cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A063524 Characteristic function of 1.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Labos Elemer, Jul 30 2001

Keywords

Comments

The identity function for Dirichlet multiplication (see Apostol).
Sum of the Moebius function mu(d) of the divisors d of n. - Robert G. Wilson v, Sep 30 2006
-a(n) is the Hankel transform of A000045(n), n >= 0 (Fibonacci numbers). See A055879 for the definition of Hankel transform. - Wolfdieter Lang, Jan 23 2007
a(A000012(n)) = 1; a(A087156(n)) = 0. - Reinhard Zumkeller, Oct 11 2008
a(n) for n >= 1 is the Dirichlet convolution of following functions b(n), c(n), a(n) = Sum_{d|n} b(d)*c(n/d): a(n) = A008683(n) * A000012(n), a(n) = A007427(n) * A000005(n), a(n) = A007428(n) * A007425(n). - Jaroslav Krizek, Mar 03 2009
From Christopher Hunt Gribble, Jul 11 2013: (Start)
a(n) for 1 <= n <= 4 and conjectured for n > 4 is the number of Hamiltonian circuits in a 2n X 2n square lattice of nodes, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 1 element: When n=1, there is only 1 Hamiltonian circuit in a 2 X 2 square lattice, as illustrated below. The circuit is the same when rotated and/or reflected and so has only 1 orbital element under the symmetry group of the square.
o--o
| |
o--o (End)
Convolution property: For any sequence b(n), the sequence c(n)=b(n)*a(n) has the following values: c(1)=0, c(n+1)=b(n) for all n > 1. In other words, the sequence b(n) is shifted 1 step to the right. - David Neil McGrath, Nov 10 2014

References

  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.

Links

Crossrefs

Cf. A000007 (same sequence shifted once left).
Cf. A000005, A000010, A000012, A000027, A003763, A008683, A007427, A007428, A007425, A023900, A055615, A101296, A227005, A227257, A227301, A209077.

Programs

Formula

From Philippe Deléham, Nov 25 2008: (Start)
G.f.: x.
E.g.f.: x. (End)
a(n) = mu(n^2). - Enrique Pérez Herrero, Sep 04 2009
a(n) = floor(n/A000203(n)) for n > 0. - Enrique Pérez Herrero, Nov 11 2009
a(n) = (1-(-1)^(2^abs(n-1)))/2 = (1-(-1)^(2^((n-1)^2)))/2. - Luce ETIENNE, Jun 05 2015
a(n) = n*(A057427(n) - A057427(n-1)) = A000007(abs(n-1)). - Chayim Lowen, Aug 01 2015
a(n) = A010051(p*n) for any prime p (where A010051(0)=0). - Chayim Lowen, Aug 05 2015
From Antti Karttunen, Jun 04 2022: (Start)
For n >= 1:
a(n) = Sum_{d|n} A000010(n/d) * A023900(d), and similarly for any pair of sequences that are Dirichlet inverses of each other, like for example A000027 & A055615 and those mentioned in Krizek's Mar 03 2009 comment above.
a(n) = [A101296(n) == 1], where [ ] is the Iverson bracket.
Fully multiplicative with a(p^e) = 0. (End)

A019590 Fermat's Last Theorem: a(n) = 1 if x^n + y^n = z^n has a nontrivial solution in integers, otherwise a(n) = 0.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) is the Hankel transform of A000045(n), n>=1 (Fibonacci numbers). See A055879 for the definition of Hankel transform. - Wolfdieter Lang, Jan 23 2007
1, -1, 0, 0, 0, ... is the convolutional inverse of the all-ones sequence. - Tanya Khovanova, Jun 29 2007
Also parity of the Euler totient function A000010. - Omar E. Pol, Jan 15 2012
a(n-1) gives the row sums of A048994. - Wolfdieter Lang, May 09 2017
Decimal expansion of 11/10. - Franklin T. Adams-Watters, Mar 08 2019

