A232642 -id:A232642 - cp's OEIS frontend

A232643 Inverse permutation of the sequence of positive integers at A232642.

1, 2, 4, 3, 6, 5, 9, 8, 14, 7, 12, 11, 19, 10, 17, 16, 27, 15, 25, 24, 40, 13, 22, 21, 35, 20, 33, 32, 53, 18, 30, 29, 48, 28, 46, 45, 74, 26, 43, 42, 69, 41, 67, 66, 108, 23, 38, 37, 61, 36, 59, 58, 95, 34, 56, 55, 90, 54, 88, 87, 142, 31, 51, 50, 82, 49

Offset: 1

Clark Kimberling, Nov 28 2013

Clark Kimberling and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from Clark Kimberling)

Cf. A232559, A232639, A232642, A000045.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a232643 = (+ 1) . fromJust . (`elemIndex` a232642_list)
-- Reinhard Zumkeller, May 14 2015

z = 14; g[1] = {1}; g[2] = {2}; g[n_] := Riffle[g[n - 1] + 1, 2 g[n - 1] + 2]; j[2] = Join[g[1], g[2]]; j[n_] := Join[j[n - 1], g[n]]; g1[n_] := DeleteDuplicates[DeleteCases[g[n], Alternatives @@ j[n - 1]]]; g1[1] = g[1]; g1[2] = g[2]; t = Flatten[Table[g1[n], {n, 1, z}]]  (* A232642 *)
Table[Length[g1[n]], {n, 1, z}]  (* A000045 *)
Flatten[Table[Position[t, n], {n, 1, 200}]]  (* A232643 *)

b-File corrected by Reinhard Zumkeller, May 14 2015

A033484 a(n) = 3*2^n - 2.

Original entry on oeis.org

1, 4, 10, 22, 46, 94, 190, 382, 766, 1534, 3070, 6142, 12286, 24574, 49150, 98302, 196606, 393214, 786430, 1572862, 3145726, 6291454, 12582910, 25165822, 50331646, 100663294, 201326590, 402653182, 805306366, 1610612734, 3221225470

Offset: 0

N. J. A. Sloane

Number of nodes in rooted tree of height n in which every node (including the root) has valency 3.
Pascal diamond numbers: reflect Pascal's n-th triangle vertically and sum all elements. E.g., a(3)=1+(1+1)+(1+2+1)+(1+1)+1. - Paul Barry, Jun 23 2003
Number of 2 X n binary matrices avoiding simultaneously the right-angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2 and j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). - Sergey Kitaev, Nov 11 2004
Binomial and inverse binomial transform are in A001047 (shifted) and A122553. - R. J. Mathar, Sep 02 2008
a(n) = (Sum_{k=0..n-1} a(n)) + (2*n + 1); e.g., a(3) = 22 = (1 + 4 + 10) + 7. - Gary W. Adamson, Jan 21 2009
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is a proper subset of y or y is a proper subset of x and x and y are disjoint, or 1) x equals y. Then a(n) = |R|. - Ross La Haye, Mar 19 2009
Equals the Jacobsthal sequence A001045 convolved with (1, 3, 4, 4, 4, 4, 4, ...). - Gary W. Adamson, May 24 2009
Equals the eigensequence of a triangle with the odd integers as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010
An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 58, 154, 178 and 184, lead to this sequence. For the corner squares these vectors lead to the companion sequence A097813. - Johannes W. Meijer, Aug 15 2010
a(n+2) is the integer with bit string "10" * "1"^n * "10".
a(n) = A027383(2n). - Jason Kimberley, Nov 03 2011
a(n) = A153893(n)-1 = A083416(2n+1). - Philippe Deléham, Apr 14 2013
a(n) = A082560(n+1,A000079(n)) = A232642(n+1,A128588(n+1)). - Reinhard Zumkeller, May 14 2015
a(n) is the sum of the entries in the n-th and (n+1)-st rows of Pascal's triangle minus 2. - Stuart E Anderson, Aug 27 2017
Also the number of independent vertex sets and vertex covers in the complete tripartite graph K_{n,n,n}. - Eric W. Weisstein, Sep 21 2017
Apparently, a(n) is the least k such that the binary expansion of A000045(k) ends with exactly n+1 ones. - Rémy Sigrist, Sep 25 2021
a(n) is the number of root ancestral configurations for a pair consisting of a matching gene tree and species tree with the modified lodgepole shape and n+1 cherry nodes. - Noah A Rosenberg, Jan 16 2025

			Binary: 1, 100, 1010, 10110, 101110, 1011110, 10111110, 101111110, 1011111110, 10111111110, 101111111110, 1011111111110, 10111111111110,
G.f. = 1 + 4*x + 10*x^2 + 22*x^3 + 46*x^4 + 94*x^5 + 190*x^6 + 382*x^7 + ...

