, conjectures that (i) a(n) <= 3.5*n for all n and (ii) the sequence covers 2/3 of all natural numbers. - N. J. A. Sloane, Jun 30 2018 [(i) is true: the indices of records for a(n)/n are n = 1, 2, 3, 4, 6, 7, and 2*3^k+2 for k >= 1, with record values 0, 1/2, 1, 1, 3/2, 7/6, 13/7, and (7*3^k-1)/(2*3^k+2) for k >= 1, so a(n) <= 3.5*n. (ii) needs further justification: the lower natural density is lim_{k->+oo} #{terms <= 7*3^k-2}/(7*3^k-2) = lim_{k->+oo} (4*3^k-1)/(7*3^k-2) = 4/7, and the upper natural density is lim_{k->+oo} #{terms <= 5*3^k-2}/(5*3^k-2) = lim_{k->+oo} (4*3^k-1)/(5*3^k-2) = 4/5. - Jianing Song, Oct 07 2022]
FORMULA
a(n+1) = a(n) +- n with - iff n is even but not n = 2 + 2*3^k. (Cf. comment from May 09 2013.) - M. F. Hasler, Apr 05 2019 a(2*3^k + 2*r - 1) = 5*3^k - 1 - r, a(2*3^k + 2*r) = 7*3^k - 2 + r, for k >= 0 and 1 <= r <= 2*3^k. - Jianing Song, Oct 07 2022
EXAMPLE
We begin with 0, 0+1=1, 1+2=3. 3-3=0 cannot be the next term because 0 is already in the sequence so we go to 3+3=6. The next could be 6-4=2 or 6+4=10 but we choose 2 because it is smaller.
MATHEMATICA
Contribution from Paul Raff, Mar 15 2010: (Start) A171884[L_List, max_Integer, True] := If[Length[L] == max, L, With[{n = Length[L]},
If[Last[L] - n < 1 || MemberQ[L, Last[L] - n],
If[MemberQ[L, Last[L] + n],
A171884[Append[L, Last[L] + n], max, True]],
A171884[Append[L, Last[L] - n], max, True]]]]
A171884[L_List, max_Integer, False] := With[{n = Length[L]},
If[MemberQ[L, Last[L] + n],
A171884[Append[L, Last[L] + n], max, True]]]
(End)
PROG
(PARI) A171884_upto(N, a=0, t=2)=vector(N, k, a+=if(!bitand(k, 1), k-1, t-=1, 1-k, t=k-1)) \\ or: A171884_upto(N, a)=vector(N, k, a+=if(bitand(k, 1)&&k\2!=3^valuation(k-(k>1), 3), 1-k, k-1)) \\ M. F. Hasler, Apr 05 2019
a(n) = if(n<=2, n-1, my(k=logint((n-1)\2, 3), r=n-2*3^k); if(r%2, 5*3^k-1-(r+1)/2, 7*3^k-2+r/2)) \\ Jianing Song, Oct 07 2022
CROSSREFS
Cf. A005132, which allows duplicate values. Cf. also A118201, in which every value of a(n) and of |a(n+1)-a(n)| occurs exactly once, but does not ensure that the latter is strictly increasing.
Lexicographically earliest sequence of distinct positive terms whose Recamán transform has only distinct values.
+10
2
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 26, 24, 25, 27, 28, 29, 30, 32, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67
COMMENTS
The Recamán transform of a sequence {b(n), n>0} is the sequence {r(n), n>=0} defined as follows: r(0) = 0; for n > 0, r(n) = r(n-1) - b(n) if nonnegative and not already in the sequence, otherwise r(n) = r(n-1) + b(n).
The Recamán transform of this sequence corresponds to A118201.
PROG
(PARI) See Links section.
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