A077998 -id:A077998 - cp's OEIS frontend

A121449 Expansion of (1 - 3x + 2x^2)/(1 - 4x + 3x^2 + x^3).

1, 1, 3, 8, 22, 61, 170, 475, 1329, 3721, 10422, 29196, 81797, 229178, 642125, 1799169, 5041123, 14124860, 39576902, 110891905, 310712054, 870595599, 2439354329, 6834918465, 19151015274, 53659951372, 150351841201, 421276495414, 1180390506681, 3307380699281

Offset: 0

Philippe Deléham, Sep 06 2006

From Roman Witula, Aug 07 2012: (Start)
In the cited Witula-Slota-Warzynski paper three so-called quasi-Fibonacci numbers A(n;d), B(n;d) and C(n;d), where n = 0,1,...,d \in C are discussed. These numbers are created by each of the following relations:
(1+d*c(j))^n = A(n;d) + B(n;d)*c(j) + C(n;d)*c(2*j), for every j=1,2,4, where c(j):=2*cos(2*Pi*j/7).
In fact all these "numbers" are integer polynomials of the argument d.
In the sequel for d=-1 we obtain A(n;-1)=a(n), B(n+1;-1)=-A085810(n).
Moreover, we have A(n;1)=A077998(n), B(n;1)=A006054(n+1), C(n;1)=A006054(n), and A(n;2)=A121442(n).
We note that the elements of the sequences A(n;-1), B(n;-1), and C(n;-1) satisfy the following system of recurrence equations:
A(0;-1)=1, B(0;-1)=C(0;-1)=0,
A(n+1;-1)=A(n;-1)-2*B(n;-1)+C(n;-1),
B(n+1;-1)=-A(n;-1)+B(n;-1), C(n+1;-1)=-B(n;-1)+2*C(n;-1).
It is proved that binomial transforms of the sequences: A(n;1), B(n;1), and C(n;1) are equal to the following sequences:
A(n;1)*(A(n;-1)-C(n;-1))-B(n;1)*(B(n;-1)+C(n;-1))+C(n;1)*B(n;-1), -A(n;1)*C(n;-1)+B(n;1)*(A(n;-1)-C(n;-1))-C(n;1)*(B(n;-1)-C(n;-1)), and
A(n;1)*(B(n;-1)-C(n;-1))-B(n;1)*B(n;-1)+C(n;1)*(A(n;-1)-B(n;-1)+C(n;-1)), respectively, whereas we have
A(n;-1) = Sum_{k=0..n} binomial(n,k)*(A(k;1)*A(n-k;1)-A(k;1)*B(n-k;1)-B(k;1)*C(n-k;1)-A(n-k;1)*C(k;1)+2*B(n-k;1)*C(k;1)-C(k;1)*C(n-k;1)),
B(n;-1) = Sum_{k=0..n} binomial(n,k)*(-A(k;1)*B(n-k;1)+A(k;1)*C(n-k;1)+B(k;1)*B(n-k;1)-A(n-k;1)*C(k;1)+B(n-k;1)*C(k;1)-C(k;1)*C(n-k;1)), and
C(n;-1) = Sum_{k=0..n} binomial(n,k)*(-A(k;1)*B(n-k;1)+A(n-k;1)*B(k;1)+B(k;1)*B(n-k;1)-B(k;1)*C(n-k;1)-A(n-k;1)*C(k;1)) (see identities (3.50-52) and (3.61-63) in the Witula-Slota-Warzynski paper).
(End)

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.
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. A085810, A077998, A006054.

I:=[1,1,3]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Sep 18 2015

CoefficientList[Series[(1 - 3*x + 2*x^2)/(1-4*x + 3*x^2 + x^3), {x, 0, 200}], x] (* Stefan Steinerberger, Sep 11 2006 *)
LinearRecurrence[{4,-3,-1},{1,1,3},50] (* Roman Witula, Aug 07 2012 *)

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

a(0)=a(1)=1, a(2)=3, a(n+1) = 4*a(n) - 3*a(n-1) - a(n-2) for n>=2.

