A277919 -id:A277919 - cp's OEIS frontend

A110170 First differences of the central Delannoy numbers (A001850).

1, 2, 10, 50, 258, 1362, 7306, 39650, 217090, 1196834, 6634890, 36949266, 206549250, 1158337650, 6513914634, 36718533570, 207412854786, 1173779487810, 6653482333450, 37770112857074, 214694383882498, 1221832400430482, 6961037946938250, 39697830840765090, 226596964146630658

Offset: 0

Emeric Deutsch, Jul 14 2005

Number of Delannoy paths of length n that do not start with a (1, 1) step (a Delannoy path of length n is a path from (0, 0) to (n, n), consisting of steps E = (1, 0), N = (0, 1) and D = (1, 1)). Example: a(1) = 2 because we have NE and EN. Column 0 of A110169 (also nonzero entries in each column of A110169).

For n > 0: a(n) = A128966(2*n,n). - Reinhard Zumkeller, Jul 20 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Thomas Selig, Combinatorial aspects of sandpile models on wheel and fan graphs, arXiv:2202.06487 [math.CO], 2022.
Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

Cf. A001850, A110169.

Cf. A085362, A162478, A359758, A360132.

a110170 0 = 1
a110170 n = a128966 (2 * n) n  -- Reinhard Zumkeller, Jul 20 2013

with(orthopoly): a:=proc(n) if n=0 then 1 else P(n,3)-P(n-1,3) fi end: seq(a(n),n=0..25);
a := n -> `if`(n=0, 1, 2*hypergeom([1 - n, -n], [1], 2)):
seq(simplify(a(n)), n=0..24); # Peter Luschny, May 22 2017

CoefficientList[Series[(1 - x)/Sqrt[1 - 6 * x + x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)

x='x+O('x^66); Vec((1-x)/sqrt(1-6*x+x^2)) \\ Joerg Arndt, May 16 2013

G.f.: (1-z)/sqrt(1-6*z+z^2).
a(n) = P_n(3) - P_{n-1}(3) (n >= 1), where P_j is j-th Legendre polynomial.
From Paul Barry, Oct 18 2009: (Start)
G.f.: (1-x)/(1-x-2x/(1-x-x/(1-x-x/(1-x-x/(1-... (continued fraction);
G.f.: 1/(1-2x/((1-x)^2-x/(1-x/((1-x)^2-x/(1-x/((1-x)^2-x/(1-... (continued fraction);
a(n) = Sum_{k = 0..n} (0^(n + k) + C(n + k - 1, 2k - 1)) * C(2k, k) = 0^n + Sum_{k = 0..n} C(n + k - 1, 2k - 1) * C(2k, k). (End)
D-finite with recurrence: n*(2*n-3)*a(n) = 2*(6*n^2-12*n+5)*a(n-1) - (n-2)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 2^(-1/4)*(3+2*sqrt(2))^n/sqrt(Pi*n). - Vaclav Kotesovec, Oct 18 2012
a(n) = A277919(2n,n). - John P. McSorley, Nov 23 2016
a(n) = 2*hypergeom([1 - n, -n], [1], 2) for n>0. - Peter Luschny, May 22 2017
D-finite with recurrence: n*a(n) +(-7*n+5)*a(n-1) +(7*n-16)*a(n-2) +(-n+3)*a(n-3)=0. - R. J. Mathar, Jan 15 2020
a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (n^2-k^2) * a(k). - Seiichi Manyama, Mar 28 2023
G.f.: Sum_{n >= 0} binomial(2*n, n)*x^n/(1 - x)^(2*n) = 1 + 2*x + 10*x^2 + 50*x^3 + .... - Peter Bala, Apr 17 2024

A005592 a(n) = F(2n+1) + F(2n-1) - 1.