J. Riordan, Series-parallel realization of the sum modulo 2 of n switching variables, in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 877-878.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, The Triple Riordan Group, arXiv:2412.05461 [math.CO], 2024. See pp. 3, 10.
Dennis E. Davenport, Shakuan K. Frankson, Louis W. Shapiro, and Leon C. Woodson, An Invitation to the Riordan Group, Enum. Comb. Appl. (2024) Vol. 4, No. 3, Art. #S2S1. See p. 22.
Erik D. Demaine et al., Picture-Hanging Puzzles, arXiv:1203.3602 [cs.DS], 2012, 2014. See p. 8, actually length(Sn) is 2^n+2^(n-1)-2, that is, a(n-1).
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Egor Lappo and Noah A. Rosenberg, A lattice structure for ancestral configurations arising from the relationship between gene trees and species trees, Adv. Appl. Math. 343 (2024), 65-81.
Eric Weisstein's World of Mathematics, Complete Tripartite Graph
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000045, A007283, A036563, A131110, A051597, A132776, A001045.
Cf. A000918.
Cf. A112468, A112739.
Cf. A082560, A000079, A232642, A128588.

List([0..35], n-> 3*2^n -2); # G. C. Greubel, Nov 18 2019

a033484 = (subtract 2) . (* 3) . (2 ^)
a033484_list = iterate ((subtract 2) . (* 2) . (+ 2)) 1
-- Reinhard Zumkeller, Apr 23 2013

[3*2^n-2: n in [1..36]]; // Vincenzo Librandi, Nov 22 2010

with(combinat):a:=n->stirling2(n,2)+stirling2(n+1,2): seq(a(n), n=1..35); # Zerinvary Lajos, Oct 07 2007
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=(a[n-1]+1)*2 od: seq(a[n], n=1..35); # Zerinvary Lajos, Feb 22 2008

Table[3 2^n - 2, {n, 0, 35}] (* Vladimir Joseph Stephan Orlovsky, Dec 16 2008 *)
(* Start from Eric W. Weisstein, Sep 21 2017 *)
3*2^Range[0, 35] - 2
LinearRecurrence[{3, -2}, {1, 4}, 36]
CoefficientList[Series[(1+x)/(1-3x+2x^2), {x, 0, 35}], x] (* End *)

a(n) = 3<Charles R Greathouse IV, Nov 02 2011

[3*2^n -2 for n in (0..35)] # G. C. Greubel, Nov 18 2019

G.f.: (1+x)/(1-3*x+2*x^2).
a(n) = 2*(a(n-1) + 1) for n>0, with a(0)=1.
a(n) = A007283(n) - 2.
G.f. is equivalent to (1-2*x-3*x^2)/((1-x)*(1-2*x)*(1-3*x)). - Paul Barry, Apr 28 2004
From Reinhard Zumkeller, Oct 09 2004: (Start)
A099257(a(n)) = A099258(a(n)) = a(n).
a(n) = 2*A055010(n) = (A068156(n) - 1)/2. (End)
Row sums of triangle A130452. - Gary W. Adamson, May 26 2007
Row sums of triangle A131110. - Gary W. Adamson, Jun 15 2007
Binomial transform of (1, 3, 3, 3, ...). - Gary W. Adamson, Oct 17 2007
Row sums of triangle A051597 (a triangle generated from Pascal's rule given right and left borders = 1, 2, 3, ...). - Gary W. Adamson, Nov 04 2007
Equals A132776 * [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, Nov 16 2007
a(n) = Sum_{k=0..n} A112468(n,k)*3^k. - Philippe Deléham, Feb 23 2014
a(n) = -(2^n) * A036563(1-n) for all n in Z. - Michael Somos, Jul 04 2017
E.g.f.: 3*exp(2*x) - 2*exp(x). - G. C. Greubel, Nov 18 2019

A128588 Expansion of g.f. x*(1+x+x^2)/(1-x-x^2).

Original entry on oeis.org

1, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338

Offset: 1

Gary W. Adamson, Mar 11 2007

Previous name was: A007318 * A128587.
a(n)/a(n-1) tends to phi, 1.618... = A001622.
Regardless of initial two terms, any linearly recurring sequence with signature (1,1) will yield an a(n)/a(n+1) ratio tending to phi (the Golden Ratio). - Harvey P. Dale, Mar 29 2017
Apart from the initial term, double the Fibonacci numbers. O.g.f.: x*(1+x+x^2)/(1-x-x^2). a(n) gives the number of binary strings of length n-1 avoiding the substrings 000 and 111. a(n) also gives the number of binary strings of length n-1 avoiding the substrings 010 and 101. - Peter Bala, Jan 22 2008
Row lengths of triangle A232642. - Reinhard Zumkeller, May 14 2015
a(n) is the number of binary strings of length n-1 avoiding the substrings 000 and 111. - Allan C. Wechsler, Feb 13 2025

Reinhard Zumkeller, Table of n, a(n) for n = 1..2500
J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms.
Andrei Asinowski and Cyril Banderier, On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020) Leibniz International Proceedings in Informatics (LIPIcs) Vol. 159, 1:1-1:16.
Elena Barcucci, Antonio Bernini, Stefano Bilotta, and Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016. See 1st column of Table 2 p. 11.
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, page 52.
B. Winterfjord, Binary strings and substring avoidance.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000045, A001622, A128587, A128586, A007318.
Cf. A006355, A055389.
Cf. A014217, A069403, A153819.
Cf. A232642, A242593.
Cf. A038754, A068922.
Main diagonal of A303696.

