A154325 -id:A154325 - cp's OEIS frontend

A000129 Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).

0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, 1136689, 2744210, 6625109, 15994428, 38613965, 93222358, 225058681, 543339720, 1311738121, 3166815962, 7645370045, 18457556052, 44560482149, 107578520350, 259717522849

Offset: 0

N. J. A. Sloane

Sometimes also called lambda numbers.
Also denominators of continued fraction convergents to sqrt(2): 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, 8119/5741, 19601/13860, 47321/33461, 114243/80782, ... = A001333/A000129.
Number of lattice paths from (0,0) to the line x=n-1 consisting of U=(1,1), D=(1,-1) and H=(2,0) steps (i.e., left factors of Grand Schroeder paths); for example, a(3)=5, counting the paths H, UD, UU, DU and DD. - Emeric Deutsch, Oct 27 2002
a(2*n) with b(2*n) := A001333(2*n), n >= 1, give all (positive integer) solutions to Pell equation b^2 - 2*a^2 = +1 (see Emerson reference). a(2*n+1) with b(2*n+1) := A001333(2*n+1), n >= 0, give all (positive integer) solutions to Pell equation b^2 - 2*a^2 = -1.
Bisection: a(2*n+1) = T(2*n+1, sqrt(2))/sqrt(2) = A001653(n), n >= 0 and a(2*n) = 2*S(n-1,6) = 2*A001109(n), n >= 0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first, resp. second kind. S(-1,x)=0. See A053120, resp. A049310. - Wolfdieter Lang, Jan 10 2003
Consider the mapping f(a/b) = (a + 2b)/(a + b). Taking a = b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/1, 3/2, 7/5, 17/12, 41/29, ... converging to 2^(1/2). Sequence contains the denominators. - Amarnath Murthy, Mar 22 2003
This is also the Horadam sequence (0,1,1,2). Limit_{n->oo} a(n)/a(n-1) = sqrt(2) + 1 = A014176. - Ross La Haye, Aug 18 2003
Number of 132-avoiding two-stack sortable permutations.
From Herbert Kociemba, Jun 02 2004: (Start)
For n > 0, the number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 2, s(n) = 3.
Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 4 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 1, s(n) = 2. (End)
Counts walks of length n from a vertex of a triangle to another vertex to which a loop has been added. - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
Apart from initial terms, Pisot sequence P(2,5). See A008776 for definition of Pisot sequences. - David W. Wilson
Sums of antidiagonals of A038207 [Pascal's triangle squared]. - Ross La Haye, Oct 28 2004
The Pell primality test is "If N is an odd prime, then P(N)-Kronecker(2,N) is divisible by N". "Most" composite numbers fail this test, so it makes a useful pseudoprimality test. The odd composite numbers which are Pell pseudoprimes (i.e., that pass the above test) are in A099011. - Jack Brennen, Nov 13 2004
a(n) = sum of n-th row of triangle in A008288 = A094706(n) + A000079(n). - Reinhard Zumkeller, Dec 03 2004
Pell trapezoids (cf. A084158); for n > 0, A001109(n) = (a(n-1) + a(n+1))*a(n)/2; e.g., 1189 = (12+70)*29/2. - Charlie Marion, Apr 01 2006
(0!a(1), 1!a(2), 2!a(3), 3!a(4), ...) and (1,-2,-2,0,0,0,...) form a reciprocal pair under the list partition transform and associated operations described in A133314. - Tom Copeland, Oct 29 2007
Let C = (sqrt(2)+1) = 2.414213562..., then for n > 1, C^n = a(n)*(1/C) + a(n+1). Example: C^3 = 14.0710678... = 5*(0.414213562...) + 12. Let X = the 2 X 2 matrix [0, 1; 1, 2]; then X^n * [1, 0] = [a(n-1), a(n); a(n), a(n+1)]. a(n) = numerator of n-th convergent to (sqrt(2)-1) = 0.414213562... = [2, 2, 2, ...], the convergents being [1/2, 2/5, 5/12, ...]. - Gary W. Adamson, Dec 21 2007
A = sqrt(2) = 2/2 + 2/5 + 2/(5*29) + 2/(29*169) + 2/(169*985) + ...; B = ((5/2) - sqrt(2)) = 2/2 + 2/(2*12) + 2/(12*70) + 2/(70*408) + 2/(408*2378) + ...; A+B = 5/2. C = 1/2 = 2/(1*5) + 2/(2*12) + 2/(5*29) + 2/(12*70) + 2/(29*169) + ... - Gary W. Adamson, Mar 16 2008
From Clark Kimberling, Aug 27 2008: (Start)
Related convergents (numerator/denominator):
  lower principal convergents: A002315/A001653
  upper principal convergents: A001541/A001542
  intermediate convergents: A052542/A001333
  lower intermediate convergents: A005319/A001541
  upper intermediate convergents: A075870/A002315
  principal and intermediate convergents: A143607/A002965
  lower principal and intermediate convergents: A143608/A079496
  upper principal and intermediate convergents: A143609/A084068. (End)
Equals row sums of triangle A143808 starting with offset 1. - Gary W. Adamson, Sep 01 2008
Binomial transform of the sequence:= 0,1,0,2,0,4,0,8,0,16,..., powers of 2 alternating with zeros. - Philippe Deléham, Oct 28 2008
a(n) is also the sum of the n-th row of the triangle formed by starting with the top two rows of Pascal's triangle and then each next row has a 1 at both ends and the interior values are the sum of the three numbers in the triangle above that position. - Patrick Costello (pat.costello(AT)eku.edu), Dec 07 2008
Starting with offset 1 = eigensequence of triangle A135387 (an infinite lower triangular matrix with (2,2,2,...) in the main diagonal and (1,1,1,...) in the subdiagonal). - Gary W. Adamson, Dec 29 2008
Starting with offset 1 = row sums of triangle A153345. - Gary W. Adamson, Dec 24 2008
From Charlie Marion, Jan 07 2009: (Start)
In general, denominators, a(k,n) and numerators, b(k,n), of continued fraction convergents to sqrt((k+1)/k) may be found as follows:
  let a(k,0) = 1, a(k,1) = 2k; for n > 0, a(k,2n) = 2*a(k,2n-1) + a(k,2n-2)
  and a(k,2n+1) = (2k)*a(k,2n) + a(k,2n-1);
  let b(k,0) = 1, b(k,1) = 2k+1; for n > 0, b(k,2n) = 2*b(k,2n-1) + b(k,2n-2)
  and b(k,2n+1) = (2k)*b(k,2n) + b(k,2n-1).
For example, the convergents to sqrt(2/1) start 1/1, 3/2, 7/5, 17/12, 41/29.
In general, if a(k,n) and b(k,n) are the denominators and numerators, respectively, of continued fraction convergents to sqrt((k+1)/k) as defined above, then
  k*a(k,2n)^2 - a(k,2n-1)*a(k,2n+1) = k = k*a(k,2n-2)*a(k,2n) - a(k,2n-1)^2 and
  b(k,2n-1)*b(k,2n+1) - k*b(k,2n)^2 = k+1 = b(k,2n-1)^2 - k*b(k,2n-2)*b(k,2n);
  for example, if k=1 and n=3, then a(1,n) = a(n+1) and
  1*a(1,6)^2 - a(1,5)*a(1,7) = 1*169^2 - 70*408 = 1;
  1*a(1,4)*a(1,6) - a(1,5)^2 = 1*29*169 - 70^2 = 1;
  b(1,5)*b(1,7) - 1*b(1,6)^2 = 99*577 - 1*239^2 = 2;
  b(1,5)^2 - 1*b(1,4)*b(1,6) = 99^2 - 1*41*239 = 2.
Cf. A001333, A142238, A142239, A153313, A153314, A153315, A153316, A153317, A153318.
(End)
Starting with offset 1 = row sums of triangle A155002, equivalent to the statement that the Fibonacci sequence convolved with the Pell sequence prefaced with a "1": (1, 1, 2, 5, 12, 29, ...) = (1, 2, 5, 12, 29, ...). - Gary W. Adamson, Jan 18 2009
It appears that P(p) == 8^((p-1)/2) (mod p), p = prime; analogous to [Schroeder, p. 90]: Fp == 5^((p-1)/2) (mod p). Example: Given P(11) = 5741, == 8^5 (mod 11). Given P(17) = 11336689, == 8^8 (mod 17) since 17 divides (8^8 - P(17)). - Gary W. Adamson, Feb 21 2009
