A074049 -id:A074049 - cp's OEIS frontend

A183079 Tree generated by the triangular numbers: a(1) = 1; a(2n) = nontriangular(a(n)), a(2n+1) = triangular(a(n+1)), where triangular = A000217, nontriangular = A014132.

1, 2, 3, 4, 6, 5, 10, 7, 21, 9, 15, 8, 55, 14, 28, 11, 231, 27, 45, 13, 120, 20, 36, 12, 1540, 65, 105, 19, 406, 35, 66, 16, 26796, 252, 378, 34, 1035, 54, 91, 18, 7260, 135, 210, 26, 666, 44, 78, 17, 1186570, 1595, 2145, 76, 5565, 119, 190, 25, 82621, 434

Offset: 1

Clark Kimberling, Dec 23 2010

A permutation of the positive integers.
In general, suppose that L and U are complementary sequences of positive integers such that
(1) L(1)=1; and
(2) if n>1, then n=L(k) or n=U(k) for some kThe tree generated by the sequence L is defined as follows:
  T(0,0)=1; T(1,0)=2; T(n,2j)=L(T(n-1,j));
  T(n,2j+1)=U(T(n-1,j)); for j=0,1,...,2^(n-1)-1, n>=2.
The numbers, taken in the order generated, form a permutation of the positive integers.

			First levels of the tree:
                                    1
                                    |
                 ...................2...................
                3                                       4
      6......../ \........5                   10......./ \........7
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  21      9          15       8          55       14         28      11
231 27  45 13     120  20   36 12    1540  65  105  19    406  35  66  16
Beginning with 3 and 4, the numbers are generated in pairs, such as (3,4), (6,5), (10,7), (21,9),...
In all such pairs, the first number belongs to A000217; the second, to A014132.

Reinhard Zumkeller, Rows n = 1..14 of triangle, flattened
Index entries for sequences that are permutations of the natural numbers

Cf. A000217, A014132, A074049.
Cf. A220347 (inverse), A220348.
Cf. A183089, A183209 (similar permutations), also A257798.

a183079 n k = a183079_tabf !! (n-1) !! (k-1)
a183079_row n = a183079_tabf !! n
a183079_tabf = [1] : iterate (\row -> concatMap f row) [2]
   where f x = [a000217 x, a014132 x]
a183079_list = concat a183079_tabf
-- Reinhard Zumkeller, Dec 12 2012

tr[n_]:=n*(n+1)/2; nt[n_]:= n+Round@ Sqrt[2*n];a[1]=1; a[n_Integer] := a[n] = If[ EvenQ@n, nt@a[n/2], tr@ a@ Ceiling[n/2]]; a/@Range[58] (* Giovanni Resta, May 20 2015 *)

;; With memoizing definec-macro.
(definec (A183079 n) (cond ((<= n 1) n) ((even? n) (A014132 (A183079 (/ n 2)))) (else (A000217 (A183079 (/ (+ n 1) 2))))))
;; Antti Karttunen, May 18 2015

Let L(n) be the n-th triangular number (A000217).
Let U(n) be the n-th non-triangular number (A014132).
The tree-array T(n,k) is then given by rows:
T(0,0)=1; T(1,0)=2;
T(n,2j)=L(T(n-1,j));
T(n,2j+1)=U(T(n-1,j));
for j=0,1,...,2^(n-1)-1, n>=2.
a(1) = 1; after which: a(2n) = A014132(a(n)), a(2n+1) = A000217(a(n+1)). - Antti Karttunen, May 20 2015

Formula added to the name and a new tree illustration to the Example section by Antti Karttunen, May 20 2015

A048680 Nonnegative integers A001477 expanded with rewrite 0->0, 01->1, then interpreted as Zeckendorffian expansions (as numbers of Fibonacci number system).

Original entry on oeis.org

0, 1, 2, 4, 3, 6, 7, 12, 5, 9, 10, 17, 11, 19, 20, 33, 8, 14, 15, 25, 16, 27, 28, 46, 18, 30, 31, 51, 32, 53, 54, 88, 13, 22, 23, 38, 24, 40, 41, 67, 26, 43, 44, 72, 45, 74, 75, 122, 29, 48, 49, 80, 50, 82, 83, 135, 52, 85, 86, 140, 87, 142, 143, 232, 21, 35, 36, 59, 37, 61

Offset: 0

Antti Karttunen, Jul 14 1999

A permutation of the nonnegative integers (A001477). Inverse permutation to A048679, i.e. A048679[ A048680[ n ] ] = n for all n and vice versa.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Ron Knott, About Fibonacci Number System
Index entries for sequences that are permutations of the natural numbers

Cf. A003714, A005203, A048678, A048679.

