A072638 -id:A072638 - cp's OEIS frontend

A071653 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using the bivariate form of A001477 as the packing bijection N x N -> N.

0, 1, 3, 2, 10, 6, 5, 7, 4, 66, 28, 21, 36, 15, 14, 9, 12, 56, 22, 8, 16, 29, 11, 2278, 435, 253, 703, 136, 120, 55, 91, 1653, 276, 45, 153, 465, 78, 77, 35, 27, 44, 20, 25, 18, 68, 2212, 407, 30, 232, 667, 121, 19, 13, 23, 106, 46, 38, 79, 1597, 254, 17, 37, 137, 436

Offset: 0

Antti Karttunen, May 30 2002

nonn

A071653(A014137(n-1)) = A072638(n) for all n > 0. - Paul D. Hanna, Jan 04 2007

Also seems that A071653(A014137(n)-1) = A006894(n) for all n > 0. - Antti Karttunen, Jul 30 2012

Antti Karttunen, Table of n, a(n) for n = 0..2055
A. Karttunen, Alternative Catalan Orderings
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A071654. Cf. also A014486, A001477, A071651, A071652.

A108225 a(0) = 0, a(1) = 2; for n >= 2, a(n) = (a(n-1) + a(n-2))*(a(n-1) - a(n-2) + 1)/2.

Original entry on oeis.org

0, 2, 3, 5, 12, 68, 2280, 2598062, 3374961778893, 5695183504492614029263280, 16217557574922386301420536972254869595782763547562

Offset: 0

N. J. A. Sloane, Jun 16 2005

nonn

From a posting by Antreas P. Hatzipolakis to the Yahoo news group "Hyacinthos", circa Jun 10 2005.
The next term has 99 digits. - Harvey P. Dale, Jun 09 2011
a(n) for n>0 gives the rank of the unlabeled binary rooted tree, among those with n+1 leaves, that has the largest rank according to the bijection of Colijn and Plazzotta (2018) between unlabeled binary rooted trees and positive integers. - Noah A Rosenberg, Jun 03 2022

Robert Israel, Table of n, a(n) for n = 0..14
C. Colijn and G. Plazzotta, A metric on phylogenetic tree shapes, Syst. Biol., 67 (2018), 113-126.
Luc Devroye, Michael R. Doboli, Noah A. Rosenberg, and Stephan Wagner, Tree height and the asymptotic mean of the Colijn-Plazzotta rank of unlabeled binary rooted trees, arXiv:2409.18956 [math.CO], 2024. See p. 3.
N. A. Rosenberg, On the Colijn-Plazzotta numbering scheme for unlabeled binary rooted trees, Discr. Appl. Math., 291 (2021), 88-98.
J.S. Seneschal, Iteration of Semi-Complete Graphs

First differences give A103410.
Cf. A006894.
Cf. A000217, A007501, A002658, A072638.

F:=proc(n) option remember; if n <= 1 then RETURN(2*n) fi; (F(n-1)+F(n-2))*(F(n-1)-F(n-2)+1)/2; end;
a[ -2]:=-2:a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=binomial(a[n-1]+2,2) od: seq(a[n]+2, n=-2..8); # Zerinvary Lajos, Jun 08 2007

RecurrenceTable[{a[0]==0,a[1]==2,a[n]==(a[n-1]+a[n-2])(a[n-1]- a[n-2]+1)/2},a[n],{n,15}] (* Harvey P. Dale, Jun 09 2011 *)

Conjecture: a(n) = A006894(n) + 1. - R. J. Mathar, Apr 23 2007
From J.S. Seneschal, Jul 17 2025 (Start)
a(n) = A000217(a(n)) - A072638(n) = A072638(n-1) + 2.
a(n) = A002658(n-1) + a(n-1) for n > 1. (End)

A204009 a(n) is a binary vector for selecting distinct terms from A000124 that when summed give n; it uses the greedy algorithm.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 1000, 1001, 1010, 1011, 10000, 10001, 10010, 10011, 10100, 100000, 100001, 100010, 100011, 100100, 100101, 1000000, 1000001, 1000010, 1000011, 1000100, 1000101, 1000110, 10000000, 10000001, 10000010, 10000011, 10000100

Offset: 0

Frank M Jackson, Jan 09 2012

nonn

a(n) is a binary vector for selecting terms from A000124 that when summed give n. It uses the greedy algorithm to select from multiple solutions.

