A083667 -id:A083667 - cp's OEIS frontend

A008776 Pisot sequences E(2,6), L(2,6), P(2,6), T(2,6).

2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886, 5083731656658, 15251194969974

Offset: 0

N. J. A. Sloane, David W. Wilson

Definitions of Pisot and related sequences:
Pisot sequence E(x, y): a(0) = x, a(1) = y, a(n) = floor(a(n-1)^2/a(n-2) + 1/2) = nearest integer to a(n-1)^2/a(n-2), with 0 < x < y.
Pisot sequence L(x, y): a(0) = x, a(1) = y, a(n) = ceiling(a(n-1)^2/a(n-2)).
Pisot sequence P(x, y): a(0) = x, a(1) = y, a(n) = ceiling(a(n-1)^2/a(n-2) - 1/2).
Pisot sequence T(x, y): a(0) = x, a(1) = y, a(n) = floor(a(n-1)^2/a(n-2)).
Pisot/Shallit sequence S(x, y): a(0) = x, a(1) = y, a(n) = floor(a(n-1)^2/a(n-2)+1).
A025192 is the main entry for the sequence of numbers 2*3^n.
Number of tilings of a 4 X (4n+4) rectangle into T tetrominoes.
Numbers n such that 3^n = n/2 mod n. Cf. A066601 3^n mod n. - Zak Seidov, Aug 26 2006, Nov 20 2008
For n >= 1, a(n) is equal to the number of functions f:{1,2...,n}->{1,2,3} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3} we have f(x) != y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007
a(n+1) is the number of compositions of n when there are 2 types of each natural number. - Milan Janjic, Aug 13 2010
2*Sum_{n>=2} 1/A083667(n) = 2*Sum_{n>=2} 2^(-n)*3^(-((n*(n-1))/2)) = Sum_{n>=1} 1/Product_{k=1..n} A008776(k) = Sum_{n>=1} 1/Product_{k=1..n} 2*3^k = 0.17609845431233461692099660022134... . - Alexander R. Povolotsky, Aug 08 2011
Number of monic squarefree polynomials over F_3 of degree n+1. - Charles R Greathouse IV, Feb 07 2012
a(n) is the sum of the elements of the n-th power of the matrix {{1, 2}, {2, 1}}. - Griffin N. Macris, Mar 25 2016
Let D(m) denote the set of divisors of a number m, and consider s1(m) and s2(m) the sums of those divisors that are congruent to 1 and 2 (mod 3) respectively. This sequence lists the numbers m such that s1(m) = 1 and s2(m) = 2. - Michel Lagneau, Feb 09 2017
a(n) is the multiplicative order of k modulo 3^(n+1), where k is any number congruent to 2 or 5 modulo 9. Note that for n > 0, k is a primitive root modulo 3^(n+1) if and only if k == 2, 5 (mod 9). - Jianing Song, Apr 20 2021

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 203).

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..200
Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See pp. 2, 26.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 170
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Tanya Khovanova, Recursive Sequences
Craig Knecht, Sphinx tiling of a repetitive shape.
C. Moore, Some Polyomino Tilings of the Plane, arXiv:math/9905012 [math.CO], 1999.
C. Pisot, La répartition modulo 1 et les nombres algébriques, Ann. Scu. Norm. Sup. Pisa 2 ser, vol 7. no 3-4 (1938) p 205-248.
Index entries for linear recurrences with constant coefficients, signature (3).

Apart from initial term, same as A025192.
Cf. A080643.
Cf. A000244.