Equals eigensequence of triangle A154325. - Gary W. Adamson, Feb 12 2009
Another combinatorial interpretation of a(n-1) arises from a simple tiling scenario. Namely, a(n-1) gives the number of ways of tiling a 1 X n rectangle with indistinguishable 1 X 2 rectangles and 1 X 1 squares that come in two varieties, say, A and B. For example, with C representing the 1 X 2 rectangle, we obtain a(4)=12 from AAA, AAB, ABA, BAA, ABB, BAB, BBA, BBB, AC, BC, CA and CB. - Martin Griffiths, Apr 25 2009
a(n+1) = 2*a(n) + a(n-1), a(1)=1, a(2)=2 was used by Theon from Smyrna. - Sture Sjöstedt, May 29 2009
The n-th Pell number counts the perfect matchings of the edge-labeled graph C_2 x P_(n-1), or equivalently, the number of domino tilings of a 2 X (n-1) cylindrical grid. - Sarah-Marie Belcastro, Jul 04 2009
As a fraction: 1/79 = 0.0126582278481... or 1/9799 = 0.000102051229...(1/119 and 1/10199 for sequence in reverse). - Mark Dols, May 18 2010
Limit_{n->oo} (a(n)/a(n-1) - a(n-1)/a(n)) tends to 2.0. Example: a(7)/a(6) - a(6)/a(7) = 169/70 - 70/169 = 2.0000845... - Gary W. Adamson, Jul 16 2010
Numbers k such that 2*k^2 +- 1 is a square. - Vincenzo Librandi, Jul 18 2010
Starting (1, 2, 5, ...) = INVERTi transform of A006190: (1, 3, 10, 33, 109, ...). - Gary W. Adamson, Aug 06 2010
[u,v] = [a(n), a(n-1)] generates all Pythagorean triples [u^2-v^2, 2uv, u^2+v^2] whose legs differ by 1. - James R. Buddenhagen, Aug 14 2010
An elephant sequence, see A175654. For the corner squares six A[5] vectors, with decimal values between 21 and 336, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A078057. - Johannes W. Meijer, Aug 15 2010
Let the 2 X 2 square matrix A=[2, 1; 1, 0] then a(n) = the (1,1) element of A^(n-1). - Carmine Suriano, Jan 14 2011
Define a t-circle to be a first-quadrant circle tangent to the x- and y-axes. Such a circle has coordinates equal to its radius. Let C(0) be the t-circle with radius 1. Then for n > 0, define C(n) to be the next larger t-circle which is tangent to C(n - 1). C(n) has radius A001333(2n) + a(2n)*sqrt(2) and each of the coordinates of its point of intersection with C(n + 1) is a(2n + 1) + (A001333(2n + 1)*sqrt(2))/2. See similar Comments for A001109 and A001653, Sep 14 2005. - Charlie Marion, Jan 18 2012
A001333 and A000129 give the diagonal numbers described by Theon from Smyrna. - Sture Sjöstedt, Oct 20 2012
Pell numbers could also be called "silver Fibonacci numbers", since, for n >= 1, F(n+1) = ceiling(phi*F(n)), if n is even and F(n+1) = floor(phi*F(n)), if n is odd, where phi is the golden ratio, while a(n+1) = ceiling(delta*a(n)), if n is even and a(n+1) = floor(delta*a(n)), if n is odd, where delta = delta_S = 1+sqrt(2) is the silver ratio. - Vladimir Shevelev, Feb 22 2013
a(n) is the number of compositions (ordered partitions) of n-1 into two sorts of 1's and one sort of 2's. Example: the a(3)=5 compositions of 3-1=2 are 1+1, 1+1', 1'+1, 1'+1', and 2. - Bob Selcoe, Jun 21 2013
Between every two consecutive squares of a 1 X n array there is a flap that can be folded over one of the two squares. Two flaps can be lowered over the same square in 2 ways, depending on which one is on top. The n-th Pell number counts the ways n-1 flaps can be lowered. For example, a sideway representation for the case n = 3 squares and 2 flaps is \\., .//, \./, ./., .\., where . is an empty square. - Jean M. Morales, Sep 18 2013
Define a(-n) to be a(n) for n odd and -a(n) for n even. Then a(n) = A005319(k)*(a(n-2k+1) - a(n-2k)) + a(n-4k) = A075870(k)*(a(n-2k+2) - a(n-2k+1)) - a(n-4k+2). - Charlie Marion, Nov 26 2013
An alternative formulation of the combinatorial tiling interpretation listed above: Except for n=0, a(n-1) is the number of ways of partial tiling a 1 X n board with 1 X 1 squares and 1 X 2 dominoes. - Matthew Lehman, Dec 25 2013
Define a(-n) to be a(n) for n odd and -a(n) for n even. Then a(n) = A077444(k)*a(n-2k+1) + a(n-4k+2). This formula generalizes the formula used to define this sequence. - Charlie Marion, Jan 30 2014
a(n-1) is the top left entry of the n-th power of any of the 3 X 3 matrices [0, 1, 1; 1, 1, 1; 0, 1, 1], [0, 1, 1; 0, 1, 1; 1, 1, 1], [0, 1, 0; 1, 1, 1; 1, 1, 1] or [0, 0, 1; 1, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n+1) counts closed walks on K2 containing two loops on the other vertex. Equivalently the (1,1) entry of A^(n+1) where the adjacency matrix of digraph is A=(0,1;1,2). - David Neil McGrath, Oct 28 2014
For n >= 1, a(n) equals the number of ternary words of length n-1 avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
This is a divisibility sequence (i.e., if n|m then a(n)|a(m)). - Tom Edgar, Jan 28 2015
A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - Michael Somos, Jan 03 2017
a(n) is the number of compositions (ordered partitions) of n-1 into two kinds of parts, n and n', when the order of the 1 does not matter, or equivalently, when the order of the 1' does not matter. Example: When the order of the 1 does not matter, the a(3)=5 compositions of 3-1=2 are 1+1, 1+1'=1+1, 1'+1', 2 and 2'. (Contrast with entry from Bob Selcoe dated Jun 21 2013.) - Gregory L. Simay, Sep 07 2017
Number of weak orderings R on {1,...,n} that are weakly single-peaked w.r.t. the total ordering 1 < ... < n and for which {1,...,n} has exactly one minimal element for the weak ordering R. - J. Devillet, Sep 28 2017
Also the number of matchings in the (n-1)-centipede graph. - Eric W. Weisstein, Sep 30 2017
Let A(r,n) be the total number of ordered arrangements of an n+r tiling of r red squares and white tiles of total length n, where the individual tile lengths can range from 1 to n. A(r,0) corresponds to a tiling of r red squares only, and so A(r,0)=1. Let A_1(r,n) = Sum_{j=0..n} A(r,j) and let A_s(r,n) = Sum_{j=0..n} A_(s-1)(r,j). Then A_0(1,n) + A_2(3,n-4) + A_4(5,n-8) + ... + A_(2j) (2j+1, n-4j) = a(n) without the initial 0. - Gregory L. Simay, May 25 2018
(1, 2, 5, 12, 29, ...) is the fourth INVERT transform of (1, -2, 5, -12, 29, ...), as shown in A073133. - Gary W. Adamson, Jul 17 2019
Number of 2-compositions of n restricted to odd parts (and allowed zeros); see Hopkins & Ouvry reference. - Brian Hopkins, Aug 17 2020
Also called the 2-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence. - Michael A. Allen, Jan 23 2023
Named by Lucas (1878) after the English mathematician John Pell (1611-1685). - Amiram Eldar, Oct 02 2023
a(n) is the number of compositions of n when there are F(i) parts of size i, with i,n >= 1, F(n) the Fibonacci numbers, A000045(n) (see example below). - Enrique Navarrete, Dec 15 2023