References

  • A. D. Aczel, Fermat's Last Theorem, Four Walls Eight Windows NY 1996.
  • A. C. Clarke, The Last Theorem, Gollancz SF 2004.
  • B. Cipra, What's Happening in the Mathematical Sciences 1994 Vol. 2, "A Truly Remarkable Proof" pp. 3-8 AMS Providence RI.
  • B. Cipra, What's Happening in the Mathematical Sciences 1995-6 Vol. 3, 'Fermat's Theorem-At Last' pp. 2-13 AMS Providence RI.
  • J. Coates and S.-T. Yau (Eds), Elliptic Curves, Modular Forms and Fermat's last Theorem, International Press Boston MA 1998.
  • G. Cornell, J. H. Silverman and G. Stevens (Eds), Modular Forms and Fermat's last Theorem, Springer NY 2000.
  • K. J. Devlin, Mathematics: The New Golden Age, Chapter 10, Columbia Univ. Press NY 1999.
  • L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 731.
  • H. M. Edwards, Fermat's Last Theorem, Springer, 1977.
  • G. Giorello & C. Sinigaglia, "Fermat: De défis en conjectures", Les génies de la science No. 32 Aug-Oct 2007, pp. 82-100, Pour la Science, Paris.
  • C. Goldstein, "Le Théorème de Fermat", La Recherche, Vol. Mar 25 1994, pp. 268-275, Paris.
  • C. Goldstein, "Le Théorème de Fermat enfin démontré", Chapter IX pp. 111-129 in 'Histoire Des Nombres', La Recherche, Tallandier, Paris 2007.
  • Y. Hellegouarch, "Fermat Vaincu", Quadrature No. 22 pp. 37-55 Editions du choix Argenteuil (France) 1995.
  • Y. Hellegouarch, "Fermat enfin démontré", Pour la Science, No. 220, 1996 pp. 92-97 Paris.
  • Y. Hellegouarch, Invitation aux mathématiques de Fermat-Wiles, Dunod Paris 2001.
  • Y. Hellegouarch, "L'Enigme du Theoreme de Fermat" pp. 31-41 in 'Qu'est-ce que l'Univers?", Université de tous les savoirs, Vol. 4 (Edit. Y. Michaud), Odile Jacob Paris 2001.
  • Y. Hellegouarch, Invitation to the Mathematics of Fermat-Wiles, Academic Press NY 2001.
  • P. Hoffman, The Man Who Loved Only Numbers, pp. 183-200, Hyperion NY 1998
  • W. Lindsay, Fermat's Last Theorem, A Perfect Proof, Lulu Press, Morrisville NC 2005.
  • L. J. Mordell, Three lectures on Fermat's last theorem, Cambr. Univ. Press 1921 (Reprinted by The Scholarly Pub. Office, Univ. of Michigan Library 2005).
  • C. J. Mozzochi, The Fermat Diary, AMS Providence RI 2000.
  • C. J. Mozzochi, The Fermat Proof, New Bern NC 2004.
  • V. K. Murty, Seminar on Fermat's Last Theorem, Amer. Math. Soc. Providence RI 1995.
  • P. Odifreddi, The Mathematical Century, Chapter 2.14 "Number Theory: Wiles' Proof of Fermat's Last Theorem (1995)" p. 82 Princeton Univ. Press NJ 2004.
  • I. Peterson, The Mathematical Tourist, pp. 234-238, W. H. Freeman/Owl Book NY 2001.
  • I. Peterson, "A Marginal Note" in Islands of Truths, pp. 280-285, W. H. Freeman NY 1990.
  • A. van der Poorten, Notes on Fermat's Last Theorem, Wiley NY 1996
  • J. Propp, Who Proved Fermat's Last Theorem? Princeton Univ. Press NJ 2005.
  • P. Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, 1979.
  • P. Ribenboim, Fermat's Last Theorem for Amateurs, Springer Verlag NY 1999.
  • R. Schoof, "Wiles' proof of the Taniyama-Weil conjecture for semi-stable elliptic curves over Q", Chap. 14 in 'Ou En Sont Les Mathématiques ?' Soc. Math. de France (SMF), Vuibert, Paris 2002.
  • S. Singh, Fermat's Enigma, Walker and Co. NY 1997.
  • I. Stewart, From Here To Infinity, Chapter 3 "Marginal Interest" pp. 25-48 OUP Oxford 1996.
  • I. Stewart and D. Tall, Algebraic Number Theory and Fermat's Last Theorem, A. K. Peters Natick MA 2001.
  • G. R. Talbott, Fermat's Last Theorem, Lotus Press WI 1991.
  • R. Van Vo, Fermat's Last Theorem, AuthorHouse, Bloomington IN 2002.
  • J. Vigouroux et al., Une aventure mathématique, le théorème de Fermat, BT2 series No. 6, PEMF Mouans-Sartoux(France) 1998.