7*a(n) = (2-c(4))*(1-c(1))^n + (2-c(1))*(1-c(2))^n + (2-c(2))*(1-c(4))^n = (s(2))^2*(1-c(1))^n + (s(4))^2*(1-c(2))^n + (s(1))^2*(1-c(4))^n, where c(j):=2*cos(2*Pi*j/7) and s(j):=2*sin(2*Pi*j/7) -- it is the special case, for d=-1, of the Binet's formula for the respective quasi-Fibonacci number A(n;d) discussed in the Witula-Slota-Warzynski paper. - Roman Witula, Aug 07 2012

More terms from Stefan Steinerberger, Sep 11 2006

a(27)-a(29) from Vincenzo Librandi, Sep 18 2015

A187070 Let i be in {1,2,3}, let r >= 0 be an integer and n=2r+i-1. Then a(n)=a(2r+i-1) gives the quantity of H_(7,3,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt((2*cos(Pi/7))^2-1).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 5, 6, 11, 14, 25, 31, 56, 70, 126, 157, 283, 353, 636, 793, 1429, 1782, 3211, 4004, 7215, 8997, 16212, 20216, 36428, 45425, 81853, 102069, 183922, 229347, 413269, 515338, 928607, 1157954, 2086561, 2601899

Offset: 0

L. Edson Jeffery, Mar 05 2011

Theory. (Start)
1. Definitions. Let T_(7,j,0) denote the rhombus with sides of unit length (=1), interior angles given by the pair (j*Pi/7,(7-j)*Pi/7) and Area(T_(7,j,0))=sin(j*Pi/7), j in {1,2,3}. Associated with T_(7,j,0) are its angle coefficients (j, 7-j) in which one coefficient is even while the other is odd. A half-tile is created by cutting T_(7,j,0) along a line extending between its two corners with even angle coefficient; let H_(7,j,0) denote this half-tile. Similarly, a T_(7,j,r) tile is a linearly scaled version of T_(7,j,0) with sides of length x^r and Area(T_(7,j,r))=x^(2*r)*sin(j*Pi/7), r>=0 an integer, where x is the positive, constant square root x=sqrt[(2*cos(j*Pi/7))^2 - 1]; likewise let H_(7,j,r) denote the corresponding half-tile. Often H_(7,i,r) (i in {1,2,3}) can be subdivided into an integral number of each equivalence class H_(7,j,0). But regardless of whether or not H_(7,j,r) subdivides, in theory such a proposed subdivision for each j can be represented by the matrix M=(m_(i,j)), i,j=1,2,3, in which the entry m_(i,j) gives the quantity of H_(7,j,0) tiles that should be present in a subdivided H_(7,i,r) tile. The number x^(2*r) (the square of the scaling factor) is an eigenvalue of M=(U_2)^r, where
  U_2= (0 0 1)
       (0 1 1)
       (1 1 1).
2. The sequence. Let r>=0, and let C_r be the r-th "block" defined by C_r={a(2*r),a(2*r+1),a(2*r+2)}. Note that C_r-2*C_(r-1)-C_(r-2)+C_(r-3)={0,0,0}. Let n=2*r+i-1. Then a(n)=a(2*r+i-1)=m_(i,3), where M=(m_(i,j))=(U_2)^r was defined above. Hence the block C_r corresponds component-wise to the third column of M, and a(n)=m_(i,3) gives the quantity of H_(7,3,0) tiles that should appear in a subdivided H_(7,i,r) tile. (End)
Combining blocks A_r, B_r and C_r, from A187068, A187069 and this sequence, respectively, as matrix columns [A_r,B_r,C_r] generates the matrix (U_2)^r, and a negative index (-1)*r yields the corresponding inverse [A_(-r),B_(-r),C_(-r)]=(U_2)^(-r) of (U_2)^r. Therefore, the three sequences need not be causal.
Since a(2*r+2)=a(2*(r+1)) for all r, this sequence arises by concatenation of third-column entries m_(1,3) and m_(2,3) from successive matrices M=(U_2)^r.
This sequence is a trivial extension of A038196.