			G.f. = x + 2*x^2 + 5*x^3 + 12*x^4 + 29*x^5 + 70*x^6 + 169*x^7 + 408*x^8 + 985*x^9 + ...
From _Enrique Navarrete_, Dec 15 2023: (Start)
From the comment on compositions with Fibonacci number of parts, F(n), there are F(1)=1 type of 1, F(2)=1 type of 2, F(3)=2 types of 3, F(4)=3 types of 4, F(5)=5 types of 5 and F(6)=8 types of 6.
The following table gives the number of compositions of n=6 with Fibonacci number of parts:
Composition, number of such compositions, number of compositions of this type:
6,           1,     8;
5+1,         2,    10;
4+2,         2,     6;
3+3,         1,     4;
4+1+1,       3,     9;
3+2+1,       6,    12;
2+2+2,       1,     1;
3+1+1+1,     4,     8;
2+2+1+1,     6,     6;
2+1+1+1+1,   5,     5;
1+1+1+1+1+1, 1,     1;
for a total of a(6)=70 compositions of n=6. (End).

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Index entries for "core" sequences.
Index entries for sequences related to Chebyshev polynomials.
Index to divisibility sequences.
Index entries for linear recurrences with constant coefficients, signature (2,1).

Partial sums of A001333.
2nd row of A172236.
a(n) = A054456(n-1, 0), n>=1 (first column of triangle).
Cf. A002203, A096669, A096670, A097075, A097076, A051927, A005409.
Cf. A175181 (Pisano periods), A214028 (Entry points), A214027 (number of zeros in a fundamental period).
A077985 is a signed version.
INVERT transform of Fibonacci numbers (A000045).
Cf. A038207.
The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
Cf. A034867, A131980, A133156, A143808, A135387, A153346, A001622, A006497, A014176 (growth power), A098316, A154325, A021083, A243399, A008555.
Cf. A048739.
Cf. A073133.
Cf. A041085.
Cf. A157103, A352361.
Sequences with g.f. 1/(1-k*x-x^2) or x/(1-k*x-x^2): A000045 (k=1), this sequence (k=2), A006190 (k=3), A001076 (k=4), A052918 (k=5), A005668 (k=6), A054413 (k=7), A041025 (k=8), A099371 (k=9), A041041 (k=10), A049666 (k=11), A041061 (k=12), A140455 (k=13), A041085 (k=14), A154597 (k=15), A041113 (k=16), A178765 (k=17), A041145 (k=18), A243399 (k=19), A041181 (k=20).

a := [0,1];; for n in [3..10^3] do a[n] := 2 * a[n-1] + a[n-2]; od; A000129 := a; # Muniru A Asiru, Oct 16 2017

a000129 n = a000129_list !! n
a000129_list = 0 : 1 : zipWith (+) a000129_list (map (2 *) $ tail a000129_list)
-- Reinhard Zumkeller, Jan 05 2012, Feb 05 2011

[0] cat [n le 2 select n else 2*Self(n-1) + Self(n-2): n in [1..35]]; // Vincenzo Librandi, Aug 08 2015

A000129 := proc(n) option remember; if n <=1 then n; else 2*procname(n-1)+procname(n-2); fi; end;
a:= n-> (<<2|1>, <1|0>>^n)[1, 2]: seq(a(n), n=0..40); # Alois P. Heinz, Aug 01 2008
A000129 := n -> `if`(n<2, n, 2^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], -1)):
seq(simplify(A000129(n)), n=0..31); # Peter Luschny, Dec 17 2015

CoefficientList[Series[x/(1 - 2*x - x^2), {x, 0, 60}], x] (* Stefan Steinerberger, Apr 08 2006 *)
Expand[Table[((1 + Sqrt[2])^n - (1 - Sqrt[2])^n)/(2Sqrt[2]), {n, 0, 30}]] (* Artur Jasinski, Dec 10 2006 *)
LinearRecurrence[{2, 1}, {0, 1}, 60] (* Harvey P. Dale, Jan 04 2012 *)
a[ n_] := With[ {s = Sqrt@2}, ((1 + s)^n - (1 - s)^n) / (2 s)] // Simplify; (* Michael Somos, Jun 01 2013 *)
Table[Fibonacci[n, 2], {n, 0, 20}] (* Vladimir Reshetnikov, May 08 2016 *)
Fibonacci[Range[0, 20], 2] (* Eric W. Weisstein, Sep 30 2017 *)
a[ n_] := ChebyshevU[n - 1, I] / I^(n - 1); (* Michael Somos, Oct 30 2021 *)

a[0]:0$
a[1]:1$
a[n]:=2*a[n-1]+a[n-2]$
A000129(n):=a[n]$
makelist(A000129(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

makelist((%i)^(n-1)*ultraspherical(n-1,1,-%i),n,0,24),expand; /* Emanuele Munarini, Mar 07 2018 */

for (n=0, 4000, a=contfracpnqn(vector(n, i, 1+(i>1)))[2, 1]; if (a > 10^(10^3 - 6), break); write("b000129.txt", n, " ", a)); \\ Harry J. Smith, Jun 12 2009