List([0..30], n-> 2*3^n); # G. C. Greubel, Sep 11 2019

a008776 = (* 2) . (3 ^)
a008776_list = iterate (* 3) 2  -- Reinhard Zumkeller, Oct 19 2015

[2*3^n: n in [0..30]]; // G. C. Greubel, Sep 11 2019

# E(x,y) is f(n,x,y,1/2), T(x,y) is f(n,x,y,0), and S(x,y) is f(n,x,y,1).
f:=proc(n,x,y,r) option remember;
if n=0 then x
elif n=1 then y
else floor(f(n-1,x,y,r)^2/f(n-2,x,y,r) + r); fi; end;
[seq(f(n,2,6,1/2),n=0..30)];
# N. J. A. Sloane, Jul 30 2016

Table[EulerPhi[3^n], {n, 0, 100}] (* Artur Jasinski, Nov 19 2008 *)
Table[MatrixPower[{{1,2},{1,2}},n][[1]][[2]],{n,0,44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
NestList[3#&,2,50] (* Harvey P. Dale, Nov 28 2022 *)

a(n)=3^n<<1 \\ corrected by Michel Marcus, Aug 03 2015

def A008776(n): return 3**n<<1 # Chai Wah Wu, Apr 02 2025

[2*3^n for n in (0..30)] # G. C. Greubel, Sep 11 2019

a(n) = 2*3^n.
a(n) = 3*a(n-1).
G.f.: 2/(1-3*x). - Philippe Deléham, Oct 08 2007
a(n-1) = phi(3^n). - Artur Jasinski, Nov 19 2008
E.g.f.: 2*exp(3*x). - Mohammad K. Azarian, Jan 15 2009
From Paul Curtz, Jan 20 2009: (Start)
a(n) = A048473(n) + 1.
a(n) = A052919(n+1)-1.
a(n) = A115099(n) - 2.
a(n) = A100774(n) + 2. (End)
If p[i]=2, (i >= 1), and if A is Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i <= j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n >= 1, a(n-1)=det A. - Milan Janjic, Apr 29 2010
G.f.: ((1/2)/G(0)-1)/x^2 where G(k) = 1 - 2^k/(2 - 4*x/(2*x - 2^k/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 22 2012
G.f.: -G(0)/x where G(k) = 1 - 1/(1-2*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 25 2013
G.f.: (1 - 1/Q(0))/x where Q(k) = 1 - x*(2*k-2)/(1 - x*(2*k+5)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013
G.f.: W(0), where W(k) = 1 + 1/(1 - x*(2*k+3)/(x*(2*k+4) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013

Jasinski formula corrected by Charles R Greathouse IV, Feb 18 2011

A054872 Number of (12345, 13245, 21345, 23145, 31245, 32145)-avoiding permutations.

Original entry on oeis.org

1, 1, 2, 6, 24, 114, 600, 3372, 19824, 120426, 749976, 4762644, 30723792, 200778612, 1326360048, 8842981848, 59425117152, 402092408346, 2737156004376, 18732169337604, 128806616999184, 889479590046108, 6165939982059600, 42891532191557736, 299307319060137504

Offset: 0

Elisa Pergola (elisa(AT)dsi.unifi.it), May 26 2000

nonn

Hankel transform is A083667, the number of different antisymmetric relations on n labeled points. - Paul Barry, Jun 26 2008
Conjectured to be the number of permutations of length n+1 avoiding the partially ordered pattern (POP) {5>1, 1>2, 1>3, 1>4} of length 5. That is, conjectured to be the number of length n+1 permutations having no subsequences of length 5 in which the fifth element is the largest and the first element is the next largest - Sergey Kitaev, Dec 13 2020
This conjecture has been proven. There are six sets of permutations avoiding six size five permutations including the two sets discussed in this sequence that are known to match this sequence. A further two are conjectured to match this sequence. - Christian Bean, Jul 23 2024

			G.f. = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 114*x^5 + 600*x^6 + 3372*x^7 + 19824*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n=1..200 from Vincenzo Librandi)
Michael H. Albert, Christian Bean, Anders Claesson, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, PermPAL database.
Elena Barcucci, Alberto Del Lungo, Elisa Pergola, and Renzo Pinzani, Permutations avoiding an increasing number of length-increasing forbidden subsequences, Discrete Mathematics and Theoretical Computer Science 4, 2000, 31-44.
Christian Bean, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding, The Electronic Journal of Combinatorics, Volume 31, Issue 3 (2024); arXiv preprint, arXiv:2312.07716 [math.CO], 2023.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Eric Weisstein's World of Mathematics, Legendre Polynomial.

Cf. A000108, A047891.

Set j=3 in the following: f := (x,j)->1-(j+1)*x- sqrt(1-2*(j+1)*x+(j-1)^2*x^2); t := (x,j)->sum(k!*x^k, k=1..(j-1)); s := (x,j)->x^(j-2)*(j-1)!*(f(x,j))/(2)+ t(x,j);

Table[SeriesCoefficient[x*(2-2*x-(1-8*x+4*x^2)^(1/2)),{x,0,n}],{n,1,20}] (* Vaclav Kotesovec, Oct 11 2012 *)
Table[2^(n-1) (LegendreP[n-1, 2] - LegendreP[n-3, 2])/(2n-3), {n, 1, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)

x='x+O('x^50); Vec(x*(2-2*x-(1-8*x+4*x^2)^(1/2))) \\ Altug Alkan, Nov 02 2015

G.f.: 1 + x*(2 - 2*x - (1 - 8*x + 4*x^2)^(1/2)). - corrected by Vaclav Kotesovec, Oct 11 2012
a(n) = 2*A047891(n-1), n>=2. - Philippe Deléham, Aug 17 2007
Recurrence: (n-1)*a(n) = 4*(2*n-5)*a(n-1) - 4*(n-4)*a(n-2). - Vaclav Kotesovec, Oct 11 2012
a(n) ~ sqrt(26*sqrt(3)-45)*(4+2*sqrt(3))^n/(sqrt(8*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 11 2012
From Vladimir Reshetnikov, Nov 01 2015: (Start)
a(n) = 2^(n-1)*(LegendreP_{n-1}(2) - LegendreP_{n-3}(2))/(2*n-3).
For n > 2, a(n) = 6*hypergeom([2-n,3-n], [2], 3).
(End)
G.f. satisfies: A(x) = x * Sum_{n>=0} ( A(x)/x + 4*x + x/A(x) )^n / (2*4^n). - Paul D. Hanna, Mar 24 2016
G.f. satisfies: A(x) = x * Sum_{n>=0} ( A(x)/x + 4*x - x/A(x) )^n / 4^n. - Paul D. Hanna, Mar 24 2016

a(0)=1 prepended by Alois P. Heinz, Dec 13 2020

A110520 Expansion of 1/(1-2xc(3*x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 2, 10, 68, 538, 4652, 42628, 406856, 4001914, 40285724, 413049580, 4298523704, 45288486436, 482122686008, 5178044596168, 56038403289488, 610508962548538, 6690154684006268, 73693477140179548, 815508203755227608

Offset: 0

Paul Barry, Jul 24 2005

Row sums of number triangle A110519.

Hankel transform is A135397. Hankel transform of the aerated sequence is A083667. - Paul Barry, Sep 15 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. Abate and W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, example section 3.

Cf. A000108, A039599, A094385.

Flatten[{1,Table[Sum[Sum[j*Binomial[2n-j-1,n-j]*Binomial[j,k]*3^(n-j)/n,{j,0,n}],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Oct 18 2012 *)

a(0)=1, a(n) = Sum_{k=0..n} Sum_{j=0..n} j*C(2n-j-1, n-j)*C(j, k)*3^(n-j)/n, n > 0.
a(n) = Sum_{k=0..n} A039599(n,k)*(-1)^k*3^(n-k). - Philippe Deléham, Dec 11 2007
a(n) = Sum_{k=0..n} A094385(n,k)*2^k. - Philippe Deléham, Feb 26 2009
From Gary W. Adamson, Jul 12 2011: (Start)
a(n) = the top left term in M^n, M = the infinite square production matrix:
  2, 2, 0, 0, 0, 0, ...
  3, 3, 3, 0, 0, 0, ...
  3, 3, 3, 3, 0, 0, ...
  3, 3, 3, 3, 3, 0, ...
  3, 3, 3, 3, 3, 3, ...
  ... (End)
n*a(n) + 2*(9-4*n)*a(n-1) + 24*(3-2*n)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011
a(n) ~ 3*12^n/(8*sqrt(Pi)n^(3/2)). - Vaclav Kotesovec, Oct 18 2012