			Suppose r=3. Then
C_r = C_3 = {a(2*r),a(2*r+1),a(2*r+2)} = {a(6),a(7),a(8)} = {3,5,6},
corresponding to the entries in the third column of
M = (U_2)^3 = (1 2 3)
              (2 4 5)
              (3 5 6).
Choose i=2 and set n=2*r+i-1. Then a(n) = a(2*r+i-1) = a(6+2-1) = a(7) = 5, which equals the entry in row 2 and column 3 of M. Hence a subdivided H_(7,2,3) tile should contain a(7) = m_(2,3) = 5 H_(7,3,0) tiles.

G. C. Greubel, Table of n, a(n) for n = 0..1000
L. Edson Jeffery, Unit-primitive matrices
Roman Witula, D. Slota and A. 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 (0,2,0,1,0,-1).

Cf. A038196, A187068, A187069.

a[0] = a[1] = 0; a[2] = a[3] = a[4] = 1; a[?Negative] = 0; a[n] := a[n] = 2*a[n-2] + a[n-4] - a[n-6]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jan 02 2013 *)

x='x+O('x^50); concat([0,0], Vec(x^2*(1+x-x^2)/(1-2*x^2-x^4+x^6))) \\ G. C. Greubel, Jul 05 2017

Recurrence: a(n) = 2*a(n-2) + a(n-4) - a(n-6).
G.f.: x^2*(1+x-x^2)/(1-2*x^2-x^4+x^6).
a(2*n)=A106803(n); a(2*n+1)=A006054(n+1); a(2*n+2)=A077998(n).
Closed-form: a(n) = (1/14)*[[X_1+Y_1*(-1)^(n-1)]*[(w_2)^2-(w_3)^2]*(w_1)^(n-1)+[X_2+Y_2*(-1)^(n-1)]*[(w_3)^2-(w_1)^2]*(w_2)^(n-1)+[X_3+Y_3*(-1)^(n-1)]*[(w_1)^2-(w_2)^2]*(w_3)^(n-1)], where w_k = sqrt[(2cos(k*Pi/7))^2-1], X_k = (w_k)^3+(w_k)^2-w_k and Y_k = -(w_k)^3+(w_k)^2+w_k, k=1,2,3.

A231187 Decimal expansion of the length ratio (largest diagonal)/side in the regular 7-gon (or heptagon).

Original entry on oeis.org

2, 2, 4, 6, 9, 7, 9, 6, 0, 3, 7, 1, 7, 4, 6, 7, 0, 6, 1, 0, 5, 0, 0, 0, 9, 7, 6, 8, 0, 0, 8, 4, 7, 9, 6, 2, 1, 2, 6, 4, 5, 4, 9, 4, 6, 1, 7, 9, 2, 8, 0, 4, 2, 1, 0, 7, 3, 1, 0, 9, 8, 8, 7, 8, 1, 9, 3, 7, 0, 7, 3, 0, 4, 9, 1, 2, 9, 7, 4, 5, 6, 9, 1, 5, 1, 8, 8, 5, 0, 1, 4, 6, 5, 3, 1, 7, 0, 7, 4, 3, 3, 3, 4, 1

Offset: 1

Wolfdieter Lang, Nov 21 2013

The length ratio (largest diagonal)/side in the regular 7-gon (heptagon) is sigma(7) = S(2, rho(7)) = -1 + rho(7)^2, with rho(7) = 2*cos(Pi/7), which is approx. 1.8019377358 (see A160389 for its decimal expansion, and A049310 for the Chebyshev S-polynomials). sigma(7), approx. 2.2469796, is also the reciprocal of one of the solutions of the minimal polynomial C(7, x) = x^3 - x^2 - 2*x + 1 of rho(7) (see A187360), namely 1/(2*cos(3*Pi/7)).
sigma(7) is the limit of a(n+1)/a(n) for n->infinity for the sequences A006054 and A077998 which can be considered as analogs of the Fibonacci sequence in the pentagon. Thus sigma(7) plays in the heptagon the role of the golden section in the pentagon.
See the Steinbach link.

			2.24697960371746706105000976800847962126454946179280421073109887819...

Peter Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
I. J. Zucker, G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Cam. Phil. Soc. 131 (2001) 309 eq. (8.10).
Index entries for algebraic numbers, degree 3.

Cf. A160389, A006054, A077998, A255240, A255241, A255249, A116425.

