A005021 -id:A005021 - cp's OEIS frontend

A080937 Number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2*n steps with all values <= 5.

1, 1, 2, 5, 14, 42, 131, 417, 1341, 4334, 14041, 45542, 147798, 479779, 1557649, 5057369, 16420730, 53317085, 173118414, 562110290, 1825158051, 5926246929, 19242396629, 62479659622, 202870165265, 658715265222, 2138834994142, 6944753544643, 22549473023585

Offset: 0

Henry Bottomley, Feb 25 2003

With interpolated zeros (1,0,1,0,2,...), counts closed walks of length n at start or end node of P_6. The sequence (0,1,0,2,...) counts walks of length n between the start and second node. - Paul Barry, Jan 26 2005
HANKEL transform of sequence and the sequence omitting a(0) is the sequence A130716. This is the unique sequence with that property. - Michael Somos, May 04 2012
From Wolfdieter Lang, Mar 30 2020: (Start)
a(n) is also the upper left entry of the n-th power of the 3 X 3 tridiagonal matrix M_3 = Matrix([1,1,0], [1,2,1], [0,1,2]) from A332602: a(n) = ((M_3)^n)[1,1].
Proof: (M_3)^n = b(n-2)*(M_3)^2 - (6*b(n-3) - b(n-4))*M_3 + b(n-3)*1_3, for n >= 0, with b(n) = A005021(n), for n >= -4. For the proof of this see a comment in A005021. Hence (M_3)^n[1,1] = 2*b(n-2) - 5*b(n-3) + b(n-4), for n >= 0. This proves the 3 X 3 part of the conjecture in A332602 by Gary W. Adamson.
The formula for a(n) given below in terms of r = rho(7) = A160389 proves that a(n)/a(n-1) converges to rho(7)^2 = A116425 = 3.2469796..., because r - 2/r = 0.6920... < 1, and r^2 - 3 = 0.2469... < 1. This limit was conjectured in A332602 by Gary W. Adamson.
(End)

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 131*x^6 + 417*x^7 + 1341*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..600
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 27.
Jean-Luc Baril, Toufik Mansour, José L. Ramírez, and Mark Shattuck, Catalan words avoiding a pattern of length four, Univ. de Bourgogne (France, 2024). See p. 3.
Jean-Luc Baril and Helmut Prodinger, Enumeration of partial Lukasiewicz paths, arXiv:2205.01383 [math.CO], 2022.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Giulio Cerbai, Anders Claesson, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
Wei Chen, Enumeration of Set Partitions Refined by Crossing and Nesting Numbers, MS Thesis, Department of Mathematics. Simon Fraser University, Fall 2014. Table 4.1, line k=2.
Johann Cigler, Number of bounded Dyck paths with "negative length", MathOverflow question, Sep 26 2020.
Michael Dairyko, Lara Pudwell, Samantha Tyner, and Casey Wynn, Non-contiguous pattern avoidance in binary trees, Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227.
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
Paul Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641 [math.CO], 2011.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
Stefan Felsner and Daniel Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
Daniel Heldt, On the mixing time of the face flip-and up/down Markov chain for some families of graphs, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.
Matthew Hyatt and Jeffrey Remmel, The classification of 231-avoiding permutations by descents and maximum drop, arXiv preprint arXiv:1208.1052 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 24 2012
Aleksandar Ilic and Andreja Ilic, On the number of restricted Dyck paths, Filomat 25:3 (2011), 191-201; DOI: 10.2298/FIL1103191I.
Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012 (Corollary 3, case k=5, pages 10-11). - From _N. J. A. Sloane_, May 09 2012
Erkko Lehtonen and Tamás Waldhauser, Associative spectra of graph algebras I. Foundations, undirected graphs, antiassociative graphs, arXiv:2011.07621 [math.CO], 2020. See also J. of Algebraic Combinatorics (2021) Vol. 53, 613-638.
Toufik Mansour and Mark Shattuck, Some enumerative results related to ascent sequences, arXiv preprint arXiv:1207.3755 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 22 2012
Sophie Morier-Genoud, Valentin Ovsienko, and Serge Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons, arXiv:1008.3359 [math.AG], 2010-2011. - From _N. J. A. Sloane_, Dec 26 2012
Sophie Morier-Genoud, Valentin Ovsienko, and Serge Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons, Annales de l'institut Fourier, 62 no. 3 (2012), 937-987. - From _N. J. A. Sloane_, Dec 26 2012
David Nečas and Ivan Ohlídal, Consolidated series for efficient calculation of the reflection and transmission in rough multilayers, Optics Express, Vol. 22, 2014, No. 4; DOI:10.1364/OE.22.004499.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Lara Pudwell, Pattern avoidance in trees, slides from a talk, mentions many sequences, 2012. - From _N. J. A. Sloane_, Jan 03 2013
Lara Pudwell and Andrew Baxter, Ascent sequences avoiding pairs of patterns, 2014.
Santiago Rojas-Rojas, Camila Muñoz, Edgar Barriga, Pablo Solano, Aldo Delgado, and Carla Hermann-Avigliano, Analytic Evolution for Complex Coupled Tight-Binding Models: Applications to Quantum Light Manipulation, arXiv:2310.12366 [quant-ph], 2023. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (5,-6,1).