A083670 Number of different antisymmetric relations on n unlabeled points.

Original entry on oeis.org

1, 2, 7, 44, 558, 16926, 1319358, 269695440, 146202099255, 212360894456310, 834625722216941739, 8954592469138636320960, 264305834899495393164591240, 21607243912704793462806305720502, 4921054357098031770205099867497197328

Offset: 0

Goetz Pfeiffer (Goetz.Pfeiffer(AT)nuigalway.ie), May 02 2003

Andrew Howroyd, Table of n, a(n) for n = 0..50
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

Cf. A083667 (labeled antisymmetric relations).

Cf. A001174, A101460.

f := function(n) local s, m, c, t, x, a, j; s := 0; m := [1..n]; c := Combinations(m,2); t := Tuples(m,2); for x in ConjugacyClasses(SymmetricGroup(n)) do a := Representative(x); j := Length(Cycles(a,m)); s := s+Size(x)*2^j*3^(Length(Cycles(a,t,OnPairs))-Length(Cycles(a,c,OnSets))-j); od; return s/Factorial(n); end;

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[Quotient[v[[i]] - 1, 2], {i, 1, Length[v]}];
a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p]*2^Length[p], {p, IntegerPartitions[n]}]; s/n!];
a /@ Range[0, 14] (* Jean-François Alcover, Jan 07 2021, after Andrew Howroyd *)

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, (v[i]-1)\2)}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)*2^#p); s/n!} \\ Andrew Howroyd, Oct 24 2019

Euler transform of A101460. - Andrew Howroyd, Oct 24 2019

A127946 Hankel transform of central coefficients of (1+k*x-3x^2)^n, k arbitrary integer.

Original entry on oeis.org

1, -6, -108, 5832, 944784, -459165024, -669462604992, 2928229434235008, 38424226636031774976, -1512608105754026853705216, -178635992073339063368878599168, 63289660175631590117213474413627392, 67269440586795655766964092111705109663744

Offset: 0

Paul Barry, Feb 08 2007

Hankel transform of A098333. The Hankel transform of e.g.f. Bessel_I(0,2*sqrt(-3)x) and its k-th binomial transforms, are given by a(n). In general, the Hankel transform of e.g.f. Bessel_I(0,2*sqrt(m)x) and its binomial transforms is 2^n*m^C(n+1,2).

Let T_n denote the n X n matrix with T_n(i,j) = 3^min(i,j); then a(n) = ((-1)^floor((n+1)/2))*det(T_(n+1))/3. - Lechoslaw Ratajczak, May 16 2021

G. C. Greubel, Table of n, a(n) for n = 0..63

a(n) = A083667(n+1)/2.

[2^n*(-3)^Binomial(n+1,2): n in [0..30]]; // G. C. Greubel, May 03 2018

A127946[0] = 1; A127946[n_] := {1, -1, -1, 1}[[Mod[n, 4] + 1]] * 2^n * 3^(n(n + 1)/2); Table[A127946[n], {n, 0, 12}] (* Jean-François Alcover, Oct 04 2016 *)
Table[2^n*(-3)^Binomial[n+1,2], {n,0,30}] (* G. C. Greubel, May 03 2018 *)

a(n)=if((n-1)%4<2,-1,1)*2^n*3^(n*(n+1)/2) \\ Charles R Greathouse IV, Oct 04 2016

a(n) = (cos(Pi*n/2) - sin(Pi*n/2))*6^n*3^C(n,2) = 2^n*(-3)^C(n+1,2).