First[RealDigits[N[Csc[Pi/14]/2,104]]] (* Stefano Spezia, Jun 26 2022 *)

 5/cos(3*Pi/7) \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(x^3-2*x^2-x+1)[3] \\ Charles R Greathouse IV, Feb 04 2025

sigma(7) = -1 + (2*cos(Pi/7))^2 = 1/(2*cos(3*Pi/7)).
Equals A116425 -1.
From Geoffrey Caveney, Apr 23 2014: (Start)
sigma(7) = exp(asinh(cos(Pi/7))).
cos(Pi/7) + sqrt(1+cos(Pi/7)^2). (End)
From Peter Bala, Oct 12 2021: (Start)
Minimal polynomial x^3 - 2*x^2 - x + 1.
Equals 2*(cos(3*Pi/7) - cos(6*Pi/7)). The other zeros of the minimal polynomial are 2*(cos(Pi/7) - cos(2*Pi/7)) = A255240 and 2*(cos(5*Pi/7) - cos(10*Pi/7)) = 1 - A160389.
The quadratic mapping z -> z^2 - 2*z cyclically permutes the zeros of the minimal polynomial. The inverse cyclic permutation is given by the mapping z -> 2 + z - z^2.
Equals Product_{n >= 0} (7*n+3)*(7*n+4)/((7*n+1)*(7*n+6)) = 1 + Product_{n >= 0} (7*n+3)*(7*n+4)/((7*n+2)*(7*n+5)) = 1 + A255249 = 1/A255241. (End)
Equals 1/(2*sin(Pi/14)) = 1 + 2*sin(3*Pi/14). - Gary W. Adamson, Jun 25 2022
Equals (2*cos(Pi/7)) * (2*cos(2*Pi/7)) = (i^(2/7) + i^(-2/7)) * (i^(4/7) + i^(-4/7)) = 1 + i^(4/7) + i^(-4/7). - Gary W. Adamson, Jul 16 2022
Equals 2F1(1/7,2/7;1/2;1) [Zucker] - R. J. Mathar, Jun 24 2024

A096976 Number of walks of length n on P_3 plus a loop at the end.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 5, 5, 14, 19, 42, 66, 131, 221, 417, 728, 1341, 2380, 4334, 7753, 14041, 25213, 45542, 81927, 147798, 266110, 479779, 864201, 1557649, 2806272, 5057369, 9112264, 16420730, 29587889, 53317085, 96072133, 173118414, 311945595, 562110290

Offset: 0

Paul Barry, Jul 16 2004

Counts closed walks of length n at the start of P_3 to which a loop has been added at the other extremity. a(n+1) counts walks between the first node and the last. Let A be the adjacency matrix of the graph P_3 with a loop added at the end. A is a 'reverse Jordan matrix' [0,0,1;0,1,1;1,1,0]. a(n) is obtained by taking the (1,1) element of A^n.
Sequence is also related to matrices associated with rhombus substitution tilings showing 7-fold rotational symmetry. Let A_{7,1} be the 3 X 3 unit-primitive matrix (see [Jeffery]) A_{7,1}=[0,1,0; 1,0,1; 0,1,1]; then a(n)=[A_{7,1}^n](1,1). - _L. Edson Jeffery, Jan 05 2012
a(n+2) is the (1,1) element of the n-th power of each of the two 3 X 3 matrices: [0,1,1; 1,0,0; 1,0,1], [0,1,1; 1,1,0; 1,0,0]. - Christopher Hunt Gribble, Apr 03 2014

			G.f. = 1 + x^2 + 2*x^4 + x^5 + 5*x^6 + 5*x^7 + 14*x^8 + 19*x^9 + ... - _Michael Somos_, Dec 12 2023

Michael De Vlieger, Table of n, a(n) for n = 0..3914
Robin Chapman and Nicholas C. Singer, Eigenvalues of a bidiagonal matrix, Amer. Math. Monthly, 111 (2004) 441.
Tomislav Došlić, Mate Puljiz, Stjepan Šebek, and Josip Žubrinić, On a variant of Flory model, arXiv:2210.12411 [math.CO], 2022.
L. E. Jeffery, Unit-primitive matrices
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Index entries for linear recurrences with constant coefficients, signature (1,2,-1).

Cf. A006053, A052547, A077998.