Links

Crossrefs

Cf. A000004, A000007, A010051, A012450.
INVERT transform gives Fibonacci numbers, A000045.
Convolution inverse of A062157. Dirichlet convolution inverse of A154269.
Cf. A229382, A229383 (near-miss counterexamples to FLT).
Cf. A048994 (row sums).
Cf. A008683.

Programs

  • PARI
    {a(n) = (n==1) + (n==2)}; /* Michael Somos, Jul 05 2009 */

Formula

a(n) = (-1)^n*Sum_{k=0..floor(n/2)} (-1)^A010060(n-2k) mod (C(n, 2k), 2). - Paul Barry, Jan 03 2005
Euler transform of length 2 sequence [1, -1]. - Michael Somos, Jul 05 2009
a(n) is multiplicative with a(2) = 1, a(2^e) = 0 if e > 1, a(p^e) = 0^e if p > 2. - Michael Somos, Jul 05 2009
G.f.: x + x^2 = x * (1 - x^2) / (1 - x). - Michael Somos, Jul 05 2009
Dirichlet g.f.: 1 + 2^(-s). - Michael Somos, Jul 05 2009
a(n) = A000035(A000010(n)). - Omar E. Pol, Oct 28 2013
a(n) = Sum_{d|n} mu(n/d) * gcd(d,2). - Ridouane Oudra, May 30 2025

A007317 Binomial transform of Catalan numbers.

Original entry on oeis.org

1, 2, 5, 15, 51, 188, 731, 2950, 12235, 51822, 223191, 974427, 4302645, 19181100, 86211885, 390248055, 1777495635, 8140539950, 37463689775, 173164232965, 803539474345, 3741930523740, 17481709707825, 81912506777200, 384847173838501, 1812610804416698
Offset: 1