Concatenation([1], List([2..40], n-> 2*Fibonacci(n))); # G. C. Greubel, Jul 10 2019

a128588 n = a128588_list !! (n-1)
a128588_list = 1 : cows where
                   cows = 2 : 4 : zipWith (+) cows (tail cows)
-- Reinhard Zumkeller, May 14 2015

[1] cat [2*Fibonacci(n): n in [2..40]]; // G. C. Greubel, Jul 10 2019

a:= n-> `if`(n<2, n, 2*(<<0|1>, <1|1>>^n)[1,2]):
seq(a(n), n=1..50);  # Alois P. Heinz, Apr 28 2018

nn=40; a=(1-x^3)/(1-x); b=x*(1-x^2)/(1-x); CoefficientList[Series[a^2 /(1-b^2), {x,0,nn}], x]  (* Geoffrey Critzer, Sep 01 2012 *)
LinearRecurrence[{1,1}, {1,2,4}, 40] (* Harvey P. Dale, Mar 29 2017 *)
Join[{1}, 2*Fibonacci[Range[2,40]]] (* G. C. Greubel, Jul 10 2019 *)

{a(n) = if( n<2, n==1, 2 * fibonacci(n))}; /* Michael Somos, Jul 18 2015 */

[1]+[2*fibonacci(n) for n in (2..40)] # G. C. Greubel, Jul 10 2019

G.f.: x*(1+x+x^2)/(1-x-x^2).
Binomial transform of A128587; a(n+2) = a(n+1) + a(n), n>3.
a(n) = A068922(n-1), n>2. - R. J. Mathar, Jun 14 2008
For n > 1: a(n+1) = a(n) + if a(n) odd then max{a(n),a(n-1)} else min{a(n),a(n-1)}, see also A038754. - Reinhard Zumkeller, Oct 19 2015
E.g.f.: 4*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5) - x. - Stefano Spezia, Feb 19 2023

New name from Joerg Arndt, Feb 16 2024

A082560 a(1)=1, a(n)=2*a(n-1) if n is odd, or a(n)=a(n/2)+1 if n is even.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 10, 4, 8, 7, 14, 6, 12, 11, 22, 5, 10, 9, 18, 8, 16, 15, 30, 7, 14, 13, 26, 12, 24, 23, 46, 6, 12, 11, 22, 10, 20, 19, 38, 9, 18, 17, 34, 16, 32, 31, 62, 8, 16, 15, 30, 14, 28, 27, 54, 13, 26, 25, 50, 24, 48, 47, 94, 7, 14, 13, 26, 12, 24, 23, 46, 11, 22, 21, 42, 20

Offset: 1

Benoit Cloitre, May 04 2003

b(1)=1, b(n)=2*b(n/2) if n is even, or b(n)=b(n-1)+1 if n is odd produces the sequence of natural numbers.

Seen as a triangle read by rows: T(1,1) = 1; T(n+1,2*k-1) = T(n,k)+1 and T(n+1,2*k) = 2*T(n,k)+2, 1 <= k <= 2^n. - Reinhard Zumkeller, May 13 2015

			.  1:                                 1
.  2:                 2                                4
.  3:        3               6                5                10
.  4:    4       8       7       14       6       12       11       22
.  5:  5  10   9  18   8  16  15   30   7  14  13   26  12   24  23   46

Reinhard Zumkeller, Rows n = 1..13 of triangle, flattened

Cf. A000079 (row lengths), A033484 (right edges), A166060 (row sums), A232642 (duplicates removed).

a082560 n k = a082560_tabf !! (n-1) !! (k-1)
a082560_row n = a082560_tabf !! (n-1)
a082560_tabf = iterate (concatMap (\x -> [x + 1, 2 * x + 2])) [1]
a082560_list = concat a082560_tabf
-- Reinhard Zumkeller, May 13 2015

a(n)=if(n<2,1,if(n%2,2*a(n-1),1+a(n/2)))

if n is in A010737 : a(n)=n-1

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A257956 Row sums of A232642, when seen as a triangle read by rows.

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A232643 Inverse permutation of the sequence of positive integers at A232642.

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A033484 a(n) = 3*2^n - 2.

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A128588 Expansion of g.f. x*(1+x+x^2)/(1-x-x^2).

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A082560 a(1)=1, a(n)=2*a(n-1) if n is odd, or a(n)=a(n/2)+1 if n is even.

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