LinearRecurrence[{1, 2, -1}, {1, 0, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)
a[ n_] := {1, 0, 0} . MatrixPower[{{1, 2, -1}, {1, 0, 0}, {0, 1, 0}}, n] . {1, 1, 3}; (* Michael Somos, Dec 12 2023 *)

{a(n) = [1, 0, 0] * [1, 2, -1; 1, 0, 0; 0, 1, 0]^n * [1, 1, 3]~}; /* Michael Somos, Dec 12 2023 */

G.f. : (1-x-x^2)/(1-x-2x^2+x^3); a(n)=a(n-1)+2a(n-2)-a(n-3).
a(n) = 5a(n-2)-6a(n-4)+a(n-6). - Floor van Lamoen, Nov 02 2005
a(n) = A077998(-n) for all n in Z. - Michael Somos, Dec 12 2023

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

A105306 Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n, having the top of the rightmost column at height k.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 8, 12, 9, 4, 1, 16, 28, 25, 14, 5, 1, 32, 64, 66, 44, 20, 6, 1, 64, 144, 168, 129, 70, 27, 7, 1, 128, 320, 416, 360, 225, 104, 35, 8, 1, 256, 704, 1008, 968, 681, 363, 147, 44, 9, 1, 512, 1536, 2400, 2528, 1970, 1182, 553, 200, 54, 10, 1, 1024

Offset: 1

Emeric Deutsch, Apr 25 2005

From Gary W. Adamson, Apr 24 2005: (Start)
Let A be the array
  1, 0, 0, 0, 0, 0, ...
  0, 1, 0, 0, 0, 0, ...
  1, 0, 1, 0, 0, 0, ...
  0, 2, 0, 1, 0, 0, ...
  1, 0, 3, 0, 1, 0, ...
  0, 3, 0, 4, 0, 1, ...
  ...
where columns are bin(n,k) with alternating zeros. (Row sums = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...(Fibonacci numbers).) Let P = infinite lower triangular Pascal triangle matrix (A007318). Form P * A: this gives the rows of the present sequence. [Comment corrected by Philippe Deléham, Dec 09 2008] (End)
T(n,k) is the number of nondecreasing Dyck paths of semilength n, having height of rightmost peak equal to k. Example: T(4,1)=4 because we have UDUDUDUD, UDUUDDUD, UUDDUDUD and UUUDDDUD, where U=(1,1) and D=(1,-1). Sum of row n = Fibonacci(2n-1) (A001519). Basically the same as A062110.
T(n,k) is the number of permutations of [n] with length n-k that avoid the patterns 321 and 3412. - Bridget Tenner, Sep 28 2005
T(2*n-1,n)/n = A001003(n-1) (little Schroeder numbers). Proof with Lagrange inversion of inverse of g.f. of A001003.
Row sums = odd-indexed Fibonacci numbers.
Diagonal sums: A077998. - Philippe Deléham, Nov 16 2008
Central coefficients are A176479. Inverse is A125692. - Paul Barry, Apr 18 2010
Riordan matrix ((1-x)/(1-2x),(x-x^2)/(1-2x)). - Emanuele Munarini, Mar 22 2011
T(n,k) is the number of ideals in the fence Z(2n-1) with  n-k elements of rank 0. - Emanuele Munarini, Mar 22 2011
Triangle T(n,k), 1 <= k <= n, read by rows, given by (0,1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 30 2011
T(n,k) is the number of permutations of [n] for which k is equal to both the length and reflection length. - Bridget Tenner, Feb 22 2012

			Triangle begins:
   1;
   1,   1;
   2,   2,   1;
   4,   5,   3,   1;
   8,  12,   9,   4,   1;
  16,  28   25,  14,   5,   1;
  32,  64,  66,  44,  20,   6,   1;
  64, 144, 168, 129,  70,  27,   7,   1;
  ...
From _Paul Barry_, Apr 18 2010: (Start)
Production matrix is
   1,  1;
   1,  1,  1;
   0,  1,  1,  1;
  -1,  0,  1,  1,  1;
   0, -1,  0,  1,  1,  1;
   2,  0, -1,  0,  1,  1,  1;
   0,  2,  0, -1,  0,  1,  1,  1;
  -5,  0,  2,  0, -1,  0,  1,  1,  1;
   0, -5,  0,  2,  0, -1,  0,  1,  1,  1;
  14,  0, -5,  0,  2,  0, -1,  0,  1,  1,  1; (End)