Views

Author

N. J. A. Sloane, Mira Bernstein

Keywords

Comments

Partial sums of A002212 (the restricted hexagonal polyominoes with n cells). Number of Schroeder paths (i.e., consisting of steps U=(1,1),D=(1,-1),H=(2,0) and never going below the x-axis) from (0,0) to (2n-2,0), with no peaks at even level. Example: a(3)=5 because among the six Schroeder paths from (0,0) to (4,0) only UUDD has a peak at an even level. - Emeric Deutsch, Dec 06 2003
Number of binary trees of weight n where leaves have positive integer weights. Non-commutative Non-associative version of partitions of n. - Michael Somos, May 23 2005
Appears also as the number of Euler trees with total weight n (associated with even switching class of matrices of order 2n). - David Garber, Sep 19 2005
Number of symmetric hex trees with 2n-1 edges; also number of symmetric hex trees with 2n-2 edges. A hex tree is a rooted tree where each vertex has 0, 1, or 2 children and, when only one child is present, it is either a left child, or a median child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read reference). A hex tree is symmetric if it is identical with its reflection in a bisector through the root. - Emeric Deutsch, Dec 19 2006
The Hankel transform of [1, 2, 5, 15, 51, 188, ...] is [1, 1, 1, 1, 1, ...], see A000012 ; the Hankel transform of [2, 5, 15, 51, 188, 731, ...] is [2, 5, 13, 34, 89, ...], see A001519. - Philippe Deléham, Dec 19 2006
a(n) = number of 321-avoiding partitions of [n]. A partition is 321-avoiding if the permutation obtained from its canonical form (entries in each block listed in increasing order and blocks listed in increasing order of their first entries) is 321-avoiding. For example, the only partition of [5] that fails to be 321-avoiding is 15/24/3 because the entries 5,4,3 in the permutation 15243 form a 321 pattern. - David Callan, Jul 22 2008
The sequence 1,1,2,5,15,51,188,... has Hankel transform A001519. - Paul Barry, Jan 13 2009
From Gary W. Adamson, May 17 2009: (Start)
Equals INVERT transform of A033321: (1, 1, 2, 6, 21, 79, 311, ...).
Equals INVERTi transform of A002212: (1, 3, 10, 36, 137, ...).
Convolved with A026378, (1, 4, 17, 75, 339, ...) = A026376: (1, 6, 30, 144, ...)
(End)
a(n) is the number of vertices of the composihedron CK(n). The composihedra are a sequence of convex polytopes used to define maps of certain homotopy H-spaces. They are cellular quotients of the multiplihedra and cellular covers of the cubes. - Stefan Forcey (sforcey(AT)gmail.com), Dec 17 2009
a(n) is the number of Motzkin paths of length n-1 in which the (1,0)-steps at level 0 come in 2 colors and those at a higher level come in 3 colors. Example: a(4)=15 because we have 2^3 = 8 paths of shape UHD, 2 paths of shape HUD, 2 paths of shape UDH, and 3 paths of shape UHD; here U=(1,1), H=(1,0), and D=(1,-1). - Emeric Deutsch, May 02 2011
REVERT transform of (1, 2, -3, 5, -8, 13, -21, 34, ... ) where the entries are Fibonacci numbers, A000045. Equivalently, coefficients in the series reversion of x(1-x)/(1+x-x^2). This means that the substitution of the gf (1-x-(1-6x+5x^2)^(1/2))/(2(1-x)) for x in x(1-x)/(1+x-x^2) will simplify to x. - David Callan, Nov 11 2012
The number of plane trees with nodes that have positive integer weights and whose total weight is n. - Brad R. Jones, Jun 12 2014
From Tom Copeland, Nov 02 2014: (Start)
Let P(x) = x/(1+x) with comp. inverse Pinv(x) = x/(1-x) = -P[-x], and C(x)= [1-sqrt(1-4x)]/2, an o.g.f. for the shifted Catalan numbers A000108, with inverse Cinv(x) = x * (1-x).
Fin(x) = P[C(x)] = C(x)/[1 + C(x)] is an o.g.f. for the Fine numbers, A000957 with inverse Fin^(-1)(x) = Cinv[Pinv(x)] = Cinv[-P(-x)].
Mot(x) = C[P(x)] = C[-Pinv(-x)] gives an o.g.f. for shifted A005043, the Motzkin or Riordan numbers with comp. inverse Mot^(-1)(x) = Pinv[Cinv(x)] = (x - x^2) / (1 - x + x^2) (cf. A057078).
BTC(x) = C[Pinv(x)] gives A007317, a binomial transform of the Catalan numbers, with BTC^(-1)(x) = P[Cinv(x)] = (x-x^2) / (1 + x - x^2).
Fib(x) = -Fin[Cinv(Cinv(-x))] = -P[Cinv(-x)] = x + 2 x^2 + 3 x^3 + 5 x^4 + ... = (x+x^2)/[1-x-x^2] is an o.g.f. for the shifted Fibonacci sequence A000045, so the comp. inverse is Fib^(-1)(x) = -C[Pinv(-x)] = -BTC(-x) and Fib(x) = -BTC^(-1)(-x).
Generalizing to P(x,t) = x /(1 + t*x) and Pinv(x,t) = x /(1 - t*x) = -P(-x,t) gives other relations to lattice paths, such as the o.g.f. for A091867, C[P[x,1-t]], and that for A104597, Pinv[Cinv(x),t+1].
(End)
Starting with offset 0, a(n) is also the number of Schröder paths of semilength n avoiding UH (an up step directly followed by a long horizontal step). Example: a(2)=5 because among the six possible Schröder paths of semilength 2 only UHD contains UH. - Valerie Roitner, Jul 23 2020

Examples

			a(3)=5 since {3, (1+2), (1+(1+1)), (2+1), ((1+1)+1)} are the five weighted binary trees of weight 3.
G.f. = x + 2*x^2 + 5*x^3 + 15*x^4 + 51*x^5 + 188*x^6 + 731*x^7 + 2950*x^8 + 12235*x^9 + ... _Michael Somos_, Jan 17 2018

References

  • J. Brunvoll et al., Studies of some chemically relevant polygonal systems: mono-q-polyhexes, ACH Models in Chem., 133 (3) (1996), 277-298, Eq. 15.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

See A181768 for another version. - N. J. A. Sloane, Nov 12 2010
First column of triangle A104259. Row sums of absolute values of A091699.
Number of vertices of multiplihedron A121988.
m-th binomial transform of the Catalan numbers: A126930 (m = -2), A005043 (m = -1), A000108 (m = 0), A064613 (m = 2), A104455 (m = 3), A104498 (m = 4) and A154623 (m = 5).
Cf. A055879, A033321, A026376, A026378, A059346, A000045, A000957, A057078, A091867, A104597, A249548, A162326.