Cf. A000007, A000012, A001519, A007051, A011782, A024175, A080937, A080938.
Cf. A033191 which essentially provide the same sequence for different limits and tend to A000108.
Cf. A005021, A094790, A094789.
Cf. A211216, A005021, A160389, A116425, A332602.

I:=[1,1,2]; [n le 3 select I[n] else 5*Self(n-1)-6*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jan 09 2016

a:= n-> (<<0|1|0>, <0|0|1>, <1|-6|5>>^n. <<1, 1, 2>>)[1, 1]:
seq(a(n), n=0..35);  # Alois P. Heinz, Nov 09 2012

nn=56;Select[CoefficientList[Series[(1-4x^2+3x^4)/(1-5x^2+6x^4-x^6), {x,0,nn}], x],#>0 &] (* Geoffrey Critzer, Jan 26 2014 *)
LinearRecurrence[{5,-6,1},{1,1,2},30] (* Jean-François Alcover, Jan 09 2016 *)

a=vector(99); a[1]=1; a[2]=2;a[3]=5; for(n=4,#a,a[n]=5*a[n-1]-6*a[n-2] +a[n-3]); a \\ Charles R Greathouse IV, Jun 10 2011

{a(n) = if( n<0, n = -n; polcoeff( (1 - 3*x + x^2) / (1 - 6*x + 5*x^2 - x^3) + x * O(x^n), n), polcoeff( (1 - 4*x + 3*x^2) / (1 - 5*x + 6*x^2 - x^3) + x * O(x^n), n))} /* Michael Somos, May 04 2012 */

a(n) = A080934(n,5).
G.f.: (1-4*x+3*x^2)/(1-5*x+6*x^2-x^3). - Ralf Stephan, May 13 2003
a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3). - Herbert Kociemba, Jun 11 2004
a(n) = A096976(2*n). - Floor van Lamoen, Nov 02 2005
a(n) = (4/7-4/7*cos(1/7*Pi)^2)*(4*(cos(Pi/7))^2)^n + (1/7-2/7*cos(1/7*Pi) + 4/7*cos(1/7*Pi)^2)*(4*(cos(2*Pi/7))^2)^n + (2/7+2/7*cos(1/7*Pi))*(4*(cos(3*Pi/7))^2)^n for n>=0. - Richard Choulet, Apr 19 2010
G.f.: 1 / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x))))). - Michael Somos, May 04 2012
a(-n) = A038213(n). a(n + 2) * a(n) - a(n + 1)^2 = a(1 - n). Convolution inverse is A123183 with A123183(0)=1. - Michael Somos, May 04 2012
From Wolfdieter Lang, Mar 30 2020: (Start)
In terms of the algebraic number r = rho(7) = A160389 of degree 3 the formula given by Richard Choulet becomes a(n) = (1/7)*(r)^(2*n)*(C1(r) + C2(r)*(r - 2/r)^(2*n) + C3(r)*(r^2 - 3)^(2*n)), with C1(r) = 4 - r^2, C2(r) = 1 - r + r^2, and C3 = 2 + r.
a(n) = ((M_3)^n)[1,1] = 2*b(n-2) - 5*b(n-3) + b(n-4), for n >= 0, with the 3 X 3 tridiagonal matrix M_3 = Matrix([1,1,0], [1,2,1], [0,1,2]) from A332602, and b(n) = A005021(n) (with offset n >= -4). (End)

A094789 Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 7 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 1, s(2n+1) = 4.

Original entry on oeis.org

1, 4, 14, 47, 155, 507, 1652, 5373, 17460, 56714, 184183, 598091, 1942071, 6305992, 20475625, 66484244, 215873462, 700937471, 2275930827, 7389902771, 23994866364, 77910846021, 252974934692, 821404463698, 2667083556359

Offset: 1

Herbert Kociemba, Jun 11 2004

In general, a(n) = (2/m)*Sum_{r=1..m-1} sin(r*j*Pi/m)*sin(r*k*Pi/m)*(2*cos(r*Pi/m))^(2n+1) counts (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = j, s(2n+1) = k.
With interpolated zeros (0,0,0,1,0,4,0,14,...) counts walks of length n between the start and fourth nodes on P_6. - Paul Barry, Jan 26 2005
The Hankel transforms of this sequence or of this sequence with the first term omitted give 1, -2, 1, 1, -2, 1, ... . - Wathek Chammam, Nov 16 2011
Diagonal of the square array A216201. - Philippe Deléham, Mar 28 2013

G. C. Greubel, Table of n, a(n) for n = 1..1000
W. Chammam, F. Marcellán and R. Sfaxi, Orthogonal polynomials, Catalan numbers, and a general Hankel determinant evaluation, Linear Algebra Appl. (2011), in press.
S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons, Annales de l'institut Fourier, 62 no. 3 (2012), 937-987; arXiv:1008.3359 [math.AG], 2010-2011. - _N. J. A. Sloane_, Dec 26 2012
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (5,-6,1).

Cf. A094790, A080937, A005021.

I:=[1,4,14]; [n le 3 select I[n] else 5*Self(n-1)-6*Self(n-2)+Self(n-3): n in [1..45]]; // Vincenzo Librandi, Nov 10 2014

f[n_] := FullSimplify[ TrigToExp[(2/7)Sum[ Sin[Pi*k/7]Sin[4Pi*k/7](2Cos[Pi*k/7])^(2n + 1), {k, 1, 6}]]]; Table[ f[n], {n, 25}] (* Robert G. Wilson v, Jun 18 2004 *)
LinearRecurrence[{5,-6,1}, {1,4,14}, 50] (* Roman Witula, Aug 09 2012 *)
CoefficientList[Series[(x - 1) / (- 1 + 5 x - 6 x^2 + x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 10 2014 *)

Vec(x*(x-1)/(-1 + 5*x - 6*x^2 + x^3) + O(x^40)) \\ Michel Marcus, Nov 10 2014

a(n) = (2/7)*Sum_{k = 1..6} sin(Pi*k/7)*sin(4*Pi*k/7)*(2*cos(Pi*k/7))^(2n + 1).
a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3).
G.f.: x*(x-1)/(-1 + 5*x - 6*x^2 + x^3). - Corrected by Vincenzo Librandi, Nov 10 2014
a(n) = 2^n*B(n; 1/2) = (1/7)*((c(1) - c(4))*(c(4))^(2n) + (c(2) - c(1))*(c(1))^(2n) + (c(4) - c(2))*(c(2))^(2n)), where c(j) := 2*cos(2*Pi*j/7). Here B(n; d), n in N, d in C denotes the respective quasi-Fibonacci number - see A121449 and Witula-Slota-Warzynski paper for details (see also A052975, A085810, A077998, A006054, A121442). - Roman Witula, Aug 09 2012
a(n+1) = A216201(n,n+2) = A216201(n,n+3). - Philippe Deléham, Mar 28 2013

A094790 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 7 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2*n, s(0) = 1, s(2n) = 3.

Original entry on oeis.org

1, 3, 9, 28, 89, 286, 924, 2993, 9707, 31501, 102256, 331981, 1077870, 3499720, 11363361, 36896355, 119801329, 388991876, 1263047761, 4101088878, 13316149700, 43237262993, 140390505643, 455845099957, 1480119728920

Offset: 1

Herbert Kociemba, Jun 11 2004

In general a(n) = (2/m)*Sum_{r=1..m-1} sin(r*j*Pi/m)*sin(r*k*Pi/m)*(2*cos(r*Pi/m))^(2n)) counts (s(0), s(1), ..., s(2n)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = j, s(2n) = k.
With interpolated zeros (0,0,1,0,3,0,9,...), counts walks of length n between the first and third nodes of P_6. - Paul Barry, Jan 26 2005
Counts all paths of length (2*n+1), n >= 0, starting at the initial node and ending on the nodes 1, 2, 3, 4 and 5 on the path graph P_6, see the Maple program. - Johannes W. Meijer, May 29 2010
With offset 0 = the INVERT transform of A055588. - Gary W. Adamson, Apr 01 2011

Michael De Vlieger, Table of n, a(n) for n = 1..1956
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Index entries for linear recurrences with constant coefficients, signature (5,-6,1).

Cf. A005021, A028495, A052975, A055588, A078038, A080937, A094789.

[n le 3 select 3^(n-1) else 5*Self(n-1) -6*Self(n-2) +Self(n-3): n in [1..31]]; // G. C. Greubel, Feb 12 2023

with(GraphTheory):G:=PathGraph(6): A:= AdjacencyMatrix(G): nmax:=24; n2:=2*nmax+1: for n from 0 to n2 do B(n):=A^n; a(n):=add(B(n)[k,1],k=1..5); od: seq(a(2*n+1),n=0..nmax); # Johannes W. Meijer, May 29 2010

f[n_]:= FullSimplify[ TrigToExp[(2/7)Sum[ Sin[Pi*k/7]Sin[3Pi*k/7](2Cos[Pi*k/7] )^(2n), {k,6}]]];
Table[f[n], {n, 25}] (* Robert G. Wilson v, Jun 18 2004 *)
LinearRecurrence[{5,-6,1},{1,3,9},30] (* Harvey P. Dale, Nov 19 2019 *)

Vec(x*(1-2*x)/(1-5*x+6*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Jun 14 2015

@CachedFunction
def a(n): # a = A094790
    if (n<4): return 3^(n-1)
    else: return 5*a(n-1) - 6*a(n-2) + a(n-3)
[a(n) for n in range(1,41)] # G. C. Greubel, Feb 12 2023

a(n) = (2/7)*Sum_{k=1..6} sin(Pi*k/7)*sin(3*Pi*k/7)*(2*cos(Pi*k/7))^(2n).
a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3).
G.f.: x*(1-2*x)/(1 - 5*x + 6*x^2 - x^3).
a(n) = rightmost term in M^n * [1,0,0] where M = the 3 X 3 matrix [2,1,1; 1,2,0; 1,0,1]. E.g., M^3 * [1,0,0] = [19,14,9]; right term = 9 = a(3). - Gary W. Adamson, Apr 04 2006

A052975 Expansion of (1-2x)(1-x)/(1-5x+6x^2-x^3).

Original entry on oeis.org

1, 2, 6, 19, 61, 197, 638, 2069, 6714, 21794, 70755, 229725, 745889, 2421850, 7863641, 25532994, 82904974, 269190547, 874055885, 2838041117, 9215060822, 29921113293, 97153242650, 315454594314, 1024274628963, 3325798821581, 10798800928441, 35063486341682

Offset: 0

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

Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 7 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 3. - Herbert Kociemba, Jun 11 2004

Counts all paths of length (2*n), n>=0, starting at the initial node and ending on the nodes 1, 2, 3, 4 and 5 on the path graph P_6, see the second Maple program. - Johannes W. Meijer, May 29 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1047
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (5,-6,1).

Cf. A060557.

Cf. A028495, A078038 and A094790.

I:=[1,2,6]; [n le 3 select I[n] else 5*Self(n-1)-6*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Sep 18 2015

spec := [S,{S=Sequence(Prod(Union(Sequence(Prod(Sequence(Z),Z)),Sequence(Z)),Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
with(GraphTheory):G:=PathGraph(6): A:= AdjacencyMatrix(G): nmax:=25; n2:=2*nmax+1: for n from 0 to n2 do B(n):=A^n; a(n):=add(B(n)[k,1],k=1..5); od: seq(a(2*n),n=0..nmax); # Johannes W. Meijer, May 29 2010

LinearRecurrence[{5,-6,1}, {1,2,6}, 50] (* Roman Witula, Aug 09 2012 *)
CoefficientList[Series[(1 - 2 x) (1 - x)/(1 - 5 x + 6 x^2 - x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 18 2015 *)

x='x+O('x^30); Vec((1-2*x)*(1-x)/(1-5*x+6*x^2-x^3)) \\ G. C. Greubel, Apr 19 2018

G.f.: (1-2*x)*(1-x)/(1-5*x+6*x^2-x^3).
a(n) = A028495(2*n). - Floor van Lamoen, Nov 02 2005
a(n) = Sum (1/7*(2-3*_alpha+_alpha^2)*_alpha^(-1-n), _alpha=RootOf(-1+5*_Z-6*_Z^2+_Z^3))
From Herbert Kociemba, Jun 11 2004: (Start)
a(n) = (2/7)*Sum_{r=1..6} sin(r*3*Pi/7)^2*(2*cos(r*Pi/7))^(2*n).
a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3). (End)
a(n) = 2^n*A(n;1/2) = (1/7)*(s(2)^2*c(4)^(2n) + s(4)^2*c(1)^(2n) + s(1)^2*c(2)^(2n)), where c(j):=2*cos(2Pi*j/7) and s(j):=2*sin(2*Pi*j/7). Here A(n;d), n in N, d in C denotes the respective quasi-Fibonacci number - see A121449 and Witula-Slota-Warzynski paper for details (see also A094789, A085810, A077998, A006054, A121442). I note that my and the respective Herbert Kociemba's formulas are "compatible". - Roman Witula, Aug 09 2012
a(n) = A005021(n)-3*A005021(n-1)+2*A005021(n-2). - R. J. Mathar, Feb 27 2019

A215694 a(n) = 5a(n-1) - 6a(n-2) + a(n-3) with a(0)=1, a(1)=2, a(2)=7.

Original entry on oeis.org

1, 2, 7, 24, 80, 263, 859, 2797, 9094, 29547, 95968, 311652, 1011999, 3286051, 10669913, 34645258, 112492863, 365262680, 1186001480, 3850924183, 12503874715, 40599829957, 131826825678, 428039023363, 1389833992704, 4512762649020, 14652848312239, 47577499659779, 154483171074481, 501603705725970, 1628697001842743

Offset: 0

Roman Witula, Aug 21 2012

The Berndt-type sequence number 9 for the argument 2Pi/7 defined by the first trigonometric relation from section "Formula". For more connections with another sequences of trigonometric nature see comments to A215512 (a(n) is equal to the sequence b(n) in these comments) and Witula-Slota's reference (Section 3). We note that a(n)=A109682(n) for n=1,2,3,4. Moreover the following summation formula hold true: sum{k=3,..,n} a(k) = 5*a(n-1) - a(n-2) - 9, for every n=3,4,... - see comments to A215512.

The inverse binomial transform is 1,1, 4, 8, 19, 42, 95,... essentially a shifted, unsigned variant of A215112. - R. J. Mathar, Aug 22 2012

			We have 10*a(3) = 3*a(4), a(0)+a(1)+3*a(2) = a(3), a(0)+a(2)+3*a(3) = a(4), a(1)+3*a(2)+3*a(4) = a(5), and a(6) = 3*a(5)+3*a(4)-a(1).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6
Index entries for linear recurrences with constant coefficients, signature (5,-6,1).

Cf. A215512, A215695.

I:=[1,2,7]; [n le 3 select I[n] else 5*Self(n-1) - 6*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 25 2018

Mathematica
```
LinearRecurrence[{5,-6,1}, {1,2,7}, 50]
```

Vec((1-3*x+3*x^2)/(1-5*x+6*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012

sqrt(7)*a(n) = s(4)*c(1)^(2*n) + s(1)*c(2)^(2*n) + s(2)*c(4)^(2*n), where c(j):=2*cos(2*Pi*j/7) and s(j):=2*sin(2*Pi*j/7).
G.f.: (1-3*x+3*x^2)/(1-5*x+6*x^2-x^3).
a(n) = A005021(n)-3*A005021(n-1)+3*A005021(n-2). - R. J. Mathar, Aug 22 2012

A215695 a(n) = 5a(n-1) - 6a(n-2) + a(n-3) with a(0)=1, a(1)=0, a(2)=-2.

Original entry on oeis.org

1, 0, -2, -9, -33, -113, -376, -1235, -4032, -13126, -42673, -138641, -450293, -1462292, -4748343, -15418256, -50063514, -162556377, -527819057, -1713820537, -5564744720, -18068619435, -58668449392, -190495275070, -618534298433, -2008368291137, -6521130940157, -21173979252396, -68751478912175, -223234649986656, -724838355712626

Offset: 0

Roman Witula, Aug 21 2012

The Berndt-type sequence number 10 for the argument 2Pi/7 defined by the first trigonometric relation from section "Formula". For additional informations and particularly connections with another sequences of trigonometric nature - see comments to A215512 (a(n) is equal to the sequence c(n) in these comments) and Witula-Slota's reference (Section 3).

The following summation formula hold true (see comments to A215512): Sum{k=3,..,n} a(k) = 5*a(n-1) - a(n-2) + 1, n=3,4,...

			We have a(8)=3*a(7)+3*a(5)-6*a(2) and a(9)=3*a(8)+3*a(6)-3*a(4)-a(1).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6
Index entries for linear recurrences with constant coefficients, signature (5,-6,1).

Cf. A215512 (the inverse binomial transform, up to signs), A215694.

I:=[1,0,-2]; [n le 3 select I[n] else 5*Self(n-1) - 6*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 25 2018

LinearRecurrence[{5,-6,1}, {1,0,-2}, 50]

x='x+O('x^30); Vec((1-5*x+4*x^2)/(1-5*x+6*x^2-x^3)) \\ G. C. Greubel, Apr 25 2018

sqrt(7)*a(n) = s(1)*c(1)^(2*n) + s(2)*c(2)^(2*n) + s(4)*c(4)^(2*n), where c(j):=2*cos(2*Pi*j/7) and s(j):=2*sin(2*Pi*j/7).
G.f.: (1-5*x+4*x^2)/(1-5*x+6*x^2-x^3).
a(n) = A005021(n) - 5*A005021(n-1) + 4*A005021(n-2). - R. J. Mathar, Aug 22 2012

A122588 Expansion of x/(1 - 9x + 28x^2 - 35x^3 + 15x^4 - x^5).

Original entry on oeis.org

1, 9, 53, 260, 1156, 4845, 19551, 76912, 297275, 1134705, 4292145, 16128061, 60304951, 224660626, 834641671, 3094322026, 11453607152, 42344301686, 156404021389, 577291806894, 2129654436910, 7853149169635, 28949515515376, 106692395098433, 393137817645838

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 19 2006

Essentially the same as A005025. - R. J. Mathar, Aug 02 2008

Colin Barker, Table of n, a(n) for n = 1..1000
MathPuzzle, Chebyshev Polynomials.
Index entries for linear recurrences with constant coefficients, signature (9,-28,35,-15,1).

Cf. A005021, A005025, A094256, A122589.

I:=[1,9,53,260,1156]; [n le 5 select I[n] else 9*Self(n-1) -28*Self(n-2) +35*Self(n-3) -15*Self(n-4) +Self(n-5): n in [1..30]]; // G. C. Greubel, Nov 29 2021

m = 10; p[x_]:= ExpandAll[x^m*ChebyshevU[m, 1/x]]; Table[SeriesCoefficient[ Series[2^(n+m-1)*x/p[x], {x,0,30}], n], {n,1,30,2}]

def A122588_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x/(1-9*x+28*x^2-35*x^3+15*x^4-x^5) ).list()
a=A122588_list(31); a[1:] # G. C. Greubel, Nov 29 2021

G.f.: x/(1 - 9*x + 28*x^2 - 35*x^3 + 15*x^4 - x^5).

Generating function corrected by R. J. Mathar, Jul 09 2009

New name (using g.f.) and editing by Joerg Arndt, Feb 12 2015

A188843 T(n,k) is the number of n X k binary arrays without the pattern 0 1 diagonally or vertically.

Original entry on oeis.org

2, 4, 3, 8, 8, 4, 16, 21, 13, 5, 32, 55, 40, 19, 6, 64, 144, 121, 66, 26, 7, 128, 377, 364, 221, 100, 34, 8, 256, 987, 1093, 728, 364, 143, 43, 9, 512, 2584, 3280, 2380, 1288, 560, 196, 53, 10, 1024, 6765, 9841, 7753, 4488, 2108, 820, 260, 64, 11, 2048, 17711, 29524

Offset: 1

R. H. Hardin, Apr 12 2011

Table starts
4   8   16   32    64    128    256     512     1024     2048      4096
8  21   55  144   377    987   2584    6765    17711    46368    121393
13  40  121  364  1093   3280   9841   29524    88573   265720    797161
19  66  221  728  2380   7753  25213   81927   266110   864201   2806272
26 100  364 1288  4488  15504  53296  182688   625184  2137408   7303360
34 143  560 2108  7752  28101 100947  360526  1282735  4552624  16131656
43 196  820 3264 12597  47652 177859  657800  2417416  8844448  32256553
53 260 1156 4845 19551  76912 297275 1134705  4292145 16128061  60304951
64 336 1581 6954 29260 119416 476905 1874730  7283640 28048800 107286661
76 425 2109 9709 42504 179630 740025 2991495 11920740 46981740 183579396

			Some solutions for 5 X 3:
  0 0 1    1 1 0    1 1 1    0 1 0    1 1 0    1 1 0    1 1 1
  0 0 0    1 0 0    1 1 0    0 0 0    1 1 0    1 1 0    1 1 1
  0 0 0    0 0 0    1 1 0    0 0 0    1 0 0    1 1 0    0 1 1
  0 0 0    0 0 0    1 1 0    0 0 0    1 0 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

R. H. Hardin, Table of n, a(n) for n = 1..1741

Diagonal is A143388.
Column 2 is A034856(n+1).
Column 3 is A137742(n+1).
Row 2 is A001906(n+1).
Row 3 is A003462(n+1).
Row 4 is A005021.
Row 5 is A005022.
Row 6 is A005023.
Row 7 is A005024.
Row 8 is A005025.

Row recurrence
Empirical: T(n,k) = Sum_{i=1..floor((n+2)/2)} binomial(n+2-i,i)*T(n,k-i)*(-1)^(i-1).
E.g.,
empirical: T(1,k) = 2*T(1,k-1),
empirical: T(2,k) = 3*T(2,k-1) -    T(2,k-2),
empirical: T(3,k) = 4*T(3,k-1) -  3*T(3,k-2),
empirical: T(4,k) = 5*T(4,k-1) -  6*T(4,k-2) +    T(4,k-3),
empirical: T(5,k) = 6*T(5,k-1) - 10*T(5,k-2) +  4*T(5,k-3),
empirical: T(6,k) = 7*T(6,k-1) - 15*T(6,k-2) + 10*T(6,k-3) -    T(6,k-4),
empirical: T(7,k) = 8*T(7,k-1) - 21*T(7,k-2) + 20*T(7,k-3) -  5*T(7,k-4),
empirical: T(8,k) = 9*T(8,k-1) - 28*T(8,k-2) + 35*T(8,k-3) - 15*T(8,k-4) + T(8,k-5).
Columns are polynomials for n > k-3.
Empirical: T(n,1) = n + 1.
Empirical: T(n,2) = (1/2)*n^2 + (5/2)*n + 1.
Empirical: T(n,3) = (1/6)*n^3 + 2*n^2 + (35/6)*n.
Empirical: T(n,4) = (1/24)*n^4 + (11/12)*n^3 + (155/24)*n^2 + (163/12)*n - 6 for n > 1.
Empirical: T(n,5) = (1/120)*n^5 + (7/24)*n^4 + (89/24)*n^3 + (473/24)*n^2 + (1877/60)*n - 33 for n > 2.
Empirical: T(n,6) = (1/720)*n^6 + (17/240)*n^5 + (203/144)*n^4 + (647/48)*n^3 + (2659/45)*n^2 + (1379/20)*n - 143 for n > 3.
Empirical: T(n,7) = (1/5040)*n^7 + (1/72)*n^6 + (143/360)*n^5 + (53/9)*n^4 + (33667/720)*n^3 + (12679/72)*n^2 + (9439/70)*n - 572 for n > 4.
Empirical: T(n,8) = (1/40320)*n^8 + (23/10080)*n^7 + (17/192)*n^6 + (269/144)*n^5 + (43949/1920)*n^4 + (228401/1440)*n^3 + (1054411/2016)*n^2 + (9941/56)*n - 2210 for n > 5.

A215404 a(n) = 4a(n-1) - 3a(n-2) - a(n-3), with a(0)=0, a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 4, 13, 39, 113, 322, 910, 2561, 7192, 20175, 56563, 158535, 444276, 1244936, 3488381, 9774440, 27387681, 76739023, 215018609, 602469686, 1688083894, 4729907909, 13252910268, 37133833451, 104046695091, 291532369743, 816855560248, 2288778436672, 6413014696201

Offset: 0

Roman Witula, Aug 09 2012

We have a(n)=C(n;-1), A121449(n)=A(n;-1), A085810(n+1)=-B(n+1;-1), where A(n;d), B(n;d), and C(n;d), n in N, d in C, are so-called quasi-Fibonacci numbers defined and discussed in the comments to A121449 and in Witula-Slota-Warzynski's paper. It follows from formulas (3.47-49) in this paper that the value of A(n;1/3), B(n;1/3) and C(n;1/3) could be obtained from special convolution type identities for sequences a(n), A121449, and A085810.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Ilya Amburg, Krishna Dasaratha, Laure Flapan, Thomas Garrity, Chansoo Lee, Cornelia Mihaila, Nicholas Neumann-Chun, Sarah Peluse, and Matthew Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239 [math.CO], 2015.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-1).

Cf. A077998, A006054, A121449, A085810, A052975, A094789, A005021, A121442.