{a(n) = imag( (1 + quadgen( 8))^n )}; /* Michael Somos, Jun 01 2013 */

{a(n) = if( n<0, -(-1)^n, 1) * contfracpnqn( vector( abs(n), i, 1 + (i>1))) [2, 1]}; /* Michael Somos, Jun 01 2013 */

a(n)=([2, 1; 1, 0]^n)[2,1] \\ Charles R Greathouse IV, Mar 04 2014

{a(n) = polchebyshev(n-1, 2, I) / I^(n-1)}; /* Michael Somos, Oct 30 2021 */

from itertools import islice
def A000129_gen(): # generator of terms
    a, b = 0, 1
    yield from [a,b]
    while True:
        a, b = b, a+2*b
        yield b
A000129_list = list(islice(A000129_gen(),20)) # Chai Wah Wu, Jan 11 2022

[lucas_number1(n, 2, -1) for n in range(30)]  # Zerinvary Lajos, Apr 22 2009

G.f.: x/(1 - 2*x - x^2). - Simon Plouffe in his 1992 dissertation.
a(2n+1)=A001653(n). a(2n)=A001542(n). - Ira Gessel, Sep 27 2002
G.f.: Sum_{n >= 0} x^(n+1) *( Product_{k = 1..n} (2*k + x)/(1 + 2*k*x) ) = Sum_{n >= 0} x^(n+1) *( Product_{k = 1..n} (x + 1 + k)/(1 + k*x) ) = Sum_{n >= 0} x^(n+1) *( Product_{k = 1..n} (x + 3 - k)/(1 - k*x) ) may all be proved using telescoping series. - Peter Bala, Jan 04 2015
a(n) = 2*a(n-1) + a(n-2), a(0)=0, a(1)=1.
a(n) = ((1 + sqrt(2))^n - (1 - sqrt(2))^n)/(2*sqrt(2)).
For initial values a(0) and a(1), a(n) = ((a(0)*sqrt(2)+a(1)-a(0))*(1+sqrt(2))^n + (a(0)*sqrt(2)-a(1)+a(0))*(1-sqrt(2))^n)/(2*sqrt(2)). - Shahreer Al Hossain, Aug 18 2019
a(n) = integer nearest a(n-1)/(sqrt(2) - 1), where a(0) = 1. - Clark Kimberling
a(n) = Sum_{i, j, k >= 0: i+j+2k = n} (i+j+k)!/(i!*j!*k!).
a(n)^2 + a(n+1)^2 = a(2n+1) (1999 Putnam examination).
a(2n) = 2*a(n)*A001333(n). - John McNamara, Oct 30 2002
a(n) = ((-i)^(n-1))*S(n-1, 2*i), with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(-1, x)=0, S(-2, x)= -1.
Binomial transform of expansion of sinh(sqrt(2)x)/sqrt(2). E.g.f.: exp(x)sinh(sqrt(2)x)/sqrt(2). - Paul Barry, May 09 2003
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k+1)*2^k. - Paul Barry, May 13 2003
a(n-2) + a(n) = (1 + sqrt(2))^(n-1) + (1 - sqrt(2))^(n-1) = A002203(n-1). (A002203(n))^2 - 8(a(n))^2 = 4(-1)^n. - Gary W. Adamson, Jun 15 2003
Unreduced g.f.: x(1+x)/(1 - x - 3x^2 - x^3); a(n) = a(n-1) + 3*a(n-2) + a(n-2). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*2^(n-2k). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
Apart from initial terms, inverse binomial transform of A052955. - Paul Barry, May 23 2004
a(n)^2 + a(n+2k+1)^2 = A001653(k)*A001653(n+k); e.g., 5^2 + 70^2 = 5*985. - Charlie Marion Aug 03 2005
a(n+1) = Sum_{k=0..n} binomial((n+k)/2, (n-k)/2)*(1+(-1)^(n-k))*2^k/2. - Paul Barry, Aug 28 2005
a(n) = a(n-1) + A001333(n-1) = A001333(n) - a(n-1) = A001109(n)/A001333(n) = sqrt(A001110(n)/A001333(n)^2) = ceiling(sqrt(A001108(n)/2)). - Henry Bottomley, Apr 18 2000
a(n) = F(n, 2), the n-th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006
Define c(2n) = -A001108(n), c(2n+1) = -A001108(n+1) and d(2n) = d(2n+1) = A001652(n); then ((-1)^n)*(c(n) + d(n)) = a(n). [Proof given by Max Alekseyev.] - Creighton Dement, Jul 21 2005
a(r+s) = a(r)*a(s+1) + a(r-1)*a(s). - Lekraj Beedassy, Sep 03 2006
a(n) = (b(n+1) + b(n-1))/n where {b(n)} is the sequence A006645. - Sergio Falcon, Nov 22 2006
From Miklos Kristof, Mar 19 2007: (Start)
Let F(n) = a(n) = Pell numbers, L(n) = A002203 = companion Pell numbers (A002203):
For a >= b and odd b,  F(a+b) + F(a-b) = L(a)*F(b).
For a >= b and even b, F(a+b) + F(a-b) = F(a)*L(b).
For a >= b and odd b,  F(a+b) - F(a-b) = F(a)*L(b).
For a >= b and even b, F(a+b) - F(a-b) = L(a)*F(b).
F(n+m) + (-1)^m*F(n-m) = F(n)*L(m).
F(n+m) - (-1)^m*F(n-m) = L(n)*F(m).
F(n+m+k) + (-1)^k*F(n+m-k) + (-1)^m*(F(n-m+k) + (-1)^k*F(n-m-k)) = F(n)*L(m)*L(k).
F(n+m+k) - (-1)^k*F(n+m-k) + (-1)^m*(F(n-m+k) - (-1)^k*F(n-m-k)) = L(n)*L(m)*F(k).
F(n+m+k) + (-1)^k*F(n+m-k) - (-1)^m*(F(n-m+k) + (-1)^k*F(n-m-k)) = L(n)*F(m)*L(k).
F(n+m+k) - (-1)^k*F(n+m-k) - (-1)^m*(F(n-m+k) - (-1)^k*F(n-m-k)) = 8*F(n)*F(m)*F(k). (End)
a(n+1)*a(n) = 2*Sum_{k=0..n} a(k)^2 (a similar relation holds for A001333). - Creighton Dement, Aug 28 2007
a(n+1) = Sum_{k=0..n} binomial(n+1,2k+1) * 2^k = Sum_{k=0..n} A034867(n,k) * 2^k = (1/n!) * Sum_{k=0..n} A131980(n,k) * 2^k. - Tom Copeland, Nov 30 2007
Equals row sums of unsigned triangle A133156. - Gary W. Adamson, Apr 21 2008
a(n) (n >= 3) is the determinant of the (n-1) X (n-1) tridiagonal matrix with diagonal entries 2, superdiagonal entries 1 and subdiagonal entries -1. - Emeric Deutsch, Aug 29 2008
a(n) = A000045(n) + Sum_{k=1..n-1} A000045(k)*a(n-k). - Roger L. Bagula and Gary W. Adamson, Sep 07 2008
From Hieronymus Fischer, Jan 02 2009: (Start)
fract((1+sqrt(2))^n) = (1/2)*(1 + (-1)^n) - (-1)^n*(1+sqrt(2))^(-n) = (1/2)*(1 + (-1)^n) - (1-sqrt(2))^n.
See A001622 for a general formula concerning the fractional parts of powers of numbers x > 1, which satisfy x - x^(-1) = floor(x).
a(n) = round((1+sqrt(2))^n/(2*sqrt(2))) for n > 0. (End) [last formula corrected by Josh Inman, Mar 05 2024]
a(n) = ((4+sqrt(18))*(1+sqrt(2))^n + (4-sqrt(18))*(1-sqrt(2))^n)/4 offset 0. - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009