V. E. Hoggatt, Jr. and Marjorie Bicknell, editors: "A Primer for the Fibonacci Numbers", 1970, p. 87.

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened)
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
Jean-Luc Baril, Javier F. González, and José L. Ramírez, Last symbol distribution in pattern avoiding Catalan words, Univ. Bourgogne (France, 2022).
Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega, J. Rafael Sendra, Juana Sendra, and Steven E. Thornton, Bohemian Upper Hessenberg Toeplitz Matrices, arXiv:1809.10664 [cs.SC], 2018.
Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega, J. Rafael Sendra, and Juana Sendra, Upper Hessenberg and Toeplitz Bohemians, arXiv:1907.10677 [cs.SC], 2019.
Emeric Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.
Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, Enumerating symmetric peaks in non-decreasing Dyck paths, Ars Mathematica Contemporanea (2021).
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.6. See pp. 9-10, 19.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
V. V. Kruchinin and D. V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, arXiv preprint arXiv:1206.0877 [math.CO], 2012, and J. Int. Seq. 15 (2012) #12.9.3.
E. Munarini and N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
T. K. Petersen and B. E. Tenner, The depth of a permutation, arXiv:1202.4765 [math.CO], 2012-2014..
M. Pétréolle, Characterization of Cyclically Fully commutative elements in finite and affine Coxeter Groups, arXiv preprint arXiv:1403.1130 [math.GR], 2014.
Bridget Eileen Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.
S.-n. Zheng and S.-l. Yang, On the-Shifted Central Coefficients of Riordan Matrices, Journal of Applied Mathematics, Volume 2014, Article ID 848374, 8 pages.

Cf. A001519. Essentially the same array as A062110.

Row sums = A001519(n-1), n >= 1.

Maple
```
T:=proc(n,k) if k
				
```

t[n_, k_] := 2^(n-2*k-1)*Binomial[n, k]*Hypergeometric2F1[-k-1, -k, -n, -1]; t[n_, n_] = 1; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 28 2014 *)

create_list(sum(binomial(k+1,i)*binomial(n-i,k)*(-1)^i*2^(n-k-i),i,0,n-k),n,0,8,k,0,n); /* Emanuele Munarini, Mar 22 2011 */

T(n,k) = sum(binomial(k+j, k-1)*binomial(n-k-1, j), j=0..n-k-1) (0<=k<=n). (This appears to be incorrect. - Emanuele Munarini, Mar 22 2011)
G.f.: t*z*(1-z)/(1 - 2*z - t*z*(1-z)).
From Emanuele Munarini, Mar 22 2011: (Start)
T(n,k) = Sum_{i=0..n-k} binomial(k+1,i)*binomial(n-i,k)*(-1)^i*2^(n-k-i).
T(n,k) = Sum_{i=0..n-k} binomial(k+1,i)*M(i,n-k-i)*2^(n-k-i), where M(n,k) = n*(n+1)*(n+2)*...*(n+k-1)/k!. (End)
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Oct 30 2013

Entry revised by N. J. A. Sloane, Apr 27 2007

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).

A106803 Expansion of x(1-x)/(1-2x-x^2+x^3).

Original entry on oeis.org

0, 1, 1, 3, 6, 14, 31, 70, 157, 353, 793, 1782, 4004, 8997, 20216, 45425, 102069, 229347, 515338, 1157954, 2601899, 5846414, 13136773, 29518061, 66326481, 149034250, 334876920, 752461609, 1690765888, 3799116465, 8536537209

Offset: 0

Roger L. Bagula, May 17 2005

Essentially a duplicate of A077998: a(n) = A077998(n-1). - Joerg Arndt, Aug 14 2015
a(n) appears in the formula for the nonnegative powers of sigma, the ratio of the smaller diagonal in the heptagon to the side length s=2*sin(Pi/7), when expressed in the basis <1,rho,sigma>, with rho = 2*cos(Pi/7), the ratio of the smaller heptagon diagonal to the side length, as follows. sigma^n = a(n-1)*1 + B(n)*rho + a(n)*sigma, n>=0, with B(n)=A006054(n). Put a(-1):= 1. See the Steinbach reference, and a comment under A052547.
a(n-1) is the top left entry of the n-th power of the 3X3 matrix [0, 1, 0; 1, 1, 1; 0, 1, 1] or of the 3X3 matrix [0, 0, 1; 0, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014

Michael De Vlieger, Table of n, a(n) for n = 0..2845
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), p. 22-31.
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Index entries for linear recurrences with constant coefficients, signature (2,1,-1).