Equals A074049(n+1) - 1.

rewrite_0to0_1to01 := proc(n) option remember; if(n < 2) then RETURN(n); else RETURN(((2^(1+(n mod 2))) * rewrite_0to0_1to01(floor(n/2))) + (n mod 2)); fi; end; interpret_as_zeckendorf_expansion := n -> sum('(bit_i(n,i)*fib(i+2))','i'=0..floor_log_2(n));

a(n)=my(k=1,s);while(n,if(n%2,s+=fibonacci(k++));k++;n>>=1);s \\ Charles R Greathouse IV, Nov 17 2013

a(n) = interpret_as_zeckendorf_expansion(rewrite_0to0_1to01(n)) (where rewrite_0to0_1to01(n)=A048678[ n ])

A026242 a(n) = j if n is L(j), else a(n) = k if n is U(k), where L = A000201, U = A001950 (lower and upper Wythoff sequences).

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 3, 5, 6, 4, 7, 8, 5, 9, 6, 10, 11, 7, 12, 8, 13, 14, 9, 15, 16, 10, 17, 11, 18, 19, 12, 20, 21, 13, 22, 14, 23, 24, 15, 25, 16, 26, 27, 17, 28, 29, 18, 30, 19, 31, 32, 20, 33, 21, 34, 35, 22, 36, 37, 23, 38, 24, 39, 40, 25, 41

Offset: 1

Clark Kimberling

Every positive integer occurs exactly twice. a(n) is the parent of n in the tree at A074049. - Clark Kimberling, Dec 24 2010
Apparently, if n=F(m) (a Fibonacci number), one of two circumstances arise:
I. a(n)=F(m-1) and a(n-1)=F(m-2). When this happens, a(n) occurs for the first time and a(n-1) occurs for the second time;
II. a(n)=F(m-2) and a(n-1)=F(m-1). When this happens, a(n) occurs for the second time and a(n-1) occurs for the first time. - Bob Selcoe, Sep 18 2014
These are the numerators when all fractions, j/r and k/r^2, are arranged in increasing order (where r = golden ratio and j,k are positive integers). - Clark Kimberling, Mar 02 2015

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 999 terms from M. F. Hasler)
S. Mneimneh, Fibonacci in The Curriculum: Not Just a Bad Recurrence, in Proceeding SIGCSE '15 Proceedings of the 46th ACM Technical Symposium on Computer Science Education, Pages 253-258. See Figure 2.
Jeffrey Shallit, Fibonacci automaton for a(n)

Cf. A026272, A074049, A089910.

Cf. A000045 (Fibonacci numbers).

mx = 100; gr = GoldenRatio; LW[n_] := Floor[n*gr]; UW[n_] := Floor[n*gr^2]; alw = Array[LW, Ceiling[mx/gr]]; auw = Array[UW, Ceiling[mx/gr^2]]; f[n_] := If[ MemberQ[alw, n], Position[alw, n][[1, 1]], Position[auw, n][[1, 1]]]; Array[f, mx] (* Robert G. Wilson v, Sep 17 2014 *)

my(A=vector(10^4),i,j=0); while(#A>=i=A000201(j++), A[i]=j; (i=A001950(j))>#A || A[i]=j); A026242=A \\ M. F. Hasler, Sep 16 2014 and Sep 18 2014

A026242=vector(#A002251,n,abs(A002251[n]-n)) \\ M. F. Hasler, Sep 17 2014

a(n) = a(m) if a(m) has already occurred exactly once and n = a(m) + m; otherwise, a(n) = least positive integer that has not yet occurred.
a(n) = abs(A002251(n) - n).
n = a(n) + a(n-1) unless n = A089910(m); if n = A089910(m), then n = a(n) + a(n-1) - m. - Bob Selcoe, Sep 20 2014
There is a 17-state automaton that accepts the Zeckendorf (Fibonacci) representation of n and a(n), in parallel.  See the file a026242.pdf. - Jeffrey Shallit, Dec 21 2023

A178528 Tree generated by the Beatty sequence of sqrt(3).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 9, 8, 11, 12, 16, 10, 14, 15, 21, 13, 18, 19, 26, 20, 28, 27, 37, 17, 23, 24, 33, 25, 35, 36, 49, 22, 30, 31, 42, 32, 44, 45, 61, 34, 47, 48, 66, 46, 63, 64, 87, 29, 40, 39, 54, 41, 56, 57, 78, 43, 59, 60, 82, 62, 85, 84, 115

Offset: 1

Clark Kimberling, Dec 23 2010

A permutation of the positive integers.

			First levels of the tree:
.....................1
.....................2
..............3..............4
..........5.......7......6.......9
........8..11..12..16..10..14..15..21

Cf. A022838 (Beatty sequence of sqrt(3)), A054406, A074049.

a = {1, 2}; row = {a[[-1]]}; r = Sqrt[3]; s = r/(r - 1); Do[a = Join[a, row = Flatten[{Floor[#*{r, s}]} & /@ row]], {n, 5}]; a (* Ivan Neretin, Nov 09 2015 *)

Let r=sqrt(3) and s=r/(r-1). The tree-array T(n,k) is then
given by rows: T(0,0)=1; T(1,0)=2;
T(n,2*j)=floor(r*T(n-1,j));
T(n,2*j+1)=floor(s*T(n-1,j));
for j=0,1,...,2^(n-1)-1, n>=2.

A183170 First of two trees generated by the Beatty sequence of sqrt(2).

Original entry on oeis.org

1, 3, 4, 10, 5, 13, 14, 34, 7, 17, 18, 44, 19, 47, 48, 116, 9, 23, 24, 58, 25, 61, 62, 150, 26, 64, 66, 160, 67, 163, 164, 396, 12, 30, 32, 78, 33, 81, 82, 198, 35, 85, 86, 208, 87, 211, 212, 512, 36, 88, 90, 218, 93, 225, 226, 546, 94, 228

Offset: 1

Clark Kimberling, Dec 28 2010

This tree grows from (L(1),U(1))=(1,3). The other tree, A183171, grows from (L(2),U(2))=(2,6). Here, L is the Beatty sequence A001951 of r=sqrt(2); U is the Beatty sequence A001952 of s=r/(r-1). The two trees are complementary; that is, every positive integer is in exactly one tree. (L and U are complementary, too.) The sequence formed by taking the terms of this tree in increasing order is A183172.

			First levels of the tree:
.......................1
.......................3
..............4...................10
.........5..........13........14........34
.......7..17......18..44....19..47....48..116

Ivan Neretin, Table of n, a(n) for n = 1..8192

Cf. A183171, A183172, A001951, A001952, A178528, A074049.

a = {1, 3}; row = {a[[-1]]}; r = Sqrt[2]; s = r/(r - 1); Do[a = Join[a, row = Flatten[{Floor[#*{r, s}]} & /@ row]], {n, 5}]; a (* Ivan Neretin, May 25 2015 *)

See the formula at A178528, but use r=sqrt(2) instead of r=sqrt(3).

A183542 First of two complementary trees generated by the Wythoff sequences.

Original entry on oeis.org

1, 2, 5, 4, 8, 9, 16, 7, 13, 14, 24, 15, 26, 27, 45, 12, 21, 22, 37, 23, 39, 40, 66, 25, 42, 43, 71, 44, 73, 74, 121, 20, 34, 55, 58, 36, 60, 61, 100, 38, 63, 64, 105, 65, 107, 108, 176, 41, 68, 69, 113, 70, 115, 116, 189, 72, 118, 119, 194, 120, 196, 197, 320

Offset: 1

Clark Kimberling, Jan 05 2011

Begin with the main tree A074049 generated by the Wythoff sequences:
...................1
...................2
...........3.................5
.......4.......7........8........13
.....6..10...11..18....12..20...21..34
Every n >2 is in the subtree from 3 or the subtree from 5.  Therefore, on subtracting 2 from all entries in those subtrees, we obtain complementary trees: A183342 and A183543.

			First three levels:
...................1
.............2............3
..........4.....8......9.....16

Cf. A074049, A183543, A183544,

A183079 (definition of tree generated by a sequence).

See the formulas at A074049 and A183544.

A183543 Second of two complementary trees generated by the Wythoff sequences.

Original entry on oeis.org

3, 6, 11, 10, 18, 19, 32, 17, 29, 30, 50, 31, 52, 53, 87, 28, 47, 48, 79, 49, 81, 82, 134, 51, 84, 85, 139, 86, 141, 142, 231, 46, 76, 77, 126, 78, 128, 129, 210, 80, 131, 132, 215, 133, 217, 218, 354, 83, 136, 137, 223, 138, 225, 226, 367, 140, 228, 229, 372, 230, 374, 375, 608

Offset: 1

Clark Kimberling, Jan 05 2011

See A183542.

			First three levels:
...............3
.......6.............11
....10...18.......19....32

Cf. A074049, A183542, A183545.

See the formulas as A074049 and A183545.

A183080 Tree generated by the Beatty sequence of 3-sqrt(2).

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 7, 13, 6, 10, 12, 21, 11, 18, 20, 35, 9, 16, 15, 27, 19, 32, 33, 56, 17, 29, 28, 48, 31, 54, 55, 94, 14, 24, 25, 43, 23, 40, 42, 73, 30, 51, 50, 86, 52, 89, 88, 151, 26, 46, 45, 78, 44, 75, 76, 129, 49, 83, 85, 146, 87, 148, 149, 254

Offset: 1

Clark Kimberling, Dec 23 2010

A permutation of the positive integers. See the note at A183079.

			First five rows:
1
2
3 5
4 8 7 13
6 10 12 21 11 18 20 35

Cf. A183079, A178528, A074049.

a = {1, 2}; row = {a[[-1]]}; r = 3 - Sqrt[2]; s = r/(r - 1); Do[a = Join[a, row = Flatten[{Floor[#*{r, s}]} & /@ row]], {n, 5}]; a (* Ivan Neretin, Nov 09 2015 *)

Let L(n)=floor(n*r), U(n)=floor(n*s), where r=3-sqrt(2) and s=r/(r-1).

The tree-array T(n,k) is then given by rows: T(0,0) = 1; T(1,0) = 2; T(n,2j) = L(T(n-1),j); T(n,2j+1) = U(T(n-1),j); for j=0,1,...,2^(n-1)-1, n>=2.

A183081 Tree generated by the Beatty sequence of 4-sqrt(5).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 8, 11, 10, 13, 12, 16, 15, 20, 14, 18, 19, 25, 17, 23, 22, 30, 21, 27, 28, 36, 26, 34, 35, 46, 24, 32, 31, 41, 33, 43, 44, 57, 29, 39, 40, 53, 38, 50, 52, 69, 37, 48, 47, 62, 49, 64, 63, 83, 45, 60, 59, 78, 61, 80, 81

Offset: 1

Clark Kimberling, Dec 23 2010

A permutation of the positive integers. See the note at A183079.

			Top 5 rows:
1
2
3 4
5 6 7 9
8 11 10 13 12 16 15 20

Cf. A074049, A178528, A183079.

a = {1, 2}; row = {a[[-1]]}; r = 4 - Sqrt[5]; s = r/(r - 1); Do[a = Join[a, row = Flatten[{Floor[#*{r, s}]} & /@ row]], {n, 5}]; a (* Ivan Neretin, Nov 09 2015 *)

Let L(n)=floor(n*r), U(n)=floor(n*s), where r=4-sqrt(5) and s=r/(r-1).
The tree-array T(n,k) is then given by rows:
T(0,0) = 1; T(1,0) = 2; T(n,2j) = L(T(n-1),j); T(n,2j+1) = U(T(n-1),j);
for j=0,1,...,2^(n-1)-1, n>=2.

Showing 1-10 of 14 results. Next

cp's OEIS Frontend

A074050 Inverse permutation to A074049.

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A183079 Tree generated by the triangular numbers: a(1) = 1; a(2n) = nontriangular(a(n)), a(2n+1) = triangular(a(n+1)), where triangular = A000217, nontriangular = A014132.

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A048680 Nonnegative integers A001477 expanded with rewrite 0->0, 01->1, then interpreted as Zeckendorffian expansions (as numbers of Fibonacci number system).

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A026242 a(n) = j if n is L(j), else a(n) = k if n is U(k), where L = A000201, U = A001950 (lower and upper Wythoff sequences).

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A178528 Tree generated by the Beatty sequence of sqrt(3).

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A183170 First of two trees generated by the Beatty sequence of sqrt(2).

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A183542 First of two complementary trees generated by the Wythoff sequences.

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A183543 Second of two complementary trees generated by the Wythoff sequences.

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