Programs

  • Maple
    G := (1-sqrt(1-4*z/(1-z)))*1/2: Gser := series(G, z = 0, 30): seq(coeff(Gser, z, n), n = 1 .. 26); # Emeric Deutsch, Aug 12 2007
seq(round(evalf(JacobiP(n-1,1,-n-1/2,9)/n,99)),n=1..25); # Peter Luschny, Sep 23 2014
  • Mathematica
    Rest@ CoefficientList[ InverseSeries[ Series[(y - y^2)/(1 + y - y^2), {y, 0, 26}], x], x] (* then A(x)=y(x); note that InverseSeries[Series[y-y^2, {y, 0, 24}], x] produces A000108(x) *) (* Len Smiley, Apr 10 2000 *)
Range[0, 25]! CoefficientList[ Series[ Exp[ 3x] (BesselI[0, 2x] - BesselI[1, 2x]), {x, 0, 25}], x] (* Robert G. Wilson v, Apr 15 2011 *)
a[n_] := Sum[ Binomial[n, k]*CatalanNumber[k], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 07 2012 *)
Rest[CoefficientList[Series[3/2 - (1/2) Sqrt[(1 - 5 x)/(1 - x)], {x, 0, 40}], x]] (* Vincenzo Librandi, Nov 03 2014 *)
Table[Hypergeometric2F1[1/2, -n+1, 2, -4], {n, 1, 30}] (* Vaclav Kotesovec, May 12 2022 *)
  • PARI
    {a(n) = my(A); if( n<2, n>0, A=vector(n); for(j=1,n, A[j] = 1 + sum(k=1,j-1, A[k]*A[j-k])); A[n])}; /* Michael Somos, May 23 2005 */
  • PARI
    {a(n) = if( n<1, 0, polcoeff( serreverse( (x - x^2) / (1 + x - x^2) + x * O(x^n)), n))}; /* Michael Somos, May 23 2005 */
  • PARI
    /* Offset = 0: */ {a(n)=local(A=1+x);for(i=1,n, A=sum(m=0,n, x^m*sum(k=0,m,A^k)+x*O(x^n))); polcoeff(A,n)} \\ Paul D. Hanna

