A008998 -id:A008998 - cp's OEIS frontend

A013609 Triangle of coefficients in expansion of (1+2*x)^n.

1, 1, 2, 1, 4, 4, 1, 6, 12, 8, 1, 8, 24, 32, 16, 1, 10, 40, 80, 80, 32, 1, 12, 60, 160, 240, 192, 64, 1, 14, 84, 280, 560, 672, 448, 128, 1, 16, 112, 448, 1120, 1792, 1792, 1024, 256, 1, 18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512, 1, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024

Offset: 0

N. J. A. Sloane

T(n,k) is the number of lattice paths from (0,0) to (n,k) with steps (1,0) and two kinds of steps (1,1). The number of paths with steps (1,0) and s kinds of steps (1,1) corresponds to the expansion of (1+s*x)^n. - Joerg Arndt, Jul 01 2011
Also sum of rows in A046816. - Lior Manor, Apr 24 2004
Also square array of unsigned coefficients of Chebyshev polynomials of second kind. - Philippe Deléham, Aug 12 2005
The rows give the number of k-simplices in the n-cube. For example, 1, 6, 12, 8 shows that the 3-cube has 1 volume, 6 faces, 12 edges and 8 vertices. - Joshua Zucker, Jun 05 2006
Triangle whose (i, j)-th entry is binomial(i, j)*2^j.
With offset [1,1] the triangle with doubled numbers, 2*a(n,m), enumerates sequences of length m with nonzero integer entries n_i satisfying sum(|n_i|) <= n. Example n=4, m=2: [1,3], [3,1], [2,2] each in 2^2=4 signed versions: 2*a(4,2) = 2*6 = 12. The Sum over m (row sums of 2*a(n,m)) gives 2*3^(n-1), n >= 1. See the W. Lang comment and a K. A. Meissner reference under A024023. - Wolfdieter Lang, Jan 21 2008
n-th row of the triangle = leftmost column of nonzero terms of X^n, where X = an infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (2,2,2,...) in the subdiagonal. - Gary W. Adamson, Jul 19 2008
Numerators of a matrix square-root of Pascal's triangle A007318, where the denominators for the n-th row are set to 2^n. - Gerald McGarvey, Aug 20 2009
From Johannes W. Meijer, Sep 22 2010: (Start)
The triangle sums (see A180662 for their definitions) link the Pell-Jacobsthal triangle, whose mirror image is A038207, with twenty-four different sequences; see the crossrefs.
This triangle may very well be called the Pell-Jacobsthal triangle in view of the fact that A000129 (Kn21) are the Pell numbers and A001045 (Kn11) the Jacobsthal numbers.
(End)
T(n,k) equals the number of n-length words on {0,1,2} having n-k zeros. - Milan Janjic, Jul 24 2015
T(n-1,k-1) is the number of 2-compositions of n with zeros having k positive parts; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 16 2020
T(n,k) is the number of chains 0=x_0Geoffrey Critzer, Oct 01 2022
Excluding the initial 1, T(n,k) is the number of k-faces of a regular n-cross polytope. See A038207 for n-cube and A135278 for n-simplex. - Mohammed Yaseen, Jan 14 2023

			Triangle begins:
  1;
  1,  2;
  1,  4,   4;
  1,  6,  12,    8;
  1,  8,  24,   32,   16;
  1, 10,  40,   80,   80,    32;
  1, 12,  60,  160,  240,   192,    64;
  1, 14,  84,  280,  560,   672,   448,    128;
  1, 16, 112,  448, 1120,  1792,  1792,   1024,    256;
  1, 18, 144,  672, 2016,  4032,  5376,   4608,   2304,    512;
  1, 20, 180,  960, 3360,  8064, 13440,  15360,  11520,   5120,  1024;
  1, 22, 220, 1320, 5280, 14784, 29568,  42240,  42240,  28160, 11264,  2048;
  1, 24, 264, 1760, 7920, 25344, 59136, 101376, 126720, 112640, 67584, 24576, 4096;
From _Peter Bala_, Apr 20 2012: (Start)
The triangle can be written as the matrix product A038207*(signed version of A013609).
  |.1................||.1..................|
  |.2...1............||-1...2..............|
  |.4...4...1........||.1..-4...4..........|
  |.8..12...6...1....||-1...6...-12...8....|
  |16..32..24...8...1||.1..-8....24.-32..16|
  |..................||....................|
(End)

B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
G. Hotz, Zur Reduktion von Schaltkreispolynomen im Hinblick auf eine Verwendung in Rechenautomaten, El. Datenverarbeitung, Folge 5 (1960), pp. 21-27.

T. D. Noe, Rows n=0..50 of triangle, flattened
Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
H. J. Brothers, Pascal's Prism: Supplementary Material, PDF version.
John Cartan, Cartan's triangle shows the relationship to the n-cube.
R. Ehrenborg and M. Readdy, Characterization of the factorial functions of Eulerian binomial and Sheffer posets
J. Goldman and J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 1-rook coefficients of k rooks on the 2xn board, all heights 2.
W. G. Harter, Representations of multidimensional symmetries in networks, J. Math. Phys., 15 (1974), 2016-2021.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
D. A. Zaitsev, A generalized neighborhood for cellular automata, Theoretical Computer Science, 666 (2017), 21-35.

Cf. A007318, A013610, etc.
Appears in A167580 and A167591. - Johannes W. Meijer, Nov 23 2009
From Johannes W. Meijer, Sep 22 2010: (Start)
Triangle sums (see the comments): A000244 (Row1); A000012 (Row2); A001045 (Kn11); A026644 (Kn12); 4*A011377 (Kn13); A000129 (Kn21); A094706 (Kn22); A099625 (Kn23); A001653 (Kn3); A007583 (Kn4); A046717 (Fi1); A007051 (Fi2); A077949 (Ca1); A008998 (Ca2); A180675 (Ca3); A092467 (Ca4); A052942 (Gi1); A008999 (Gi2); A180676 (Gi3); A180677 (Gi4); A140413 (Ze1); A180678 (Ze2); A097117 (Ze3); A055588 (Ze4).
(End)
Cf. A024023, A038207, A046816, A105728, A115068, A135278, A171977, A180662.
T(2n,n) gives A059304.

a013609 n = a013609_list !! n
a013609_list = concat $ iterate ([1,2] *) [1]
instance Num a => Num [a] where
   fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

a013609 n k = a013609_tabl !! n !! k
a013609_row n = a013609_tabl !! n
a013609_tabl = iterate (\row -> zipWith (+) ([0] ++ row) $
                                zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Jul 22 2013, Feb 27 2013

[2^k*Binomial(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Sep 17 2021

bin2:=proc(n,k) option remember; if k<0 or k>n then 0 elif k=0 then 1 else 2*bin2(n-1,k-1)+bin2(n-1,k); fi; end; # N. J. A. Sloane, Jun 01 2009

Flatten[Table[CoefficientList[(1 + 2*x)^n, x], {n, 0, 10}]][[1 ;; 59]] (* Jean-François Alcover, May 17 2011 *)
BinomialROW[n_, k_, t_] := Sum[Binomial[n, k]*Binomial[k, j]*(-1)^(k - j)*t^j, {j, 0, k}]; Column[Table[BinomialROW[n, k, 3], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Jan 28 2019 *)

a(n,k):=coeff(expand((1+2*x)^n),x^k);
create_list(a(n,k),n,0,6,k,0,n); /* Emanuele Munarini, Nov 21 2012 */

/* same as in A092566 but use */
steps=[[1,0], [1,1], [1,1]]; /* note double [1,1] */
/* Joerg Arndt, Jul 01 2011 */

flatten([[2^k*binomial(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Sep 17 2021

G.f.: 1 / (1 - x*(1+2*y)).
T(n,k) = 2^k*binomial(n,k).
T(n,k) = 2*T(n-1,k-1) + T(n-1,k). - Jon Perry, Nov 22 2005
Row sums are 3^n = A000244(n). - Joerg Arndt, Jul 01 2011
T(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i). - Mircea Merca, Apr 28 2012
E.g.f.: exp(2*y*x + x). - Geoffrey Critzer, Nov 12 2012
Riordan array (x/(1 - x), 2*x/(1 - x)). Exp(2*x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(2*x)*(1 + 6*x + 12*x^2/2! + 8*x^3/3!) = 1 + 8*x + 40*x^2/2! + 160*x^3/3! + 560*x^4/4! + .... The same property holds more generally for Riordan arrays of the form (f(x), 2*x/(1 - x)). - Peter Bala, Dec 21 2014
T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(n,k) * binomial(k,j) * 3^j. - Kolosov Petro, Jan 28 2019
T(n,k) = 2*(n+1-k)*T(n,k-1)/k, T(n,0) = 1. - Alexander R. Povolotsky, Oct 08 2023
For n >= 1, GCD(T(n,1), ..., T(n,n)) = GCD(T(n,1),T(n,n)) = GCD(2*n,2^n) = A171977(n). - Pontus von Brömssen, Nov 01 2024

A052980 Expansion of (1 - x)/(1 - 2*x - x^3).

Original entry on oeis.org

1, 1, 2, 5, 11, 24, 53, 117, 258, 569, 1255, 2768, 6105, 13465, 29698, 65501, 144467, 318632, 702765, 1549997, 3418626, 7540017, 16630031, 36678688, 80897393, 178424817, 393528322, 867954037, 1914332891, 4222194104, 9312342245, 20539017381, 45300228866

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

a(n) counts permutations of length n which embed into the (infinite) increasing oscillating sequence given by 4,1,6,3,8,5,...,2k+2,2k-1,...; these are also the permutations which avoid {321, 2341, 3412, 4123}. - Vincent Vatter, May 23 2008
a(n) is the top left entry of the n-th power of any of the 3X3 matrices [1, 1, 0; 1, 1, 1; 1, 0, 0] or [1, 1, 1; 1, 1, 0; 0, 1, 0] or [1, 1, 1; 0, 0, 1; 1, 0, 1] or [1, 0, 1; 1, 0, 0; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n) is the number of possible tilings of a 2 X n board, using dominoes and L-shaped trominoes. - Michael Tulskikh, Aug 21 2019
a(n) = A190512(n-1) for n>0. - Greg Dresden, Feb 28 2020

Kenneth Edwards and Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kassie Archer and Noel Bourne, Pattern avoidance in compositions and powers of permutations, arXiv:2505.05218 [math.CO], 2025. See p. 4.
Robert Brignall, Nik Ruškuc, and Vincent Vatter, Simple permutations: decidability and unavoidable substructures, Theoretical Computer Science 391 (2008), 150-163.
Greg Dresden and Michael Tulskikh, Tilings of 2 X n boards with dominos and L-shaped trominos, Washington & Lee University (2021).
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1053
Djamila Oudrar, Sur l'énumération de structures discrètes, une approche par la théorie des relations, Thesis (in French), arXiv:1604.05839 [math.CO], 2016.
Djamila Oudrar and Maurice Pouzet, Profile and hereditary classes of ordered relational structures, arXiv preprint arXiv:1409.1108 [math.CO], 2014 [The first version of this document erroneously gives the A-number as A005298]
Vincent Vatter, Small permutation classes, arXiv:0712.4006 [math.CO], 2007-2016.
Index entries for linear recurrences with constant coefficients, signature (2,0,1).

See A190512 and A110513 for other versions of this sequence.
Column k=2 of A219987.
Cf. A008998.

I:=[1,1,2]; [n le 3 select I[n] else 2*Self(n-1)+Self(n-3): n in [1..40]];

R:=PowerSeriesRing(Integers(), 32); Coefficients(R!( (1 - x)/(1 - 2*x - x^3))); // Marius A. Burtea, Feb 14 2020

spec := [S,{S=Sequence(Prod(Union(Prod(Z,Z,Z),Z),Sequence(Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

CoefficientList[Series[(1 - x)/(1 - 2 x - x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 05 2014 *)

Vec((1-x)/(1-2*x-x^3)+O(x^99)) \\ Charles R Greathouse IV, Nov 20 2011

Recurrence: a(0)=1, a(1)=1, a(2)=2; thereafter a(n) = 2*a(n-1)+a(n-3).
a(n) = Sum(1/59*(4+3*_alpha^2+17*_alpha)*_alpha^(-1-n), _alpha = RootOf(-1+2*_Z+_Z^3)).
a(n) = A008998(n) - A008998(n-1). - R. J. Mathar, Feb 04 2014
Let u1 = 2.20556943... denote the real root of x^3-2*x^2-1. There is an explicit constant c1 = 0.460719842... such that for n>0, a(n) = nearest integer to c1*u1^n. - N. J. A. Sloane, Nov 07 2016
a(2n) = a(n)^2 - a(n-1)^2 + (1/2)*(a(n+2) - a(n+1) - a(n))^2. - Greg Dresden and Michael Tulskikh, Aug 20 2019
a(n) = 2^(n-1) + Sum_{i=3..n}(2^(n-i)*a(i-3)). - Greg Dresden, Aug 27 2019
a(n+1) = (Sum_{i >= 0} 2^(n-3i-2)*(4*binomial(n-2i, i) + binomial(n-2i-2, i))). - Michael Tulskikh, Feb 14 2020
a(n) = A008998(n-1) + A008998(n-3). - Michael Tulskikh, Feb 14 2020

A008999 a(n) = 2*a(n-1) + a(n-4).

Original entry on oeis.org

1, 2, 4, 8, 17, 36, 76, 160, 337, 710, 1496, 3152, 6641, 13992, 29480, 62112, 130865, 275722, 580924, 1223960, 2578785, 5433292, 11447508, 24118976, 50816737, 107066766, 225581040, 475281056, 1001378849

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 479
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014
Index entries for linear recurrences with constant coefficients, signature (2,0,0,1).

Cf. A008998.

a:=[1,2,4,8];; for n in [5..40] do a[n]:=2*a[n-1]+a[n-4]; od; a; # G. C. Greubel, Jun 12 2019

I:=[1, 2, 4, 8]; [n le 4 select I[n] else 2*Self(n-1)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, May 09 2012

A008999 := proc(n) option remember; if n <= 3 then 2^n else 2*A008999(n-1)+A008999(n-4); fi; end;

LinearRecurrence[{2,0,0,1},{1,2,4,8},40] (* Harvey P. Dale, May 09 2012 *)
CoefficientList[Series[1/(1-2x-x^4),{x,0,40}],x] (* Vincenzo Librandi, May 09 2012 *)

a(n):=sum(sum(binomial(n-m+(-3)*j,j)*binomial(n-3*j,m),j,0,(n-m)/3),m,0,n); /* Vladimir Kruchinin, May 23 2011 */

my(x='x+O('x^40)); Vec(1/(1-2*x-x^4)) \\ G. C. Greubel, Jun 12 2019

(1/(1-2*x-x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 12 2019

G.f.: 1/(1-2*x-x^4). - Philippe Deléham, Dec 02 2006
a(n) = Sum_{m=0..n} Sum_{j=0..(n-m)/3} binomial(n-m+(-3)*j,j)*binomial(n-3*j,m). - Vladimir Kruchinin, May 23 2011
O.g.f.: exp( Sum {n>=1} ( (1 + sqrt(1 + x^2))^n + (1 - sqrt(1 + x^2))^n ) * x^n/n ). Cf. A008998. - Peter Bala, Dec 22 2014

A193641 Number of arrays of -1..1 integers x(1..n) with every x(i) in a subsequence of length 1 or 2 with sum zero.

Original entry on oeis.org

1, 3, 7, 15, 33, 73, 161, 355, 783, 1727, 3809, 8401, 18529, 40867, 90135, 198799, 438465, 967065, 2132929, 4704323, 10375711, 22884351, 50473025, 111321761, 245527873, 541528771, 1194379303, 2634286479, 5810101729, 12814582761

Offset: 1

R. H. Hardin, Aug 02 2011

nonn

Column 1 of A193648.
Or yet empirical: row sums of triangle
  m/k |  0     1     2     3     4     5     6     7
  ==================================================
   0  |  1
   1  |  1     2
   2  |  1     2     4
   3  |  1     2     4     8
   4  |  1     4     4     8    16
   5  |  1     4    12     8    16    32
   6  |  1     4    12    32    16    32    64
   7  |  1     6    12    32    80    32    64   128
which is triangle for numbers 2^k*C(m,k) with triplicated diagonals. - Vladimir Shevelev, Apr 13 2012

			Some solutions for n=6:
   1   1   1   0   0   1  -1   1   0  -1  -1   0   0   0  -1  -1
  -1  -1  -1   0  -1  -1   1  -1   1   1   1   1   1   0   1   1
  -1   0   1   0   1   1   0   0  -1  -1   0  -1  -1   1  -1   1
   1   1   1   0   1   0  -1  -1   1   1   0   0  -1  -1  -1  -1
   0  -1  -1  -1  -1   0   1   1  -1   0   0   0   1   1   1   1
   0   1   1   1   1   0  -1   0   0   0   0   0   0  -1  -1  -1

R. H. Hardin, Table of n, a(n) for n = 1..200
Jean-Luc Baril and Nathanaël Hassler, Intervals in a family of Fibonacci lattices, Univ. de Bourgogne (France, 2024). See p. 7.
Tomislav Doslic and Ivana Zubac, Counting maximal matchings in linear polymers, Ars Mathematica Contemporanea 11 (2016) 255-276.

a193641 n = a193641_list !! n
a193641_list = drop 2 xs where
   xs = 1 : 1 : 1 : zipWith (+) xs (map (* 2) $ drop 2 xs)
-- Reinhard Zumkeller, Jan 01 2014

Empirical: a(n) = 2*a(n-1) + a(n-3).
Empirical: G.f.: -x*(1+x+x^2) / ( -1+2*x+x^3 ); a(n) = A008998(n-3) + A008998(n-2) + A008998(n-1). - R. J. Mathar, Feb 19 2015
Empirical: a(n) = 1 + 2*A077852(n-2) for n >= 2. - Greg Dresden, Apr 04 2021
Empirical: partial sums of A052910. - Sean A. Irvine, Jul 14 2022

A077852 Expansion of (1-x)^(-1)/(1-2*x-x^3).