If p[i] = Fibonacci(i) and if A is the Hessenberg matrix of order n defined by A[i,j] = p[j-i+1] when i<=j, A[i,j]=-1 when i=j+1, and A[i,j]=0 otherwise, then, for n >= 1, a(n) = det A. - Milan Janjic, May 08 2010
a(n) = 3*a(n-1) - a(n-2) - a(n-3), n > 2. - Gary Detlefs, Sep 09 2010
From Charlie Marion, Apr 13 2011: (Start)
a(n) = 2*(a(2k-1) + a(2k))*a(n-2k) - a(n-4k).
a(n) = 2*(a(2k) + a(2k+1))*a(n-2k-1) + a(n-4k-2). (End)
G.f.: x/(1 - 2*x - x^2) = sqrt(2)*G(0)/4; G(k) = ((-1)^k) - 1/(((sqrt(2) + 1)^(2*k)) - x*((sqrt(2) + 1)^(2*k))/(x + ((sqrt(2) - 1)^(2*k + 1))/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 02 2011
In general, for n > k, a(n) = a(k+1)*a(n-k) + a(k)*a(n-k-1). See definition of Pell numbers and the formula for Sep 04 2008. - Charlie Marion, Jan 17 2012
Sum{n>=1} (-1)^(n-1)/(a(n)*a(n+1)) = sqrt(2) - 1. - Vladimir Shevelev, Feb 22 2013
From Vladimir Shevelev, Feb 24 2013: (Start)
(1) Expression a(n+1) via a(n): a(n+1) = a(n) + sqrt(2*a^2(n) + (-1)^n);
(2) a(n+1)^2 - a(n)*a(n+2) = (-1)^n;
(3) Sum_{k=1..n} (-1)^(k-1)/(a(k)*a(k+1)) = a(n)/a(n+1);
(4) a(n)/a(n+1) = sqrt(2) - 1 + r(n), where |r(n)| < 1/(a(n+1)*a(n+2)). (End)
a(-n) = -(-1)^n * a(n). - Michael Somos, Jun 01 2013
G.f.: G(0)/(2+2*x) - 1/(1+x), where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Aug 10 2013
G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 + x)/( x*(4*k+4 + x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 30 2013
a(n) = Sum_{r=0..n-1} Sum_{k=0..n-r-1} binomial(r+k,k)*binomial(k,n-k-r-1). - Peter Luschny, Nov 16 2013
a(n) = Sum_{k=1,3,5,...<=n} C(n,k)*2^((k-1)/2). - Vladimir Shevelev, Feb 06 2014
a(2n) = 2*a(n)*(a(n-1) + a(n)). - John Blythe Dobson, Mar 08 2014
a(k*n) = a(k)*a(k*n-k+1) + a(k-1)*a(k*n-k). - Charlie Marion, Mar 27 2014
a(k*n) = 2*a(k)*(a(k*n-k)+a(k*n-k-1)) + (-1)^k*a(k*n-2k). - Charlie Marion, Mar 30 2014
a(n+1) = (1+sqrt(2))*a(n) + (1-sqrt(2))^n. - Art DuPre, Apr 04 2014
a(n+1) = (1-sqrt(2))*a(n) + (1+sqrt(2))^n. - Art DuPre, Apr 04 2014
a(n) = F(n) + Sum_{k=1..n} F(k)*a(n-k), n >= 0 where F(n) the Fibonacci numbers A000045. - Ralf Stephan, May 23 2014
a(n) = round(sqrt(a(2n) + a(2n-1)))/2. - Richard R. Forberg, Jun 22 2014
a(n) = Product_{k divides n} A008555(k). - Tom Edgar, Jan 28 2015
a(n+k)^2 - A002203(k)*a(n)*a(n+k) + (-1)^k*a(n)^2 = (-1)^n*a(k)^2. - Alexander Samokrutov, Aug 06 2015
a(n) = 2^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], -1) for n >= 2. - Peter Luschny, Dec 17 2015
a(n+1) = Sum_{k=0..n} binomial(n,k)*2^floor(k/2). - Tony Foster III, May 07 2017
a(n) = exp((i*Pi*n)/2)*sinh(n*arccosh(-i))/sqrt(2). - Peter Luschny, Mar 07 2018
From Rogério Serôdio, Mar 30 2018: (Start)
Some properties:
(1) a(n)^2 - a(n-2)^2 = 2*a(n-1)*(a(n) + a(n-2)) (see A005319);
(2) a(n-k)*a(n+k) = a(n)^2 + (-1)^(n+k+1)*a(k)^2;
(3) Sum_{k=2..n+1} a(k)*a(k-1) = a(n+1)^2 if n is odd, else a(n+1)^2 - 1 if n is even;
(4) a(n) - a(n-2*k+1) = (A077444(k) - 1)*a(n-2*k+1) + a(n-4*k+2);
(5) Sum_{k=n..n+9} a(k) = 41*A001333(n+5). (End)
From Kai Wang, Dec 30 2019: (Start)
a(m+r)*a(n+s) - a(m+s)*a(n+r) = -(-1)^(n+s)*a(m-n)*a(r-s).
a(m+r)*a(n+s) + a(m+s)*a(n+r) = (2*A002203(m+n+r+s) - (-1)^(n+s)*A002203(m-n)*A002203(r-s))/8.
A002203(m+r)*A002203(n+s) - A002203(m+s)*A002203(n+r) = (-1)^(n+s)*8*a(m-n)*a(r-s).
A002203(m+r)*A002203(n+s) - 8*a(m+s)*a(n+r) = (-1)^(n+s)*A002203(m-n)*A002203(r-s).
A002203(m+r)*A002203(n+s) + 8*a(m+s)*a(n+r) = 2*A002203(m+n+r+s)+ (-1)^(n+s)*8*a(m-n)*a(r-s). (End)
From Kai Wang, Jan 12 2020: (Start)
a(n)^2 - a(n+1)*a(n-1) = (-1)^(n-1).
a(n)^2 - a(n+r)*a(n-r) = (-1)^(n-r)*a(r)^2.
a(m)*a(n+1) - a(m+1)*a(n) = (-1)^n*a(m-n).
a(m-n) = (-1)^n (a(m)*A002203(n) - A002203(m)*a(n))/2.
a(m+n) = (a(m)*A002203(n) + A002203(m)*a(n))/2.
A002203(n)^2 - A002203(n+r)*A002203(n-r) = (-1)^(n-r-1)*8*a(r)^2.
A002203(m)*A002203(n+1) - A002203(m+1)*A002203(n) = (-1)^(n-1)*8*a(m-n).
A002203(m-n) = (-1)^(n)*(A002203(m)*A002203(n) - 8*a(m)*a(n) )/2.
A002203(m+n) = (A002203(m)*A002203(n) + 8*a(m)*a(n) )/2. (End)
From Kai Wang, Mar 03 2020: (Start)
Sum_{m>=1} arctan(2/a(2*m+1)) = arctan(1/2).
Sum_{m>=2} arctan(2/a(2*m+1)) = arctan(1/12).
In general, for n > 0,
Sum_{m>=n} arctan(2/a(2*m+1)) = arctan(1/a(2*n)). (End)
a(n) = (A001333(n+3*k) + (-1)^(k-1)*A001333(n-3*k)) / (20*A041085(k-1)) for any k>=1. - Paul Curtz, Jun 23 2021
Sum_{i=0..n} a(i)*J(n-i) = (a(n+1) + a(n) - J(n+2))/2 for J(n) = A001045(n). - Greg Dresden, Jan 05 2022
From Peter Bala, Aug 20 2022: (Start)
Sum_{n >= 1} 1/(a(2*n) + 1/a(2*n)) = 1/2.
Sum_{n >= 1} 1/(a(2*n+1) - 1/a(2*n+1)) = 1/4. Both series telescope - see A075870 and A005319.
Product_{n >= 1} ( 1 + 2/a(2*n) ) = 1 + sqrt(2).
Product_{n >= 2} ( 1 - 2/a(2*n) ) = (1/3)*(1 + sqrt(2)). (End)
G.f. = 1/(1 - Sum_{k>=1} Fibonacci(k)*x^k). - Enrique Navarrete, Dec 17 2023
Sum_{n >=1} 1/a(n) = 1.84220304982752858079237158327980838... - R. J. Mathar, Feb 05 2024
a(n) = ((3^(n+1) + 1)^(n-1) mod (9^(n+1) - 2)) mod (3^(n+1) - 1). - Joseph M. Shunia, Jun 06 2024

A006318 Large Schröder numbers (or large Schroeder numbers, or big Schroeder numbers).

Original entry on oeis.org

1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038, 3937603038, 20927156706, 111818026018, 600318853926, 3236724317174, 17518619320890, 95149655201962, 518431875418926, 2832923350929742, 15521467648875090

Offset: 0

N. J. A. Sloane

For the little Schröder numbers (or little Schroeder numbers, or small Schroeder numbers) see A001003.
The number of perfect matchings in a triangular grid of n squares (n = 1, 4, 9, 16, 25, ...). - Roberto E. Martinez II, Nov 05 2001
a(n) is the number of subdiagonal paths from (0, 0) to (n, n) consisting of steps East (1, 0), North (0, 1) and Northeast (1, 1) (sometimes called royal paths). - David Callan, Mar 14 2004
Twice A001003 (except for the first term).
a(n) is the number of dissections of a regular (n+4)-gon by diagonals that do not touch the base. (A diagonal is a straight line joining two nonconsecutive vertices and dissection means the diagonals are noncrossing though they may share an endpoint. One side of the (n+4)-gon is designated the base.) Example: a(1)=2 because a pentagon has only 2 such dissections: the empty one and the one with a diagonal parallel to the base. - David Callan, Aug 02 2004
a(n) is the number of separable permutations, i.e., permutations avoiding 2413 and 3142 (see Shapiro and Stephens). - Vincent Vatter, Aug 16 2006
Eric W. Weisstein comments that the Schröder numbers bear the same relationship to the Delannoy numbers (A001850) as the Catalan numbers (A000108) do to the binomial coefficients. - Jonathan Vos Post, Dec 23 2004
a(n) is the number of lattice paths from (0, 0) to (n+1, n+1) consisting of unit steps north N = (0, 1) and variable-length steps east E = (k, 0), with k a positive integer, that stay strictly below the line y = x except at the endpoints. For example, a(2) = 6 counts 111NNN, 21NNN, 3NNN, 12NNN, 11N1NN, 2N1NN (east steps indicated by their length). If the word "strictly" is replaced by "weakly", the counting sequence becomes the little Schröder numbers, A001003 (offset). - David Callan, Jun 07 2006
a(n) is the number of dissections of a regular (n+3)-gon with base AB that do not contain a triangle of the form ABP with BP a diagonal. Example: a(1) = 2 because the square D-C | | A-B has only 2 such dissections: the empty one and the one with the single diagonal AC (although this dissection contains the triangle ABC, BC is not a diagonal). - David Callan, Jul 14 2006
a(n) is the number of (colored) Motzkin n-paths with each upstep and each flatstep at ground level getting one of 2 colors and each flatstep not at ground level getting one of 3 colors. Example: With their colors immediately following upsteps/flatsteps, a(2) = 6 counts U1D, U2D, F1F1, F1F2, F2F1, F2F2. - David Callan, Aug 16 2006
The Hankel transform of this sequence is A006125(n+1) = [1, 2, 8, 64, 1024, 32768, ...]; example: Det([1, 2, 6, 22; 2, 6, 22, 90; 6, 22, 90, 394; 22, 90, 394, 1806]) = 64. - Philippe Deléham, Sep 03 2006
Triangle A144156 has row sums equal to A006318 with left border A001003. - Gary W. Adamson, Sep 12 2008
a(n) is also the number of order-preserving and order-decreasing partial transformations (of an n-chain). Equivalently, it is the order of the Schröder monoid, PC sub n. - Abdullahi Umar, Oct 02 2008
Sum_{n >= 0} a(n)/10^n - 1 = (9 - sqrt(41))/2. - Mark Dols, Jun 22 2010
1/sqrt(41) = Sum_{n >= 0} Delannoy number(n)/10^n. - Mark Dols, Jun 22 2010
a(n) is also the dimension of the space Hoch(n) related to Hochschild two-cocycles. - Ph. Leroux (ph_ler_math(AT)yahoo.com), Aug 24 2010
Let W = (w(n, k)) denote the augmentation triangle (as at A193091) of A154325; then w(n, n) = A006318(n). - Clark Kimberling, Jul 30 2011
Conjecture: For each n > 2, the polynomial sum_{k = 0}^n a(k)*x^{n-k} is irreducible modulo some prime p < n*(n+1). - Zhi-Wei Sun, Apr 07 2013
From Jon Perry, May 24 2013: (Start)
Consider a Pascal triangle variant where T(n, k) = T(n, k-1) + T(n-1, k-1) + T(n-1, k), i.e., the order of performing the calculation must go from left to right (A033877). This sequence is the rightmost diagonal.
Triangle begins:
  1;
  1,  2;
  1,  4,  6;
  1,  6, 16, 22;
  1,  8, 30, 68, 90;
  ... (End)
a(n) is the number of permutations avoiding 2143, 3142 and one of the patterns among 246135, 254613, 263514, 524361, 546132. - Alexander Burstein, Oct 05 2014
a(n) is the number of semi-standard Young tableaux of shape n x 2 with consecutive entries.  That is, j in P and 1 <= i<= j imply i in P. - Graham H. Hawkes, Feb 15 2015
a(n) is the number of unary-rooted size n unary-binary trees (each node has either 1 or 2 degree out). - John Bodeen, May 29 2017
Conjecturally, a(n) is the number of permutations pi of length n such that s(pi) avoids the patterns 231 and 321, where s denotes West's stack-sorting map. - Colin Defant, Sep 17 2018
a(n) is the number of n X n permutation matrices which percolate under the 2-neighbor bootstrap percolation rule (see Shapiro and Stephens). The number of general n X n matrices of weight n which percolate is given in A146971. - Jonathan Noel, Oct 05 2018
a(n) is the number of permutations of length n+1 which avoid 3142 and 3241. The permutations are precisely the permutations that are sortable by a decreasing stack followed by an increasing stack in series. - Rebecca Smith, Jun 06 2019
a(n) is the number of permutations of length n+1 avoiding the partially ordered pattern (POP) {3>1, 4>1, 1>2} of length 4. That is, the number of length n+1 permutations having no subsequences of length 4 in which the second element is the smallest, and the first element is smaller than the third and fourth elements. - Sergey Kitaev, Dec 10 2020
Named after the German mathematician Ernst Schröder (1841-1902). - Amiram Eldar, Apr 15 2021
a(n) is the number of sequences of nonnegative integers (u_1, u_2, ..., u_n) such that (i) u_i <= i for all i, and (ii) the nonzero u_i are weakly increasing. For example, a(2) = 6 counts 00, 01, 02, 10, 11, 12. See link "Some bijections for lattice paths" at A001003. - David Callan, Dec 18 2021
a(n) is the number of separable elements of the Weyl group of type B_n/C_n (see Gaetz and Gao). - Fern Gossow, Jul 31 2023
The number of domino tilings of an Aztec triangle of order n. Dually, the number perfect matchings of the edges in the cellular graph formed by a triangular grid of n squares (n = 1, 4, 9, 16, 25, ...) as in Ciucu (1996). - Michael Somos, Sep 16 2024
a(n) is the number of dissections of a convex (n+3)-sided polygon by non-intersecting diagonals such that none of the dividing diagonals passes through a chosen vertex. - Muhammed Sefa Saydam, Mar 01 2025
a(n) is the number of dissections of a convex (n+m+1)-sided polygon by non-intersecting diagonals such that the selected m consecutive sides of the polygon will be in the same subpolygon. - Muhammed Sefa Saydam, Jul 02 2025