Formula

(n+2)*a(n+2) = (6n+4)*a(n+1) - 5n*a(n).
G.f.: 3/2-(1/2)*sqrt((1-5*x)/(1-x)) [Gessel-Kim]. - N. J. A. Sloane, Jul 05 2014
G.f. for sequence doubled: (1/(2*x))*(1+x-(1-x)^(-1)*(1-x^2)^(1/2)*(1-5*x^2)^(1/2)).
a(n) = hypergeom([1/2, -n], [2], -4), n=0, 1, 2...; Integral representation as n-th moment of a positive function on a finite interval of the positive half-axis: a(n)=int(x^n*sqrt((5-x)/(x-1))/(2*Pi), x=1..5), n=0, 1, 2... This representation is unique. - Karol A. Penson, Sep 24 2001
a(1)=1, a(n)=1+sum(i=1, n-1, a(i)*a(n-i)). - Benoit Cloitre, Mar 16 2004
a(n) = Sum_{k=0..n} (-1)^k*3^(n-k)*binomial(n, k)*binomial(k, floor(k/2)) [offset 0]. - Paul Barry, Jan 27 2005
G.f. A(x) satisfies 0=f(x, A(x)) where f(x, y)=x-(1-x)(y-y^2). - Michael Somos, May 23 2005
G.f. A(x) satisfies 0=f(x, A(x), A(A(x))) where f(x, y, z)=x(z-z^2)+(x-1)y^2 . - Michael Somos, May 23 2005
G.f. (for offset 0): (-1+x+(1-6*x+5*x^2)^(1/2))/(2*(-x+x^2)).
G.f. =z*c(z/(1-z))/(1-z) = 1/2 - (1/2)sqrt(1-4z/(1-z)), where c(z)=(1-sqrt(1-4z))/(2z) is the Catalan function (follows from Michael Somos' first comment). - Emeric Deutsch, Aug 12 2007
G.f.: 1/(1-2x-x^2/(1-3x-x^2/(1-3x-x^2/(1-3x-x^2/(1-3x-x^2/(1-.... (continued fraction). - Paul Barry, Apr 19 2009
a(n) = Sum_{k, 0<=k<=n} A091965(n,k)*(-1)^k. - Philippe Deléham, Nov 28 2009
E.g.f.: exp(3x)*(I_0(2x)-I_1(2x)), where I_k(x) is a modified Bessel function of the first kind. - Emanuele Munarini, Apr 15 2011
If we prefix sequence with an additional term a(0)=1, g.f. is (3-3*x-sqrt(1-6*x+5*x^2))/(2*(1-x)). [See Kim, 2011] - N. J. A. Sloane, May 13 2011
From Gary W. Adamson, Jul 21 2011: (Start)
a(n) = upper left term in M^(n-1), M = an infinite square production matrix as follows:
2, 1, 0, 0, 0, 0, ...
1, 2, 1, 0, 0, 0, ...
1, 1, 2, 1, 0, 0, ...
1, 1, 1, 2, 1, 0, ...
1, 1, 1, 1, 2, 1, ...
1, 1, 1, 1, 1, 2, ...
... (End)
G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 - A(x)^(n+1))/(1 - A(x)); offset=0. - Paul D. Hanna, Nov 07 2011
G.f.: 1/x - 1/x/Q(0), where Q(k)= 1 + (4*k+1)*x/((1-x)*(k+1) - x*(1-x)*(2*k+2)*(4*k+3)/(x*(8*k+6)+(2*k+3)*(1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013
G.f.: (1-x - (1-5*x)*G(0))/(2*x*(1-x)), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1-x) - 2*x*(1-x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 25 2013
Asymptotics (for offset 0): a(n) ~ 5^(n+3/2)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 28 2013
G.f.: G(0)/(1-x), where G(k) = 1 + (4*k+1)*x/((k+1)*(1-x) - 2*x*(1-x)*(k+1)*(4*k+3)/(2*x*(4*k+3) + (2*k+3)*(1-x)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 29 2014
a(n) = JacobiP(n-1,1,-n-1/2,9)/n. - Peter Luschny, Sep 23 2014
0 = +a(n)*(+25*a(n+1) -50*a(n+2) +15*a(n+3)) +a(n+1)*(-10*a(n+1) +31*a(n+2) -14*a(n+3)) +a(n+2)*(+2*a(n+2) +a(n+3)) for all n in Z. - Michael Somos, Jan 17 2018
a(n+1) = (2/Pi) * Integral_{x = -1..1} (m + 4*x^2)^n*sqrt(1 - x^2) dx at m = 1. In general, the integral, qua sequence in n, gives the m-th binomial transform of the Catalan numbers. - Peter Bala, Jan 26 2020

A030529 Number of polyhexes of class PF2 with a particular symmetry.

Original entry on oeis.org

0, 0, 1, 4, 17, 66, 269, 1102, 4635, 19768, 85659, 375524, 1664015, 7438862, 33515027, 152016610, 693622315, 3181516040, 14661568795, 67850245684, 315187594779, 1469195413102, 6869889480447, 32215398047474, 151467333043437, 713881813137776, 3372142135461789
Offset: 2

Views

Author

N. J. A. Sloane

Keywords

Comments

See references for precise definition.
Column D_{2h}(b) and Eq. 50 in Cyvin et al. (1994). - Sean A. Irvine, Mar 27 2021

Links

Crossrefs

Cf. A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534.

Programs

  • PARI
    A055879(n)= my(A); if( n<1, 0, n--; A = O(x); for( k = 0, n\2, A = 1 / (1 - x - x^2 / (1 + x - x^2 * A))); polcoeff( A, n));
b(n) = (A055879(2*n+1) - A055879(2*n) - A055879(n)) / 2;
a(n) = if( n<=2, 0, b(n - 2)); \\ Michel Marcus, Apr 03 2020

Formula

a(2)=0, a(n+2) = (M(2*n+1) - M(2*n) - M(n)) / 2 where M(n) = A055879(n) [Cyvin Eq. (54)]. - Sean A. Irvine, Apr 03 2020

Extensions

More terms from Sean A. Irvine, Apr 03 2020

A030532 Number of polyhexes of class PF2 with symmetry point group C_s.