Cf. A006356, A077998, A187068, A187070, A199853.

m = {{0, 0, 1}, {1, 2, 0}, {1, 1, 0}}; v[0] = {0, 1, 1}; v[n_] := m.v[n - 1]; Table[v[n][[1]], {n, 0, 30}] (* Edited and corrected by L. Edson Jeffery, Oct 18 2017 *)
RecurrenceTable[{a[1]== 0, a[2]== 1, a[3]== 1, a[n]== 2*a[n-1]  + a[n-2] - a[n-3]}, a, {n,30}] (* G. C. Greubel, Aug 14 2015 *)

concat(0,Vec((1-x)/(x^3-2*x-x^2+1)+O(x^99))) \\ Charles R Greathouse IV, Sep 25 2012

a(n) = A077998(n-1). - R. J. Mathar, Aug 07 2008
a(n) = A187070(2*n), a(n) = A187068(2*n+2). - L. Edson Jeffery, Mar 10 2011
a(n+1) = - A199853(n+1). - G. C. Greubel, Aug 14 2015
a(n) = 2*a(n-1) + a(n-2) - a(n-3), a(0)=0, a(1)=a(2)=1. - G. C. Greubel, Aug 14 2015
a(n) = A006356(n-2) for n > 1. - Georg Fischer, Oct 21 2018

Edited by N. J. A. Sloane, Aug 08 2008

A188316 Riordan array (1/(1-x^2), x/((1-x)*(1-x^2))).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 3, 2, 1, 1, 3, 6, 3, 1, 0, 6, 10, 10, 4, 1, 1, 6, 20, 22, 15, 5, 1, 0, 10, 30, 49, 40, 21, 6, 1, 1, 10, 50, 91, 100, 65, 28, 7, 1, 0, 15, 70, 168, 216, 181, 98, 36, 8, 1, 1, 15, 105, 280, 444, 441, 301, 140, 45, 9, 1

Offset: 0

Paul Barry, Mar 28 2011

Row sums are A077998.
Diagonal sums are A052547.
Inverse is A188317.

			Triangle begins
1,
0, 1,
1, 1, 1,
0, 3, 2, 1,
1, 3, 6, 3, 1,
0, 6, 10, 10, 4, 1,
1, 6, 20, 22, 15, 5, 1,
0, 10, 30, 49, 40, 21, 6, 1,
1, 10, 50, 91, 100, 65, 28, 7, 1,
0, 15, 70, 168, 216, 181, 98, 36, 8, 1

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-3,k), T(0,0) = T(1,1) = T(2,0) = T(2,1) = T(2,2)=1, T(1,0)=0, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 16 2014

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

A121449 Expansion of (1 - 3*x + 2*x^2)/(1 - 4*x + 3*x^2 + x^3).

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A187070 Let i be in {1,2,3}, let r >= 0 be an integer and n=2*r+i-1. Then a(n)=a(2*r+i-1) gives the quantity of H_(7,3,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt((2*cos(Pi/7))^2-1).

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A231187 Decimal expansion of the length ratio (largest diagonal)/side in the regular 7-gon (or heptagon).

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A096976 Number of walks of length n on P_3 plus a loop at the end.

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

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A105306 Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n, having the top of the rightmost column at height k.

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

A121449 Expansion of (1 - 3x + 2x^2)/(1 - 4x + 3x^2 + x^3).

A187070 Let i be in {1,2,3}, let r >= 0 be an integer and n=2r+i-1. Then a(n)=a(2r+i-1) gives the quantity of H_(7,3,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt((2*cos(Pi/7))^2-1).

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

A106803 Expansion of x(1-x)/(1-2x-x^2+x^3).

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