			a(3) = 22 since the top row of Q^n = (6, 6, 6, 4, 0, 0, 0, ...); where 22 = (6 + 6 + 6 + 4).
G.f. = 1 + 2*x + 6*x^2 + 22*x^3 + 90*x^4 + 394*x^5 + 1806*x^6 + 8858*x^7 + 41586*x^8 + ...

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Index entries for "core" sequences

Apart from leading term, twice A001003 (the small Schroeder numbers). Cf. A025240.
Sequences A085403, A086456, A103137, A112478 are essentially the same sequence.
Main diagonal of A033877.
Cf. A002003, A004148, A088617, A060693, A144156.
Row sums of A104219. Bisections give A138462, A138463.
Row sums of A175124.
The sequences listed in Yang-Jiang's Table 1 appear to be A006318, A001003, A027307, A034015, A144097, A243675, A260332, A243676. - N. J. A. Sloane, Mar 28 2021

Concatenation([1],List([1..25],n->(1/n)*Sum([0..n],k->2^k*Binomial(n,k)*Binomial(n,k-1)))); # Muniru A Asiru, Nov 29 2018

a006318 n = a004148_list !! n
a006318_list = 1 : f [1] where
   f xs = y : f (y : xs) where
     y = head xs + sum (zipWith (*) xs $ reverse xs)
-- Reinhard Zumkeller, Nov 13 2012

Order := 24: solve(series((y-y^2)/(1+y),y)=x,y); # then A(x)=y(x)/x
BB:=(-1-z-sqrt(1-6*z+z^2))/2: BBser:=series(BB, z=0, 24): seq(coeff(BBser, z, n), n=1..23); # Zerinvary Lajos, Apr 10 2007
A006318_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 2*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od; convert(a,list)end: A006318_list(22); # Peter Luschny, May 19 2011
A006318 := n-> add(binomial(n+k, n-k) * binomial(2*k, k)/(k+1), k=0..n): seq(A006318(n), n=0..22); # Johannes W. Meijer, Jul 14 2013
seq(simplify(hypergeom([-n,n+1],[2],-1)), n=0..100); # Robert Israel, Mar 23 2015

a[0] = 1; a[n_Integer] := a[n] = a[n - 1] + Sum[a[k]*a[n - 1 - k], {k, 0, n - 1}]; Array[a[#] &, 30]
InverseSeries[Series[(y - y^2)/(1 + y), {y, 0, 24}], x] (* then A(x) = y(x)/x *) (* Len Smiley, Apr 11 2000 *)
CoefficientList[Series[(1 - x - (1 - 6x + x^2)^(1/2))/(2x), {x, 0, 30}], x] (* Harvey P. Dale, May 01 2011 *)
a[ n_] := 2 Hypergeometric2F1[ -n + 1, n + 2, 2, -1]; (* Michael Somos, Apr 03 2013 *)
a[ n_] := With[{m = If[ n < 0, -1 - n, n]}, SeriesCoefficient[(1 - x - Sqrt[ 1 - 6 x + x^2])/(2 x), {x, 0, m}]]; (* Michael Somos, Jun 10 2015 *)
Table[-(GegenbauerC[n+1, -1/2, 3] + KroneckerDelta[n])/2, {n, 0, 30}] (* Vladimir Reshetnikov, Nov 12 2016 *)
CoefficientList[Nest[1+x(#+#^2)&, 1+O[x], 20], x] (* Oliver Seipel, Dec 21 2024 *)

{a(n) = if( n<0, n = -1-n); polcoeff( (1 - x - sqrt( 1 - 6*x + x^2 + x^2 * O(x^n))) / 2, n+1)}; /* Michael Somos, Apr 03 2013 */

{a(n) = if( n<1, 1, sum( k=0, n, 2^k * binomial( n, k) * binomial( n, k-1)) / n)};

from gmpy2 import divexact
A006318 = [1, 2]
for n in range(3,10**3):
    A006318.append(int(divexact(A006318[-1]*(6*n-9)-(n-3)*A006318[-2],n)))
# Chai Wah Wu, Sep 01 2014

# Generalized algorithm of L. Seidel
def A006318_list(n) :
    D = [0]*(n+1); D[1] = 1
    b = True; h = 1; R = []
    for i in range(2*n) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k-1]
            h += 1;
        else :
            for k in range(1,h, 1) : D[k] += D[k-1]
            R.append(D[h-1]);
        b = not b
    return R
A006318_list(23) # Peter Luschny, Jun 02 2012