Original entry on oeis.org

1, 3, 7, 16, 36, 80, 177, 391, 863, 1904, 4200, 9264, 20433, 45067, 99399, 219232, 483532, 1066464, 2352161, 5187855, 11442175, 25236512, 55660880, 122763936, 270764385, 597189651, 1317143239, 2905050864, 6407291380, 14131726000, 31168502865, 68744297111

Offset: 0

N. J. A. Sloane, Nov 17 2002

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,1,-1).

Cf. A019489. - R. J. Mathar, Sep 19 2008

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|1|-2|3>>^n)[4,4]:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 12 2017

CoefficientList[Series[(1-x)^(-1)/(1-2x-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{3,-2,1,-1},{1,3,7,16},40] (* Harvey P. Dale, Oct 05 2012 *)

From R. J. Mathar, May 15 2008: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) - a(n-4).
a(n+1) - a(n) = A008998(n+1). (End)
a(n) = 2*a(n-1) + a(n-3) + 1. - Greg Dresden, Apr 04 2021

A117716 Triangle T(n,k) read by rows: the coefficient [x^n] of x^2/(1-(k+1)*x-x^3) in row n, columns 0 <= k <= n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 3, 4, 1, 4, 9, 16, 25, 2, 9, 28, 65, 126, 217, 3, 20, 87, 264, 635, 1308, 2415, 4, 44, 270, 1072, 3200, 7884, 16954, 32960, 6, 97, 838, 4353, 16126, 47521, 119022, 264193, 534358, 9, 214, 2601, 17676, 81265, 286434, 835569, 2117656, 4815801, 10050030

Offset: 0

Roger L. Bagula, Apr 13 2006, corrected Apr 15 2006

			Triangle begins as:
  0;
  0,  0;
  1,  1,   1;
  1,  2,   3,    4;
  1,  4,   9,   16,   25;
  2,  9,  28,   65,  126,  217;
  3, 20,  87,  264,  635, 1308,  2415;
  4, 44, 270, 1072, 3200, 7884, 16954, 32960;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000930 (column 0), A008998 (column 1), A052541 (column 2), A052927 (column 3), A001093 (row 5), A185065 (row 6), A117715, A117724.

m:=12;
R:=PowerSeriesRing(Integers(), m+2);
A117716:= func< n,k | Coefficient(R!( x^2/(1-(k+1)*x-x^3) ), n) >;
[[A117716(n,k): k in [0..n]]: n in [0..m]]; // G. C. Greubel, Jul 23 2023

A117716 := proc(n,m)
        x^2/(1-(m+1)*x-x^3) ;
        if n < 0 then
                0;
        else
                coeftayl(%,x=0,n) ;
        end if;
end proc: # R. J. Mathar, May 14 2013

T[n_, k_]:= T[n, k]= Coefficient[Series[x^2/(1-(k+1)*x-x^3), {x,0,n+ 2}], x, n];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

def A117716(n,k):
    P. = PowerSeriesRing(QQ)
    return P( x^2/(1-(k+1)*x-x^3) ).list()[n]
flatten([[A117716(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 23 2023

Edited by G. C. Greubel, Jul 23 2023

A052910 Expansion of 1 + 2/(1-2*x-x^3).