Original entry on oeis.org

1, 2, 6, 17, 46, 122, 321, 842, 2206, 5777, 15126, 39602, 103681, 271442, 710646, 1860497, 4870846, 12752042, 33385281, 87403802, 228826126, 599074577, 1568397606, 4106118242, 10749957121, 28143753122, 73681302246, 192900153617, 505019158606, 1322157322202

Offset: 0

N. J. A. Sloane

For any m, the maximum element in the continued fraction of F(2n+m)/F(m) is a(n). - Benoit Cloitre, Jan 10 2006
The continued fraction [a(n);1,a(n)-1,1,a(n)-1,...] = phi^(2n), where phi = 1.618... is the golden ratio, A001622. - Thomas Ordowski, Jun 07 2013
a(n) is the number of labeled subgraphs of the n-cycle C_n. For example, a(3)=17. There are 7 subgraphs of the triangle C_3 with 0 edges, 6 with 1 edge, 3 with 2 edges, and 1 with 3 edges (C_3 itself); here 7+6+3+1 = 17. - John P. McSorley, Oct 31 2016
a(n) equals the sum of the n-th row of triangle A277919. - John P. McSorley, Nov 25 2016

			G.f. = 1 + 2*x + 6*x^2 + 17*x^3 + 46*x^4 + 122*x^5 + 321*x^6 + 842*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 1..200 from Vincenzo Librandi)
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 41, 56.
M. D. McIlroy, The number of states of a dynamic storage system, Computer J., Vol. 25, No. 3 (1982), pp. 388-392.
M. D. McIlroy, The number of states of a dynamic storage system, Computer J., Vol. 25, No. 3 (1982), pp. 388-392. (Annotated scanned copy)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Jesús Salas and Alan D. Sokal, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial, J. Stat. Phys., Vol. 135 (2009), pp. 279-373; arXiv preprint, arXiv:0711.1738 [cond-mat.stat-mech], 2007-2009. Mentions this sequence. - N. J. A. Sloane, Mar 14 2014
Robert S. Seamons, Problem B-89, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 4, No. 2 (1966), p. 190; A Close Approximation, Solution to Problem B-89 by Douglas Lind, ibid., Vol. 5, No. 1 (1967), pp. 108-109.
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Equals A004146+1 and A005248+1.
Bisection of A014217; the other bisection is A002878, which also bisects A000032.
Cf. A000045, A065034.

a005592 n = a005592_list !! (n-1)
a005592_list = map (subtract 1) $
                   tail $ zipWith (+) a001519_list $ tail a001519_list
-- Reinhard Zumkeller, Aug 09 2013

[Fibonacci(2*n+1)+Fibonacci(2*n-1)-1: n in [1..30]]; // Vincenzo Librandi, Aug 23 2011

A005592:=-(2-2*z+z**2)/(z-1)/(z**2-3*z+1); # conjectured by Simon Plouffe in his 1992 dissertation
# second Maple program:
F:= n-> (<<0|1>, <1|1>>^n)[1,2]:
a:= n-> F(2*n+1)+F(2*n-1)-1:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 04 2016

Table[Fibonacci[2n+1]+Fibonacci[2n-1]-1,{n,30}] (* Harvey P. Dale, Aug 22 2011 *)
a[n_] := LucasL[2n]-1; Array[a, 30] (* Jean-François Alcover, Dec 09 2015 *)

a(n)=fibonacci(2*n+1)+fibonacci(2*n-1)-1 \\ Charles R Greathouse IV, Aug 23 2011

[lucas_number2(n,3,1)-1 for n in range(1,29)] # Zerinvary Lajos, Jul 06 2008

a(n) = Lucas(2*n)-1, with Lucas(n)=A000032(n).
a(n) = floor(r^(2*n)), where r = golden ratio = (1+sqrt(5))/2.
a(n) = floor(Fibonacci(5*n)/Fibonacci(3*n)). - Gary Detlefs, Mar 11 2011
a(n) = +4*a(n-1) -4*a(n-2) +1*a(n-3). - Joerg Arndt, Mar 11 2011
a(n) = A001519(2*n-1) + A001519(2*n+1) - 1. - Reinhard Zumkeller, Aug 09 2013
a(n) = 3*a(n) - a(n-1) + 1; a(n) = A004146(n) + 1, n>0. - Richard R. Forberg, Sep 04 2013
a(n) = 2*cosh(2*n*arcsinh(1/2)) - 1. - Ilya Gutkovskiy, Oct 31 2016
a(n) = floor(sqrt(5)*Fibonacci(2*n)), for n > 0 (Seamons, 1966). - Amiram Eldar, Feb 05 2022

Formulae and comments by Clark Kimberling, Nov 24 2010

a(0)=1 prepended by Alois P. Heinz, Nov 04 2016

cp's OEIS Frontend

A110170 First differences of the central Delannoy numbers (A001850).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula