A052975 -id:A052975 - cp's OEIS frontend

A028495 Expansion of g.f. (1-x^2)/(1-x-2*x^2+x^3).

1, 1, 2, 3, 6, 10, 19, 33, 61, 108, 197, 352, 638, 1145, 2069, 3721, 6714, 12087, 21794, 39254, 70755, 127469, 229725, 413908, 745889, 1343980, 2421850, 4363921, 7863641, 14169633, 25532994, 46008619, 82904974, 149389218, 269190547, 485064009, 874055885

Offset: 0

N. J. A. Sloane

Form the graph with matrix A = [0,1,1; 1,0,0; 1,0,1] (P_3 with a loop at an extremity). Then A028495 counts closed walks of length n at the degree 3 vertex. - Paul Barry, Oct 02 2004
Equals INVERT transform of (1, 1, 0, 1, 0, 1, 0, 1, ...). - Gary W. Adamson, Apr 28 2009
From Johannes W. Meijer, May 29 2010: (Start)
a(n) is the number of ways White can force checkmate in exactly (n+1) moves, n>=0, ignoring the fifty-move and the triple repetition rules, in the following chess position: White Ka1, Ra8, Bc1, Nb8, pawns a6, a7, b2, c6, d2, f6 and h6; Black Kc8, pawns b3, c7, d3, f7 and h7. (After Noam D. Elkies, see link; diagram 5).
Counts all paths of length n, n>=0, starting at the initial node on the path graph P_6, see the second Maple program. (End)
a(n) is the number of length n-1 binary words such that each maximal block of 1's has odd length.  a(4) = 6 because we have: 000, 001, 010, 100, 101, 111. - Geoffrey Critzer, Nov 17 2012
a(n) is the number of compositions of n where increments can only appear at every second position, starting with the second and third part, see example. Also, a(n) is the number of compositions of n where there is no fall between every second pair of parts, starting with the first and second part; see example. - Joerg Arndt, May 21 2013
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 1, 0; 1, 0, 1; 0, 1, 0] or of the 3 X 3 matrix [1, 0, 1; 0, 0, 1; 1, 1, 0]. - R. J. Mathar, Feb 03 2014
Range of row n of the circular Pascal array of order 7. - Shaun V. Ault, Jun 05 2014
a(n) is the number of compositions of n into parts from {1,2,4,6,8,10,...}. Example: a(4)= 6 because we have 4, 22, 211, 121, 112, and 1111. - Emeric Deutsch, Aug 17 2016
In general, a(n,m) = (2^n/(m+1))*Sum_{r=1..m} (1-(-1)^r)*cos(Pi*r/(m+1))^n*(1+cos(Pi*r/(m+1))) gives the number of paths of length n starting at the initial node on the path graph P_m. Here we have m=6. - Herbert Kociemba, Sep 15 2020
a(n-1) is the number of triangular dcc-polyominoes having area n (see Baril et al. at page 11). - Stefano Spezia, Oct 14 2023
a(n) is the number of permutations p of [n] with p(j)Alois P. Heinz, Mar 29 2024

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 19*x^6 + 33*x^7 + 61*x^8 + ...
From _Joerg Arndt_, May 21 2013: (Start)
There are a(6)=19 compositions of 6 where increments can only appear at every second position:
  01:  [ 1 1 1 1 1 1 ]
  02:  [ 1 1 1 1 2 ]
  03:  [ 1 1 2 1 1 ]
  04:  [ 1 1 2 2 ]
  05:  [ 1 1 3 1 ]
  06:  [ 1 1 4 ]
  07:  [ 2 1 1 1 1 ]
  08:  [ 2 1 2 1 ]
  09:  [ 2 1 3 ]
  10:  [ 2 2 1 1 ]
  11:  [ 2 2 2 ]
  12:  [ 3 1 1 1 ]
  13:  [ 3 1 2 ]
  14:  [ 3 2 1 ]
  15:  [ 3 3 ]
  16:  [ 4 1 1 ]
  17:  [ 4 2 ]
  18:  [ 5 1 ]
  19:  [ 6 ]
There are a(6)=19 compositions of 6 where there is no fall between every second pair of parts, starting with the first and second part:
  01:  [ 1 1 1 1 1 1 ]
  02:  [ 1 1 1 1 2 ]
  03:  [ 1 1 1 2 1 ]
  04:  [ 1 1 1 3 ]
  05:  [ 1 1 2 2 ]
  06:  [ 1 1 4 ]
  07:  [ 1 2 1 1 1 ]
  08:  [ 1 2 1 2 ]
  09:  [ 1 2 3 ]
  10:  [ 1 3 1 1 ]
  11:  [ 1 3 2 ]
  12:  [ 1 4 1 ]
  13:  [ 1 5 ]
  14:  [ 2 2 1 1 ]
  15:  [ 2 2 2 ]
  16:  [ 2 3 1 ]
  17:  [ 2 4 ]
  18:  [ 3 3 ]
  19:  [ 6 ]
(End)
19 = (1, 0, 1, 0, 1, 1) dot (1, 1, 2, 3, 6, 10) = (1 + 0 + 2 + 0 + 6 + 10). Cf. comment of Apr 28 2009. - _Gary W. Adamson_, Aug 10 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. V. Ault and C. Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Mathematics (2014).
Jean-Luc Baril, José L. Ramírez, and Fabio A. Velandia, Bijections between Directed-Column Convex Polyominoes and Restricted Compositions, September 29, 2023.
Alexandru Chirvasitu, Tara Hudson, and Aparna Upadhyay, Recursive sequences attached to modular representations of finite groups, arXiv:2105.04732 [math.RT], 2021. See (1) p. 27.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015.
Johann Cigler, Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials, arXiv:2212.02118 [math.NT], 2022.
Nachum Dershowitz, Between Broadway and the Hudson, arXiv:2006.06516 [math.CO], 2020.
Tomislav Došlić, Mate Puljiz, Stjepan Šebek, and Josip Žubrinić, On a variant of Flory model, arXiv:2210.12411 [math.CO], 2022.
Noam D. Elkies, New Directions in Enumerative Chess Problems, The Electronic Journal of Combinatorics, 11 (2), 2004-2005; arXiv:math/0508645 [math.CO], 2005. - _Johannes W. Meijer_, May 29 2010
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 14.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1056
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1057
Mohammed L. Nadji, Moussa Ahmia, Daniel F. Checa, and José L. Ramírez, Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants, J. Int. Seq. (2025) Vol. 28, Issue 3, Article 25.3.6. See pp. 9, 15.
László Németh and Dragan Stevanović, Graph solution of system of recurrence equations, Research Gate, 2023. See Table 2 at p. 6.
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 (1,2,-1).

Cf. A052534, A052547, A096976, A276053, A068911, A078038, A094790, A000045, A038754, A028495, A030436, A061551, A178381.

Cf. A006053, A052975, A060557, A094718.

spec := [S,{S=Sequence(Union(Prod(Sequence(Prod(Z,Z)),Z,Z),Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
with(GraphTheory): P:=6: G:= PathGraph(P): A:=AdjacencyMatrix(G): nmax:=34; for n from 0 to nmax do B(n):=A^n; a(n):=add(B(n)[1,k], k=1..P) od: seq(a(n), n=0..nmax); # Johannes W. Meijer, May 29 2010
a := (-1)^(3/7) - (-1)^(4/7):
b := (-1)^(5/7) - (-1)^(2/7):
c := (-1)^(1/7) - (-1)^(6/7):
f := n -> (a^n * (2 + a) + b^n * (2 + b) + c^n * (2 + c))/7:
seq(simplify(f(n)), n=0..36); # Peter Luschny, Sep 16 2020

LinearRecurrence[{1, 2, -1}, {1, 1, 2}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)
CoefficientList[Series[(1-x^2)/(1-x-2x^2+x^3),{x,0,40}],x] (* Harvey P. Dale, Dec 23 2018 *)
a[n_,m_]:= 2^(n+1)/(m+1) Module[{x=(Pi r)/(m+1)},Sum[Cos[x]^n (1+Cos[x]),{r,1,m,2}]]
Table[a[n,6],{n,0,40}]//Round (* Herbert Kociemba, Sep 15 2020 *) (* Herbert Kociemba, Sep 14 2020 *)

{a(n) = if( n<0, n = -1-n; polcoeff( (1 - x^2) / (1 - 2*x - x^2 + x^3) + x * O(x^n), n), polcoeff( (1 - x^2) / (1 - x - 2*x^2 + x^3) + x * O(x^n), n))} /* Michael Somos, Apr 05 2012 */

a(n)=([0,1,0;0,0,1;-1,2,1]^n*[1;1;2])[1,1] \\ Charles R Greathouse IV, Aug 25 2016

Recurrence: {a(0)=1, a(1)=1, a(2)=2, a(n)-2*a(n+1)-a(n+2)+a(n+3)=0}.
a(n) = Sum_(1/7*(1+2*_alpha)*_alpha^(-1-n), _alpha=RootOf(_Z^3-2*_Z^2-_Z+1)).
a(n) = A094718(6, n). - N. J. A. Sloane, Jun 12 2004
a(n) = a(n-1) + Sum_{k=1..floor(n/2)} a(n-2*k). - Floor van Lamoen, Oct 29 2005
a(n) = 5*a(n-2) - 6*a(n-4) + a(n-6). - Floor van Lamoen, Nov 02 2005
a(n) = A006053(n+2) - A006053(n). - R. J. Mathar, Nov 16 2007
a(2*n) = A052975(n), a(2*n+1) = A060557(n). - Johannes W. Meijer, May 29 2010
G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 + x / (1 - x))))). - Michael Somos, Apr 05 2012
a(-1 - n) = A052534(n). - Michael Somos, Apr 05 2012
a(n) = (2^n/7)*Sum_{r=1..6} (1-(-1)^r)*cos(Pi*r/7)^n*(1+cos(Pi*r/7)). - Herbert Kociemba, Sep 15 2020

More terms from James Sellers, Jun 05 2000

A085810 Number of three-choice paths along a corridor of height 5, starting from the lower side.

Original entry on oeis.org

1, 2, 5, 13, 35, 96, 266, 741, 2070, 5791, 16213, 45409, 127206, 356384, 998509, 2797678, 7838801, 21963661, 61540563, 172432468, 483144522, 1353740121, 3793094450, 10628012915, 29779028189, 83438979561, 233790820762, 655067316176, 1835457822857, 5142838522138, 14409913303805

Offset: 1

Philippe Deléham, Jul 25 2003

From Svjetlan Feretic, Jun 01 2013: (Start)
A three-choice path is a path whose steps lie in the set {(1,1), (1,0), (1,-1)}.
The paths under consideration "live" in a corridor like 0<=y<=5. Thus, the ordinate of a vertex of a path can take six values (0,1,2,3,4,5), but the height of the corridor is five.
a(1)=1 is the number of paths with zero steps, a(2)=2 is the number of paths with one step, a(3)=5 is the number of paths with two steps, ...
Narrower corridors produce A000012, A000079, A000129, A001519, A057960. An infinitely wide corridor would produce A005773.
(End)
Diagonal sums of A114164. - Paul Barry, Nov 15 2005
C(n):= a(n)*(-1)^n appears in the following formula for the nonpositive powers of rho*sigma, where rho:=2*cos(Pi/7) and sigma:=sin(3*Pi/7)/sin(Pi/7) = rho^2-1 are the ratios of the smaller and larger diagonal length to the side length in a regular 7-gon (heptagon). See the Steinbach reference where the basis  <1,rho,sigma> is used in an extension of the rational field. (rho*sigma)^(-n) = C(n) + B(n)*rho + A(n)*sigma,n>=0, with B(n)= A181880(n-2)*(-1)^n, and A(n)= A116423(n+1)*(-1)^(n+1). For the nonnegative powers see A120757(n), |A122600(n-1)| and A181879(n), respectively. See also a comment under A052547.
a(n) is also the number of bi-wall directed polygons with n cells. (The definition of bi-wall directed polygons is given in the article on A122737.)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Svjetlan Feretic, Generating functions for bi-wall directed polygons, in: Proc. of the Seventh Int. Conf. on Lattice Path Combinatorics and Applications (eds. S. Rinaldi and S. G. Mohanty), Siena, 2010, 147-151.
Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), p. 22-31 (formula 5).
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. A000012, A000079, A000129, A001519, A057960, A005773.

I:=[1,2,5]; [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}, {1,2,5}, 50] (* Roman Witula, Aug 09 2012 *)
CoefficientList[Series[(1 - 2 x)/(1 - 4 x + 3 x^2 + x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 18 2015 *)

x='x+O('x^30); Vec((1-2*x)/(1-4*x+3*x^2+x^3)) \\ G. C. Greubel, Apr 19 2018

a(n) = 4*a(n-1) - 3*a(n-2) - a(n-3).
From Paul Barry, Nov 15 2005: (Start)
G.f.: (1-2*x)/(1-4*x+3*x^2+x^3).
a(n) = Sum_{k=0..floor(n/2)} (Sum_{j=0..n-k} C(n-k, j)*C(j+k, 2k));
a(n) = Sum_{k=0..floor(n/2)} (Sum_{j=0..n-k} C(n-k, k+j)*C(k, k-j)*2^(n-2k-j));
a(n) = Sum_{k=0..floor(n/2)} (Sum_{j=0..n-2*k} C(n-j, n-2*k-j)*C(k, j)(-1)^j*2^(n-2*k-j)). (End)
a(n-1) = -B(n;-1) = (1/7)*((c(4)-c(1))*(1-c(1))^n + (c(1)-c(2))*(1-c(2))^n + (c(2)-c(4))*(1-c(4))^n), where a(-1):=0, c(j):=2*cos(2*Pi*j/7). Moreover, B(n;d), n in N, d in C, denotes the respective quasi-Fibonacci number defined in comments to A121449 or in Witula-Slota-Warzynski's paper (see also A077998, A006054, A052975, A094789, A121442). - Roman Witula, Aug 09 2012

Name corrected and clarified, and offset 1 from Svjetlan Feretic, Jun 01 2013

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

A216201 Square array T, read by antidiagonals : T(n,k) = 0 if n-k>=3 or if k-n>=4, T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 0, 0, 4, 6, 3, 0, 0, 4, 10, 9, 0, 0, 0, 0, 14, 19, 9, 0, 0, 0, 0, 14, 33, 28, 0, 0, 0, 0, 0, 0, 47, 61, 28, 0, 0, 0, 0, 0, 0, 47, 108, 89, 0, 0, 0, 0, 0, 0, 0, 0, 155, 197, 89, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 12 2013

			Square array begins:
1, 1, 1,  1,  0,   0,   0,   0,   0,   0, 0, 0, 0, ... row n = 0
1, 2, 3,  4,  4,   0,   0,   0,   0,   0, 0, 0, 0, ... row n = 1
1, 3, 6, 10, 14,  14,   0,   0,   0,   0, 0, 0, 0, ... row n = 2
0, 3, 9, 19, 33,  47,  47,   0,   0,   0, 0, 0, 0, ... row n = 3
0, 0, 9, 28, 61, 108, 155, 155,   0,   0, 0, 0, 0, ... row n = 4
0, 0, 0, 28, 89, 197, 352, 507, 507,   0, 0, 0, 0, ... row n = 5
0, 0, 0,  0, 89, 286, 638,1147,1652,1652, 0, 0, 0, ... row n = 6
...

E. Lucas, Théorie des nombres, Tome 1, Albert Blanchard, Paris, 1958, p.89

E. Lucas, Théorie des nombres, Tome 1, Jacques Gabay, Paris, 1991, p.89

Cf. A052975, A060557, A078038, A095789, A094790

T(n,n) = A052975(n).
T(n,n+1) = A060557(n).
T(n+1,n) = T(n+2,n) = A094790(n+1).
T(n,n+2) = T(n,n+3) = A094789(n+1).
Sum_{k, 0<=k<=n} T(n-k,k) = (-1)^n*A078038(n).

A121442 Expansion of (1-x^2)/(1-x-9*x^2+x^3).

Original entry on oeis.org

1, 1, 9, 17, 97, 241, 1097, 3169, 12801, 40225, 152265, 501489, 1831649, 6192785, 22176137, 76079553, 269472001, 932011841, 3281180297, 11399814865, 39998425697, 139315579185, 487901595593, 1701743382561, 5953542163713, 20781331011169, 72661467102025

Offset: 0

Philippe Deléham, Sep 06 2006

From Roman Witula, Aug 08 2012: (Start)
We have a(n)=A(n;2), where A(n;2), B(n;2) and C(n;2) are the special cases of so-called quasi-Fibonacci numbers A(n;d), B(n;d), and C(n;d) for the value of argument d=2 - for details see Witula's comments to A121449 or the paper of Witula-Slota-Warzynski's. The sequences A(n;2), B(n;2) and C(n;2) are defined by the following system of recurrence equations:
A(0;2)=1, B(0;2)=C(0;2)=0,
A(n+1;2)=A(n;2)+4*B(n;2)-2*C(n;2), B(n+1;2)=2*A(n;2)+B(n;2), and C(n+1;2)=2*B(n;2)-C(n;2).
We note that A(n;1)=A077998(n), B(n;1)=A006054(n+1), and C(n;1)=A006054(n). We know (see formulas (3.61-63) in Witula et al.'s paper) that the sequences: (-2)^(-n)*(A(n;1)*(A(n;2)-C(n;2))-B(n;1)*(B(n;2)+C(n;2))+C(n;1)*B(n;2)), (-2)^(-n)*(-A(n;1)*C(n;2)+B(n;1)*(A(n;2)-C(n;2))-C(n;1)*(B(n;2)-C(n;2))), and (-2)^(-n)*(A(n;1)*(B(n;2)-C(n;2))-B(n;1)*B(n;2)+C(n;1)*(A(n;2)-B(n;2)+C(n;2))) are the binomial transforms of the sequences (-2)^(-n)*A(n;1), (-2)^(-n)*B(n;1), and (-2)^(-n)*C(n;1) respectively. Moreover the elements of the sequences A(n;1/2)=2^(-n)*A052975, B(n;1/2)=2^(-n)*A094789, and C(n;1/2) could be described by certain convolutions type identities for the elements of A(n;2), B(n;2), and C(n;2) (see identities (3.58-60) in Witula et al.'s paper). (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
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 (1, 9, -1).

Cf. A077998, A006054, A121449, A085810.

I:=[1,1,9]; [n le 3 select I[n] else Self(n-1)+9*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Sep 18 2015

LinearRecurrence[{1,9,-1},{1,1,9},50] (* Roman Witula, Aug 08 2012 *)
CoefficientList[Series[(1 - x^2)/(1 - x - 9 x^2 + x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 18 2015 *)

Vec((1-x^2)/(1-x-9*x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

a(0)=a(1)=1, a(2)=9, a(n+1) = a(n)+9*a(n-1)-a(n-2) for n>=2.

7*a(n) = (2-c(4))*(1-2*c(1))^n + (2-c(1))*(1-2*c(2))^n + (2-c(2))*(1-2*c(4))^n = (s(2))^2*(1-2*c(1))^n + (s(4))^2*(1-2*c(2))^n + (s(1))^2*(1-2*c(4))^n, where c(j):=2*Cos(2Pi*j/7) and s(j):=2*Sin(2Pi*j/7) - it is the special case, for d=2, of the Binet's formula for the respective quasi-Fibonacci number A(n;d) discussed in Witula-Slota-Warzynski's paper (see also A121449). - Roman Witula, Aug 08 2012

Corrected by T. D. Noe, Oct 25 2006

More terms from Vincenzo Librandi, Sep 18 2015

A060557 Row sums of triangle A060556.

Original entry on oeis.org

1, 3, 10, 33, 108, 352, 1145, 3721, 12087, 39254, 127469, 413908, 1343980, 4363921, 14169633, 46008619, 149389218, 485064009, 1574993356, 5113971944, 16604963593, 53915979657, 175064088671

Offset: 0

Wolfdieter Lang, Apr 06 2001

Equals the INVERT transform of A045623: (1, 2, 5, 12, 28, ...). - Gary W. Adamson, Oct 26 2010

Harry J. Smith, Table of n, a(n) for n = 0..500
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).

a(n)=A028495(2n+1).
Cf. A053975.
Cf. A052975 (row sums of triangle A060102).
Cf. A045623. - Gary W. Adamson, Oct 26 2010

a[0] = 1; a[1] = 3; a[2] = 10; a[n_] := a[n] = 5*a[n-1] - 6*a[n-2] + a[n-3]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Jul 05 2013, after Floor van Lamoen *)
LinearRecurrence[{5,-6,1},{1,3,10},30] (* Harvey P. Dale, Nov 29 2013 *)

{ f="b060557.txt"; a0=1; a1=3; a2=10; write(f, "0 1"); write(f, "1 3"); write(f, "2 10"); for (n=3, 500, write(f, n, " ", a=5*a2 - 6*a1 + a0); a0=a1; a1=a2; a2=a; ) } \\ Harry J. Smith, Jul 07 2009

a(n) = Sum_{m=0..n} A060556(n, m).
G.f.: (1-x)^2/(1 - 5*x + 6*x^2 - x^3).
a(n) = 5a(n-1) - 6a(n-2) + a(n-3). - Floor van Lamoen, Nov 02 2005

A060102 Bisection of triangle A060098: even-indexed members of column sequences of A060098 (not counting leading zeros).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 8, 1, 1, 16, 30, 13, 1, 1, 25, 80, 71, 19, 1, 1, 36, 175, 259, 140, 26, 1, 1, 49, 336, 742, 660, 246, 34, 1, 1, 64, 588, 1806, 2370, 1442, 399, 43, 1, 1, 81, 960, 3906, 7062, 6292, 2828, 610, 53

Offset: 0

Wolfdieter Lang, Apr 06 2001

Row sums give A052975. Column sequences without leading zeros give for m=0..5: A000012 (powers of 1), A000290 (squares), A002417(n+1), A060103-5.

Companion triangle (odd-indexed members) A060556.

			{1}; {1,1}; {1,4,1}; {1,9,8,1}; ... Pe(3,x) = 1 + 3*x.

a(n, m) = A060098(2*n-m, m).
a(n, m) = Sum_{j=0..floor((m+1)/2)} binomial((n-m)-j+2*m, 2*m)*binomial(m+1, 2*j), n >= m >= 0, otherwise zero.
G.f. for column m: (x^m)*Pe(m+1, x)/(1-x)^(2*m+1), with Pe(n, x) = Sum_{j=0..floor(n/2)} binomial(n, 2*j)*x^j (even members of row n of Pascal triangle A007318).

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).

A215492 a(n) = 21a(n-2) + 7a(n-3), with a(0)=0, a(1)=3, and a(2)=6.

Original entry on oeis.org

0, 3, 6, 63, 147, 1365, 3528, 29694, 83643, 648270, 1964361, 14199171, 45789471, 311933118, 1060973088, 6871121775, 24463966674, 151720368891, 561841152579, 3357375513429, 12860706786396, 74437773850062, 293576471108319, 1653218198356074, 6686170310225133

Offset: 0

Roman Witula, Aug 13 2012

We have a(n)=B(n;3), where B(n;d), n=1,2,..., d \in C, denote one of the quasi-Fibonacci numbers defined in the comments to A121449 and in the Witula-Slota-Warzynski paper. Its conjugate sequences A(n;3) and C(n;3) are discussed in A121458 and A215484 respectively. Similarly as in A121458 we deduce that each of the following elements a(3*n), a(3*n+1), a(3*n+2) is divided by 3*7^n for every n=0,1,... .

G. C. Greubel, Table of n, a(n) for n = 0..1000
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 (0, 21, 7).

Cf. A121458, A215484, A121449, A085810, A215404, A077998, A006054, A033304, A052975, A094789, A005021, A121442, A121458.

I:=[0,3,6]; [n le 3 select I[n] else 21*Self(n-2)+7*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Sep 18 2015

LinearRecurrence[{0,21,7}, {0,3,6}, 50]
CoefficientList[Series[(3 x + 6 x^2)/(1 - 21 x^2 - 7 x^3), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 18 2015 *)

concat(0,Vec((3+6*x)/(1-21*x^2-7*x^3)+O(x^99))) \\ Charles R Greathouse IV, Oct 01 2012

a(n) = (1/7)*((c(1)-c(4))*(1+3*c(1))^n + (c(2)-c(1))*(1+3*c(2))^n + (c(4)-c(2))*(1+3*c(4))^n), where c(j):=2*cos(2*Pi*j/7) (for the proof see Witula-Slota-Warzynski paper).

G.f.: (3*x+6*x^2)/(1-21*x^2-7*x^3).

cp's OEIS Frontend

A028495 Expansion of g.f. (1-x^2)/(1-x-2*x^2+x^3).

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A085810 Number of three-choice paths along a corridor of height 5, starting from the lower side.

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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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A216201 Square array T, read by antidiagonals : T(n,k) = 0 if n-k>=3 or if k-n>=4, T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A121442 Expansion of (1-x^2)/(1-x-9*x^2+x^3).

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A060557 Row sums of triangle A060556.

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

A215492 a(n) = 21a(n-2) + 7a(n-3), with a(0)=0, a(1)=3, and a(2)=6.