A141342 A transform of the Fibonacci numbers.

Original entry on oeis.org

1, 1, -1, 3, -13, 65, -353, 2025, -12077, 74143, -465481, 2974863, -19289821, 126594191, -839273105, 5612483619, -37814455781, 256447068841, -1749182184793, 11991887667273, -82588248514885, 571118483653841

Offset: 0

Paul Barry, Jun 26 2008

A transform of F(n+1) by the inverse of the Riordan array (1, x*(1+x)/(1-2*x)).
Equivalently, row sums of the inverse of the Riordan array (1, x/(2-sqrt(1+4*x)).
Hankel transform is alternating sign version of A083667.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A141343.

CoefficientList[Series[1/(1-2*x-2*x^2+x*Sqrt[1+8*x+4*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)

x='x+O('x^50); Vec(1/(1-2*x-2*x^2+x*sqrt(1+8*x+4*x^2))) \\ G. C. Greubel, Mar 21 2017

G.f.: 1/(1-2*x-2*x^2+x*sqrt(1+8*x+4*x^2)).
Conjecture: (n-1)*a(n) +4*(n-4)*a(n-1) + (65-29*n)*a(n-2) +12*(7-2*n)*a(n-3)+ 4*(4-n)*a(n-4) =0. - R. J. Mathar, Nov 14 2011
a(n) ~ (-1)^n * (5*sqrt(3)-14) * sqrt(2*sqrt(3)-3) * 2^(n+1/2) * (2+sqrt(3))^n / (121 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 01 2014

A341471 Number of antisymmetric, antitransitive relations on n labeled nodes.

Original entry on oeis.org

1, 1, 3, 21, 317, 9735, 583907, 66226033, 13837055261

Offset: 0

Peter Kagey, Feb 13 2021

An antisymmetric, antitransitive relation is one where xRy implies "not yRx" and xRy and yRz implies "not xRz". All antitransitive relations are irreflexive, so this sequence is counting "anti-equivalence relations".
a(n) < A047656(n).
Idea thanks to Richard Arratia, who saw, verbatim in an editorial, "False equivalences? There were almost too many to count."

			There are a(3) = 21 antisymmetric, antitransitive relations on n = 3 letters:
  - the empty relation,
  - all six relations containing only a single pair (x,y) (with x != y),
  - all twelve relations {(x1,y1), (x2,y2)} containing exactly two ordered pairs, neither of which is (y1,x1) or (y2,x2), and
  - two relations containing three ordered pairs: {(1,2), (2,3), (3,1)} and {(1,3), (3,2), (2,1)}.

Wikipedia, Binary relation

Number of relations on labeled nodes: A000110 (equivalence), A001831 (transitive and antitransitive), A002416 (unrestricted), A006125 (symmetric), A006905 (transitive), A047656 (reflexive and antisymmetric), A083667 (antisymmetric), A341473 (antitransitive).

a(6)-a(8) from Bert Dobbelaere, Feb 27 2021

cp's OEIS Frontend

A008776 Pisot sequences E(2,6), L(2,6), P(2,6), T(2,6).

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A054872 Number of (12345, 13245, 21345, 23145, 31245, 32145)-avoiding permutations.

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A110520 Expansion of 1/(1-2*x*c(3*x)), c(x) the g.f. of A000108.

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A083670 Number of different antisymmetric relations on n unlabeled points.

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A127946 Hankel transform of central coefficients of (1+k*x-3x^2)^n, k arbitrary integer.

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Magma

Mathematica

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A141342 A transform of the Fibonacci numbers.

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A341471 Number of antisymmetric, antitransitive relations on n labeled nodes.

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A110520 Expansion of 1/(1-2xc(3*x)), c(x) the g.f. of A000108.