Original entry on oeis.org

1, 2, 4, 8, 18, 40, 88, 194, 428, 944, 2082, 4592, 10128, 22338, 49268, 108664, 239666, 528600, 1165864, 2571394, 5671388, 12508640, 27588674, 60848736, 134206112, 296000898, 652850532, 1439907176, 3175815250, 7004481032

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 890
Index entries for linear recurrences with constant coefficients, signature (2,0,1).

a:=[2,4,8];; for n in [4..30] do a[n]:=2*a[n-1]+a[n-3]; od; Concatenation([1], a); # G. C. Greubel, Oct 15 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x^3)/(1-2*x-x^3) )); // G. C. Greubel, Oct 15 2019

spec := [S,{S=Sequence(Prod(Sequence(Prod(Z,Z,Z)),Union(Z,Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

Join[{1},LinearRecurrence[{2,0,1},{2,4,8},30]] (* Harvey P. Dale, Jun 07 2012 *)

my(x='x+O('x^30)); Vec((1-x^3)/(1-2*x-x^3)) \\ G. C. Greubel, Oct 15 2019

def A052910_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x^3)/(1-2*x-x^3)).list()
A052910_list(30) # G. C. Greubel, Oct 15 2019

G.f.: (1-x^3)/(1-2*x-x^3).
a(n) = 2*a(n-1) + a(n-3), with a(0)=1, a(1)=2, a(2)=4, a(3)=8.
a(n) = Sum_{alpha=RootOf(-1 + 2*z + z^3)} (2/59)*(12 -8*alpha + 9*alpha^2)*alpha^(-1-n).
a(n) = A008998(n) - A008998(n-3). - R. J. Mathar, Nov 28 2011

More terms from James Sellers, Jun 05 2000

A077926 Expansion of (1-x)^(-1)/(1+2*x+x^3).

Original entry on oeis.org

1, -1, 3, -6, 14, -30, 67, -147, 325, -716, 1580, -3484, 7685, -16949, 37383, -82450, 181850, -401082, 884615, -1951079, 4303241, -9491096, 20933272, -46169784, 101830665, -224594601, 495358987, -1092548638, 2409691878, -5314742742, 11722034123, -25853760123, 57022262989, -125766560100

Offset: 0

N. J. A. Sloane, Nov 17 2002

Index entries for linear recurrences with constant coefficients, signature (-1,2,-1,1).

Vec((1-x)^(-1)/(1+2*x+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

a(n)=-a(n-1)+2*a(n-2)-a(n-3)+a(n-4). - N-E. Fahssi, Mar 29 2008

a(n) -a(n-1) = (-1)^n*A008998(n). - R. J. Mathar, Jun 08 2020

A020708 Pisot sequences E(4,9), P(4,9).

Original entry on oeis.org

4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064, 11169, 24634, 54332, 119833, 264300, 582932, 1285697, 2835694, 6254320, 13794337, 30424368, 67103056, 148000449, 326425266, 719953588, 1587907625, 3502240516, 7724434620, 17036776865, 37575794246, 82876023112

Offset: 0

David W. Wilson

nonn

Colin Barker, Table of n, a(n) for n = 0..1000
Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT], 2016.
Index entries for linear recurrences with constant coefficients, signature (2,0,1).

This is a subsequence of A008998.

See A008776 for definitions of Pisot sequences.

Exy:=[4,9]; [n le 2 select Exy[n] else Floor(Self(n-1)^2/Self(n-2) + 1/2): n in [1..40]]; // Bruno Berselli, Feb 05 2016

RecurrenceTable[{a[0] == 4, a[1] == 9, a[n] == Floor[a[n - 1]^2/a[n - 2] + 1/2]}, a, {n, 0, 30}] (* Bruno Berselli, Feb 05 2016 *)
LinearRecurrence[{2,0,1},{4,9,20},40] (* Harvey P. Dale, Dec 19 2022 *)

Vec((4+x+2*x^2) / (1-2*x-x^3) + O(x^30)) \\ Jinyuan Wang, Mar 10 2020

a(n) = 2*a(n-1) + a(n-3) (holds at least up to n = 1000 but is not known to hold in general).
Empirical g.f.: (4+x+2*x^2) / (1-2*x-x^3). - Colin Barker, Jun 05 2016
Theorem: E(4,9) satisfies a(n) = 2 a(n - 1) + a(n - 3) for n >= 3. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger, and implies the above conjectures. - N. J. A. Sloane, Sep 09 2016

cp's OEIS Frontend

A141016 Duplicate of A008998.

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A013609 Triangle of coefficients in expansion of (1+2*x)^n.

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A052980 Expansion of (1 - x)/(1 - 2*x - x^3).

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A008999 a(n) = 2*a(n-1) + a(n-4).

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A193641 Number of arrays of -1..1 integers x(1..n) with every x(i) in a subsequence of length 1 or 2 with sum zero.

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A077852 Expansion of (1-x)^(-1)/(1-2*x-x^3).

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A117716 Triangle T(n,k) read by rows: the coefficient [x^n] of x^2/(1-(k+1)*x-x^3) in row n, columns 0 <= k <= n.

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