Original entry on oeis.org

0, 1, 6, 35, 168, 807, 3738, 17326, 79909, 369330, 1709087, 7929590, 36880231, 171981241, 804008476, 3767969067, 17699758030, 83328230588, 393123455667, 1858351021018, 8801159427825, 41756067216508, 198437454009869, 944521139813575, 4502419756667924
Offset: 4

Views

Author

N. J. A. Sloane

Keywords

Comments

See reference for precise definition.
Cyvin has incorrect a(13)=369366 and a(14)=1709123 in Table III due to using incorrect values for A026298(13) and A026298(14) in Table II.

Links

Crossrefs

Cf. A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534.

Programs

  • PARI
    L(n) = my(x = 'x + O('x^(n+4))); polcoeff((1+x)*(1-3*x^2-sqrt(1-6*x^2+5*x^4))/(2*x^2*(1-x)), n); \\ A039658
Lp(n) = my(x = 'x + O('x^(n+4))); polcoeff((1+x)*(1-6*x^2+7*x^4-(1-3*x^2)*sqrt(1-6*x^2+5*x^4))/(2*x^4*(1-x)), n); \\ A039660
M(n)= my(A); if( n<1, 0, n--; A = O(x); for( k = 0, n\2, A = 1 / (1 - x - x^2 / (1 + x - x^2 * A))); polcoeff( A, n)); \\ A055879
N(n) = polcoeff( (1 - x - sqrt(1 - 6*x + 5*x^2 + x^2 * O(x^n))) / 2, n+1); \\ A002212
Mp(n) = N(n) - sum(j=0, n-1, N(j)); \\ A039919
b(n) = N(n+3) - 6*N(n+2) - Mp(floor((n+1)/2)) + (41*N(n+1)-21*N(n)-L(n))/4 - (M(n+3)-M(n+2)+M(n)-if (!(n%2),M(n/2))+Lp(n))/2;
a(n) = if (n<=4, 0, b(n-4)); \\ Michel Marcus, Apr 05 2020

Formula

a(n+4) = N(n+3) - 6*N(n+2) - M'(floor((n+1)/2)) + (41*N(n+1)-21*N(n)-L(n))/4 - (M(n+3)-M(n+2)+M(n)-e(n)*M(n/2)+L'(n))/2 where N(n)=A002212(n), M(n)=A055879(n), M'(n)=A039919(n), L(n)=A039658(n), L'(n)=A039660(n), e(n)=1 if n is even and 0 if n is odd. - Sean A. Irvine, Apr 03 2020

Extensions

a(13) and a(14) corrected, title improved, and more terms from Sean A. Irvine, Apr 03 2020

A030519 Number of polyhexes of class PF2 with four catafusenes annealated to pyrene.

Original entry on oeis.org

2, 13, 101, 619, 3641, 20028, 106812, 554352, 2828660, 14244878, 71077246, 352184306, 1736118578, 8525182798, 41741378126, 203929434766, 994680883360, 4845761306611, 23586192274443, 114731539477465, 557859497501007, 2711772157178038, 13180227306740726
Offset: 8

Views

Author

N. J. A. Sloane

Keywords

Comments

See reference for precise definition.

Links

Crossrefs

Cf. A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534.

Programs

  • PARI
    Lp(n)=my(x = 'x + O('x^(n+4))); polcoeff((1+x)*(1-6*x^2+7*x^4-(1-3*x^2)*sqrt(1-6*x^2+5*x^4))/(2*x^4*(1-x)), n); \\ A039660
M(n)= my(A); if( n<1, 0, n--; A = O(x); for( k = 0, n\2, A = 1 / (1 - x - x^2 / (1 + x - x^2 * A))); polcoeff( A, n)); \\ A055879
N(n) = polcoeff( (1 - x - sqrt(1 - 6*x + 5*x^2 + x^2 * O(x^n))) / 2, n+1); \\ A002212
b(n) = N(n+3) - 9*N(n+2) + 25*N(n+1) - 21*N(n) + (M(n+3) - M(n+2) - 3*M(n+1) + 3*M(n) + Lp(n))/2;
a(n) = b(n-4); \\ Michel Marcus, Apr 03 2020

Formula

a(n+4) = N(n+3) - 9*N(n+2) + 25*N(n+1) - 21*N(n) + (M(n+3) - M(n+2) - 3*M(n+1) + 3*M(n) + L'(n))/2 where N(n)=A002212(n), M(n)=A055879(n), and L'(n)=A039660(n) for n >= 4. - Sean A. Irvine, Apr 02 2020

Extensions

More terms and title improved by Sean A. Irvine, Apr 02 2020

A045909 Number of unsymmetrical catafusenes with n hexagons (see reference for precise definition).

