A099257 -id:A099257

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A099258

Inverse of A099257.

+20
3

1, 3, 2, 4, 6, 7, 8, 9, 5, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Permutation of the natural numbers;

a(A033484(n)) = A033484(n);

a(A033484(n) - 1) = A033484(n)/2 = A055010(n).

LINKS

Table of n, a(n) for n=1..72.

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(n) = if n = 3*2^k - 2 then n else (if n = 3*2^k - 3 then (n + 1)/2 else n + 1).

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Oct 09 2004

STATUS

approved

A033484

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

+10
70

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

REFERENCES

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.

LINKS

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).

FORMULA

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

EXAMPLE

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 + ...

MAPLE

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

MATHEMATICA

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 *)

PROG

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

(PARI) a(n) = 3<<n-2; \\ Charles R Greathouse IV, Nov 02 2011

(Haskell)

a033484 = (subtract 2) . (* 3) . (2 ^)

a033484_list = iterate ((subtract 2) . (* 2) . (+ 2)) 1

-- Reinhard Zumkeller, Apr 23 2013

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

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

CROSSREFS

Cf. A000045, A007283, A036563, A131110, A051597, A132776, A001045.

Cf. A000918.

Cf. A112468, A112739.

Cf. A082560, A000079, A232642, A128588.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A068156

G.f.: (x+2)*(x+1)/((x-1)*(x-2)) = Sum_{n>=0} a(n)*(x/2)^n.

+10
32

1, 3, 9, 21, 45, 93, 189, 381, 765, 1533, 3069, 6141, 12285, 24573, 49149, 98301, 196605, 393213, 786429, 1572861, 3145725, 6291453, 12582909, 25165821, 50331645, 100663293, 201326589, 402653181, 805306365, 1610612733

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Number of moves to solve Hard Pagoda puzzle.

Partial sums of A111286. Binomial transform of (1,2,4,2,4,2,4 ....). - Paul Barry, Feb 28 2003

Warren W. Kokko writes that this sequence also appears to give the number of scoring sequences for the Racer Dice Game with n dice. - N. J. A. Sloane, Feb 24 2015

From Michel Lagneau, Apr 27 2015: (Start)

For n > 0, a(n) is the number of identical bowls having the same weight except for one which has a higher weight than the others which are identifiable by a weighing machine using n weighings.

Example: a(2)=9 because two weighings are sufficient:

Start with 9 bowls;

Step 1: remove 3 bowls => there are still 6 bowls;

Step 2: first weighing of 6 bowls (3 bowls on each side of the weighing machine);

Step 3: if the machine is in equilibrium, we find immediately the unknown bowl with a second weighing from the first 3 removing bowls. Else, we find immediately the unknown bowl with a second weighing from the 3 heaviest bowls.

Note: If the unknown bowl has a lower weight, the reasoning is the same, but it is necessary to know whether the unknown bowl is heavier or lighter.

In the general case, we always remove 3 bowls in step 1.

(End)

The number of ternary words of length n that avoid {11-2,22-1}. G.f. [1+(k-1)*x^2]/[1-k*x+(k-1)*x^2] at k=3. [Theorem 7.93 by Heubach and Mansour]. - R. J. Mathar, May 22 2016

Apart from the first term, column 2 of A222057. - Anton Zakharov, Oct 27 2016

REFERENCES

Richard I. Hess, Compendium of Over 7000 Wire Puzzles, privately printed, 1991.

Richard I. Hess, Analysis of Ring Puzzles, booklet distributed at 13th International Puzzle Party, Amsterdam, Aug 20 1993.

S. Heubach, T. Mansour, in Combinatorics of Compositions and words, Discr. Math. Applicat. (ed by K H Rosen), CRC Press 2010, p 300.

Warren W. Kokko, The Racer Dice Game, Manuscript, 2015.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Artur Schaefer, Endomorphisms of The Hamming Graph and Related Graphs, arXiv preprint arXiv:1602.02186 [math.CO], 2016. See Table Remark 4.5.

Index entries for linear recurrences with constant coefficients, signature (3,-2).

FORMULA

a(0) = 1, a(n) = A060482(2n+1). For n > 0, a(n+1) = 2*a(n)+3.

G.f.: (1+2*x^2)/((1-2*x)*(1-x)). - Paul Barry, Feb 28 2003

a(n) = 3*2^n+0^n-3. - Paul Barry, Sep 04 2003

a(n) = A099257(A033484(n)+1) = 2*A033484(n) + 1. - Reinhard Zumkeller, Oct 09 2004

a(n) = 3*a(n-1) - 2*a(n-2), n > 1. - Vincenzo Librandi, Nov 11 2011

a(n) = a(n-1)+ 3*2^(n-1); a(1)=3. - Ctibor O. Zizka, Apr 17 2008

E.g.f.: 1 + 3*(exp(x) - 1)*exp(x). - Ilya Gutkovskiy, May 22 2016

MATHEMATICA

Join[{1}, LinearRecurrence[{3, -2}, {3, 9}, 30]] (* Jean-François Alcover, Jan 08 2019 *)

CoefficientList[Series[(1+2x^2)/((1-2x)(1-x)), {x, 0, 40}], x] (* Harvey P. Dale, Jan 02 2022 *)

PROG

(Magma) [3*2^n+0^n-3 : n in [0..30]]; // Vincenzo Librandi, Nov 11 2011

(Sage) def a(n): return 3*2**n+0**n-3 # Torlach Rush, Jan 09 2025

CROSSREFS

A diagonal of A233308 (for n > 1).

Cf. A000079.

KEYWORD

nonn,easy

AUTHOR

Benoit Cloitre, Mar 12 2002

STATUS

approved

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