G.f.: (1 - x - (1 - 6*x + x^2)^(1/2))/(2*x).
a(n) = 2*hypergeom([-n+1, n+2], [2], -1). - Vladeta Jovovic, Apr 24 2003
For n > 0, a(n) = (1/n)*Sum_{k = 0..n} 2^k*C(n, k)*C(n, k-1). - Benoit Cloitre, May 10 2003
The g.f. satisfies (1 - x)*A(x) - x*A(x)^2 = 1. - Ralf Stephan, Jun 30 2003
For the asymptotic behavior, see A001003 (remembering that A006318 = 2*A001003). - N. J. A. Sloane, Apr 10 2011
From Philippe Deléham, Nov 28 2003: (Start)
Row sums of A088617 and A060693.
a(n) = Sum_{k = 0..n} C(n+k, n)*C(n, k)/(k+1). (End)
With offset 1: a(1) = 1, a(n) = a(n-1) + Sum_{i = 1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
a(n) = Sum_{k = 0..n} A000108(k)*binomial(n+k, n-k). - Benoit Cloitre, May 09 2004
a(n) = Sum_{k = 0..n} A011117(n, k). - Philippe Deléham, Jul 10 2004
a(n) = (CentralDelannoy(n+1) - 3 * CentralDelannoy(n))/(2*n) = (-CentralDelannoy(n+1) + 6 * CentralDelannoy(n) - CentralDelannoy(n-1))/2 for n >= 1, where CentralDelannoy is A001850. - David Callan, Aug 16 2006
From Abdullahi Umar, Oct 11 2008: (Start)
A123164(n+1) - A123164(n) = (2*n+1)*a(n) (n >= 0).
  and 2*A123164(n) = (n+1)*a(n) - (n-1)*a(n-1) (n > 0). (End)
Define the general Delannoy numbers d(i, j) as in A001850. Then a(k) = d(2*k, k) - d(2*k, k-1) and a(0) = 1, Sum_{j=0..n} ((-1)^j * (d(n, j) + d(n-1, j-1)) * a(n-j)) = 0. - Peter E John, Oct 19 2006
Given an integer t >= 1 and initial values u = [a_0, a_1, ..., a_{t-1}], we may define an infinite sequence Phi(u) by setting a_n = a_{n-1} + a_0*a_{n-1} + a_1*a_{n-2} + ... + a_{n-2}*a_1 for n >= t. For example, Phi([1]) is the Catalan numbers A000108. The present sequence is (essentially) Phi([2]). - Gary W. Adamson, Oct 27 2008
G.f.: 1/(1-2x/(1-x/(1-2x/(1-x/(1-2x/(1-x/(1-2x/(1-x/(1-2x/(1-x.... (continued fraction). - Paul Barry, Dec 08 2008
G.f.: 1/(1 - x - x/(1 - x - x/(1 - x - x/(1 - x - x/(1 - x - x/(1 - ... (continued fraction). - Paul Barry, Jan 29 2009
a(n) ~ ((3 + 2*sqrt(2))^n)/(n*sqrt(2*Pi*n)*sqrt(3*sqrt(2) - 4))*(1-(9*sqrt(2) + 24)/(32*n) + ...). - G. Nemes (nemesgery(AT)gmail.com), Jan 25 2009
Logarithmic derivative yields A002003. - Paul D. Hanna, Oct 25 2010
a(n) = the upper left term in M^(n+1), M = the production matrix:
  1, 1, 0, 0, 0, 0, ...
  1, 1, 1, 0, 0, 0, ...
  2, 2, 1, 1, 0, 0, ...
  4, 4, 2, 1, 1, 0, ...
  8, 8, 8, 2, 1, 1, ...
  ... - Gary W. Adamson, Jul 08 2011
a(n) is the sum of top row terms in Q^n, Q = an infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  1, 1, 2, 0, 0, 0, ...
  1, 1, 1, 2, 0, 0, ...
  1, 1, 1, 1, 2, 0, ...
  1, 1, 1, 1, 1, 2, ...
  ... - Gary W. Adamson, Aug 23 2011
From Tom Copeland, Sep 21 2011: (Start)
With F(x) = (1 - 3*x - sqrt(1 - 6*x + x^2))/(2*x) an o.g.f. (nulling the n = 0 term) for A006318, G(x) = x/(2 + 3*x + x^2) is the compositional inverse.
Consequently, with H(x) = 1/ (dG(x)/dx) = (2 + 3*x + x^2)^2 / (2 - x^2),
  a(n) = (1/n!)*[(H(x)*d/dx)^n] x evaluated at x = 0, i.e.,
  F(x) = exp[x*H(u)*d/du] u, evaluated at u = 0. Also, dF(x)/dx = H(F(x)). (End)
a(n-1) = number of ordered complete binary trees with n leaves having k internal vertices colored black, the remaining n - 1 - k internal vertices colored white, and such that each vertex and its rightmost child have different colors ([Drake, Example 1.6.7]). For a refinement of this sequence see A175124. - Peter Bala, Sep 29 2011
D-finite with recurrence: (n-2)*a(n-2) - 3*(2*n-1)*a(n-1) + (n+1)*a(n) = 0. - Vaclav Kotesovec, Oct 05 2012
G.f.: A(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) = (1 - G(0))/x; G(k) = 1 + x - 2*x/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 04 2012
G.f.: A(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) = (G(0) - 1)/x; G(k) = 1 - x/(1 - 2/G(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Jan 04 2012
a(n+1) = a(n) + Sum_{k=0..n} a(k)*(n-k). - Reinhard Zumkeller, Nov 13 2012
G.f.: 1/Q(0) where Q(k) = 1 + k*(1 - x) - x - x*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013
a(-1-n) = a(n). - Michael Somos, Apr 03 2013
G.f.: 1/x - 1 - U(0)/x, where U(k) = 1 - x - x/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013
G.f.: (2 - 2*x - G(0))/(4*x), where G(k) = 1 + 1/( 1 - x*(6 - x)*(2*k - 1)/(x*(6 - x)*(2*k - 1) + 2*(k + 1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013
a(n) = 1/(n + 1) * (Sum_{j=0..n} C(n+j, j)*C(n+j+1, j+1)*(Sum_{k=0..n-j} (-1)^k*C(n+j+k, k))). - Graham H. Hawkes, Feb 15 2015
a(n) = hypergeom([-n, n+1], [2], -1). - Peter Luschny, Mar 23 2015
a(n) = sqrt(2) * LegendreP(n, -1, 3) where LegendreP is the associated Legendre function of the first kind (in Maple's notation). - Robert Israel, Mar 23 2015
G.f. A(x) satisfies: A(x) = Sum_{j>=0} x^j * Sum_{k=0..j} binomial(j,k)*A(x)^k. - Ilya Gutkovskiy, Apr 11 2019
From Peter Bala, May 13 2024: (Start)
a(n) = 2 * Sum_{k = 0..floor(n/2)} binomial(n, 2*k)*binomial(2*n-2*k, n)/(n-2*k+1) for n >= 1.
a(n) = Integral_{x = 0..1} Legendre_P(n, 2*x+1) dx. (End)
G.f. A(x) = 1/(1 - x) * c(x/(1-x)^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Aug 29 2024

Edited by Charles R Greathouse IV, Apr 20 2010

cp's OEIS Frontend

A156708 Triangle read by rows, binomial transform of A154325.

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A000129 Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).

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A006318 Large Schröder numbers (or large Schroeder numbers, or big Schroeder numbers).

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A193091 Augmentation of the triangular array A158405. See Comments.

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A132046 Triangle read by rows: T(n,0) = T(n,n) = 1, and T(n,k) = 2*binomial(n,k) for 1 <= k <= n - 1.

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Mathematica

Maxima

Formula

A319840 Table read by antidiagonals: T(n, k) is the number of elements on the perimeter of an n X k matrix.

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