Original entry on oeis.org

0, 0, 1, 8, 39, 190, 861, 3943, 17867, 81496, 372378, 1709842, 7879360, 36454004, 169239651, 788311290, 3683018475, 17255646688, 81056033636, 381668479192, 1801188588566, 8517994717214, 40360814900566, 191589547228523, 911009396044132, 4338777661726040
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Column C_s of Table 6 in Cyvin et al. Computed by subtracting all the other symmetry classes from the total. - Sean A. Irvine, Mar 27 2021

Links

Crossrefs

Cf. A045906, A045907, A045908, A342554, A342570, A030529, A055879.

Formula

a(n) = A045906(n) - A045907(n) - A045908(n/2) - A342554(n/2) - A342570(n/2) - 2*A030529(n/4) - A055879(n/4) - [n=0] with sequences assumed to be 0 for non-integer arguments. - Sean A. Irvine, Mar 27 2021

Extensions

More terms from Sean A. Irvine, Mar 27 2021

A055878 Least positive sequence with Hankel transform {1,1,1,1,1,...}.

Original entry on oeis.org

1, 1, 2, 1, 6, 1, 43, 1, 7794, 1, 387355897233, 1, 58111569344245781528237519282283636, 1
Offset: 1

Views

Author

John W. Layman, Jul 15 2000

Keywords

Comments

Hankel transform {t(n)} of {a(n)} is given by t(n) = Det[{a(1), a(2), ..., a(n)}, {a(2), a(3), ..., a(n+1)}, ..., {a(n), a(n+1), ..., a(2n-1)}].

References

  • M M Mangontarum, O I Cauntongan, A P Macodi-Ringia, The Noncentral Version of the Whitney Numbers: A Comprehensive Study, International Journal of Mathematics and Mathematical Sciences, Volume 2016, Article ID 6206207, 16 pages; http://dx.doi.org/10.1155/2016/6206207. See reference 21.

Links

Crossrefs

Cf. A055877, A055564, A055879.

Extensions

Next term is too large to include.

A342554 Number of catafusenes with 2n hexagons and C symmetry.

Original entry on oeis.org

0, 0, 1, 5, 22, 97, 408, 1752, 7535, 32895, 144816, 644058, 2886967, 13037352, 59248959, 270799903, 1243939278, 5740014143, 26594306442, 123668622726, 577003771681, 2700346810482, 12672753684799, 59625966523555, 281207237408728, 1329130156859362
Offset: 0

Views

Author

Sean A. Irvine, Mar 27 2021

Keywords

Comments

Column C in Table 6 and Eq. (52) in Cyvin et al.

Links

Crossrefs

Cf. A045906, A045907, A045908, A045909, A342570.

Formula

a(n) = (1/4) * (A002212(n+1) - A002212(n) - A039658(n)) - (1/2) * (A055879(n) + A002212(n/2) - A055879(n/2)) where sequences are assumed 0 for non-integer arguments.

A342570 Number of catafusenes with 2n hexagons and D_{2h}(a) symmetry.

Original entry on oeis.org

0, 1, 1, 3, 4, 9, 13, 29, 46, 101, 167, 375, 644, 1461, 2563, 5899, 10534, 24469, 44237, 103643, 189302, 446381, 821905, 1948853, 3612868, 8605289, 16044151, 38362199, 71877226, 172423769, 324440379, 780496109, 1474118424, 3554991269, 6736507309, 16281079899
Offset: 0

Views

Author

Sean A. Irvine, Mar 27 2021

Keywords

Comments

Column D_{2h}(a) in Table 6 and Eq. (51) in Cyvin et al.

Links

Crossrefs

Cf. A045906, A045907, A045908, A045909, A342554.

Formula

a(n) = A055879(n) + (1/2) * (A039658(n) - A055879(n/2)) where A055879 is 0 for non-integer arguments.
Showing 1-10 of 14 results. Next

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