			14 can be written as 7+4+2+1, i.e., 1111, or as 11+2+1, i.e., 10011, and the latter is chosen because it uses the greedy algorithm for selection.

Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]

Cf. A000124, A072638.

complete[m_Integer] := (m(m+1)/2+1); gentable[n_Integer] := (m=n; ptable={0}; While[m!=0, (i=0; While[complete[i]<=m&&ptable[[i+1]]!=1, (AppendTo[ptable, 0]; i++)]; ptable[[i]]=1; m=m-complete[i-1])]; ptable); decimal[n_Integer] := (gentable[n]; Sum[2^(k-1)*ptable[[k]], {k, 1, Length[ptable]}]); Table[IntegerString[decimal[s], 2], {s, 0, 100}]

a(n) x A000124 = n, where x is the inner product and the binary vector is in ascending powers of 2 with infinite trailing zeros.

Edited by N. J. A. Sloane, May 20 2023

A329221 a(0)=0. If a(n)=k is the first occurrence of k then a(n+1)=a(k), otherwise a(n+1)=n-m where m is the index of the greatest prior term.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 1, 2, 3, 0, 1, 2, 3, 4, 1, 1, 2, 3, 4, 5, 2, 1, 2, 3, 4, 5, 6, 1, 1, 2, 3, 4, 5, 6, 7, 1, 1, 2, 3, 4, 5, 6, 7, 8, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 3, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 1, 2, 3, 4, 5, 6, 7, 8

Offset: 0

David James Sycamore, Nov 22 2019

Subsequence a(A000217(k+1)), k>=0 is an identical copy of the original. Erasure of the first occurrence of every k does not reproduce the original so this is not a fractal sequence. However, if a(0) and the copy subsequence are both erased, what remains is A002260. Hence this sequence contains both a copy identical to the original, and a fractal subsequence different from the original.

			a(0)=0 is the first occurrence of the term 0, therefore a(1)=a(0+1)=a(0)=0. a(1)=0 has been seen before, and 0 is the index of the greatest prior term (0), so a(2)=a(1+1)=1-0=1.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Wikipedia, Fractal sequence

Cf. A000217, A002260, A181391, A072638.

Block[{a, c, j, k, m, r, nn}, nn = 120; c[] := 0; a[0] = j = r = m = 0; Do[If[c[j] == 0, k = a[j], k = n - m - 1]; c[j]++; Set[{a[n], j}, {k, k}]; If[k > r, r = k; m = n], {n, nn}]; Array[a, nn + 1, 0] ] (* _Michael De Vlieger, Jun 30 2025 *)

a(k) = a(A000217(k+1)), k >= 0.
The n-th occurrence of k is a((k^2 + (2*n+1)*k + n*(n-1))/2), k >= 1.
The n-th occurrence of 0 is a(A072638(n)), n >= 0.

A225505 a(n) = triangular(a(n-1)+a(n-2)) if n > 1, else a(n) = n.

Original entry on oeis.org

0, 1, 1, 3, 10, 91, 5151, 13741903, 94490753712985, 4464252567106907867941273716, 9964775491460730298984873904585383048580645394630925051

Offset: 0

Alex Ratushnyak, May 09 2013

nonn

Cf. A000217, A072638.

prpr, prev = 0, 1
for i in range(1, 17):
  print(prpr, end=', ')
  n = prpr+prev
  cur = n*(n+1)//2
  prpr, prev = prev, cur

cp's OEIS Frontend

A071653 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using the bivariate form of A001477 as the packing bijection N x N -> N.

Views

Author

Keywords

Comments

Links

Crossrefs

A108225 a(0) = 0, a(1) = 2; for n >= 2, a(n) = (a(n-1) + a(n-2))*(a(n-1) - a(n-2) + 1)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A204009 a(n) is a binary vector for selecting distinct terms from A000124 that when summed give n; it uses the greedy algorithm.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A329221 a(0)=0. If a(n)=k is the first occurrence of k then a(n+1)=a(k), otherwise a(n+1)=n-m where m is the index of the greatest prior term.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A225505 a(n) = triangular(a(n-1)+a(n-2)) if n > 1, else a(n) = n.

Views

Author

Keywords

Crossrefs

Programs

Python