I:=[0,0,1]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-Self(n-3): n in [1..35]]; // Vincenzo Librandi, Sep 18 2015

LinearRecurrence[{4,-3,-1}, {0,0,1}, 50]
CoefficientList[Series[x^2/(1 - 4 x + 3 x^2 + x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 18 2015 *)

Vec(x^2/(1-4*x+3*x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012

G.f.: x^2/(1-4*x+3*x^2+x^3).

a(n) = (1/7)*((c(2)-c(4))*(1-c(1))^n + (c(4)-c(1))*(1-c(2))^n + (c(1)-c(2))*(1-c(4))^n), where c(j):=2*cos(2*Pi*j/7) - this formula is the Binet formula for a(n) (see the Binet formula (3.17) for the respective quasi-Fibonacci number C(n;d) for value d=-1 in the Witula-Slota-Warzynski paper).

A005023 Number of walks of length 2n+7 in the path graph P_8 from one end to the other.

Original entry on oeis.org

7, 34, 143, 560, 2108, 7752, 28101, 100947, 360526, 1282735, 4552624, 16131656, 57099056, 201962057, 714012495, 2523515514, 8916942687, 31504028992, 111295205284, 393151913464, 1388758662221, 4905479957435, 17327203698086, 61202661233823, 216176614077600

Offset: 1

N. J. A. Sloane

W. Feller, An Introduction to Probability Theory and its Applications, 3rd ed, Wiley, New York, 1968, p. 96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. J. Everett and P. R. Stein, The combinatorics of random walk with absorbing barriers, Discrete Math. 17 (1977), no. 1, 27-45.
C. J. Everett and P. R. Stein, The combinatorics of random walk with absorbing barriers, Discrete Math. 17 (1977), no. 1, 27-45. [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
Index entries for linear recurrences with constant coefficients, signature (7,-15,10,-1).

Cf. A094829 (first differences), A094256 (essentially the same).

I:=[7, 34, 143, 560]; [n le 4 select I[n] else 7*Self(n-1)-15*Self(n-2)+10*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 08 2013

a:=k->sum(binomial(7+2*k,9*j+k-2),j=ceil((2-k)/9)..floor((9+k)/9))-sum(binomial(7+2*k,9*j+k-1),j=ceil((1-k)/9)..floor((8+k)/9)): seq(a(k),k=1..28);
A005023:=-(-7+15*z-10*z**2+z**3)/(z-1)/(z**3-9*z**2+6*z-1); # Conjectured by Simon Plouffe in his 1992 dissertation.

CoefficientList[Series[(-z^3 + 10 z^2 - 15 z + 7)/(z^4 - 10 z^3 + 15 z^2 - 7 z + 1), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 27 2011 *)
LinearRecurrence[{7,-15,10,-1},{7,34,143,560},40] (* Harvey P. Dale, May 26 2013 *)
CoefficientList[Series[(1 / x) (1 / (1 - 7 x + 15 x^2 - 10 x^3 + x^4) - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 08 2013 *)

G.f.: 1/(1-7x+15x^2-10x^3+x^4) - 1. a(n)=7a(n-1)-15a(n-2)+10a(n-3)-a(n-4). - Emeric Deutsch, Apr 02 2004

a(k) = sum(binomial(7+2k, 9j+k-2)-binomial(7+2k, 9j+k-1), j=-infinity..infinity) (a finite sum).

Better definition from Emeric Deutsch, Apr 02 2004

cp's OEIS Frontend

A080937 Number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2*n steps with all values <= 5.

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A094789 Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 7 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 1, s(2n+1) = 4.

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A094790 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 7 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2*n, s(0) = 1, s(2n) = 3.

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A052975 Expansion of (1-2*x)*(1-x)/(1-5*x+6*x^2-x^3).

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A215694 a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3) with a(0)=1, a(1)=2, a(2)=7.

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A215695 a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3) with a(0)=1, a(1)=0, a(2)=-2.

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A052975 Expansion of (1-2x)(1-x)/(1-5x+6x^2-x^3).

A215694 a(n) = 5a(n-1) - 6a(n-2) + a(n-3) with a(0)=1, a(1)=2, a(2)=7.

A215695 a(n) = 5a(n-1) - 6a(n-2) + a(n-3) with a(0)=1, a(1)=0, a(2)=-2.

A122588 Expansion of x/(1 - 9x + 28x^2 - 35x^3 + 15x^4 - x^5).

A215404 a(n) = 4a(n-1) - 3a(n-2) - a(n-3), with a(0)=0, a(1)=0 and a(2)=1.