A094789 -id:A094789 - cp's OEIS frontend

A006053 a(n) = a(n-1) + 2*a(n-2) - a(n-3), with a(0) = a(1) = 0, a(2) = 1.

0, 0, 1, 1, 3, 4, 9, 14, 28, 47, 89, 155, 286, 507, 924, 1652, 2993, 5373, 9707, 17460, 31501, 56714, 102256, 184183, 331981, 598091, 1077870, 1942071, 3499720, 6305992, 11363361, 20475625, 36896355, 66484244, 119801329, 215873462, 388991876, 700937471

Offset: 0

N. J. A. Sloane

a(n+1) = S(n) for n>=1, where S(n) is the number of 01-words of length n, having first letter 1, in which all runlengths of 1's are odd. Example: S(4) counts 1000, 1001, 1010, 1110. See A077865. - Clark Kimberling, Jun 26 2004
For n>=1, number of compositions of n into floor(j/2) kinds of j's (see g.f.). - Joerg Arndt, Jul 06 2011
Counts walks of length n between the first and second nodes of P_3, to which a loop has been added at the end. 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 (1,2) element of A^n. - Paul Barry, Jul 16 2004
Interleaves A094790 and A094789. - Paul Barry, Oct 30 2004
a(n) appears in the formula for the nonnegative powers of rho:= 2*cos(Pi/7), 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 sigma:=rho^2-1, the ratio of the larger heptagon diagonal to the side length, as follows. rho^n = C(n)*1 + C(n+1)*rho + a(n)*sigma, n>=0, with C(n) = A052547(n-2). See the Steinbach reference, and a comment under A052547. - Wolfdieter Lang, Nov 25 2010
If with the above notations the power basis <1,rho,rho^2> of Q(rho) is used, nonnegative powers of rho are given by rho^n  = -a(n-1)*1 + A052547(n-1)*rho + a(n)*rho^2. For negative powers see A006054. - Wolfdieter Lang, May 06 2011
-a(n-1) also appears in the formula for the nonpositive powers of sigma (see the above comment for the definition, and the Steinbach basis <1,rho,sigma>) as follows: sigma^(-n) = A(n)*1 -a(n+1)*rho -A(n-1)*sigma, with A(n) = A052547(n), A(-1):=0. - Wolfdieter Lang, Nov 25 2010

			G.f. = x^2 + x^3 + 3*x^4 + 4*x^5 + 9*x^6 + 14*x^7 + 28*x^8 + 47*x^9 + ...
Regarding the description "number of compositions of n into floor(j/2) kinds of j's," the a(6)=9 compositions of 6 are (2a, 2a, 2a), (3a, 3a), (2a, 4a), (2a, 4b), (4a, 2a), (4b, 2a), (6a), (6b), (6c). - _Bridget Tenner_, Feb 25 2022

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. Witula, E. Hetmaniok and D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the 15th International Conference on Fibonacci Numbers and Their Applications (2012).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Robin Chapman and Nicholas C. Singer, Eigenvalues of a bidiagonal matrix, Amer. Math. Monthly, 111 (2004), p. 441.
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, 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.
Jia Huang, A coin flip game and generalizations of Fibonacci numbers, arXiv:2501.07463 [math.CO], 2025. See p. 9.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 433
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 p. 11.
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.
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
R. Sachdeva and A. K. Agarwal, Combinatorics of certain restricted n-color composition functions, Discrete Mathematics, 340, (2017), 361-372.
Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7, J. Integer Seq., 12 (2009), Article 09.8.5.
Index entries for linear recurrences with constant coefficients, signature (1,2,-1).

Cf. A006054, A052547, A077865, A094789, A094790, A096975, A096976.

Cf. A120757, A187065, A187066, A187067, A214683, A215076, A215100.

a006053 n = a006053_list !! n
a006053_list = 0 : 0 : 1 : zipWith (+) (drop 2 a006053_list)
   (zipWith (-) (map (2 *) $ tail a006053_list) a006053_list)
-- Reinhard Zumkeller, Oct 14 2011

[ n eq 1 select 0 else n eq 2 select 0 else n eq 3 select 1 else Self(n-1) +2*Self(n-2) -Self(n-3): n in [1..40] ]; // Vincenzo Librandi, Aug 19 2011

a[0]:=0: a[1]:=0: a[2]:=1: for n from 3 to 40 do a[n]:=a[n-1]+2*a[n-2]-a[n-3] od:seq(a[n], n=0..40); # Emeric Deutsch
A006053:=z**2/(1-z-2*z**2+z**3); # conjectured by Simon Plouffe in his 1992 dissertation

LinearRecurrence[{1,2,-1}, {0,0,1}, 50]  (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)

{a(n) = if( n<0, n = -1-n; polcoeff( -1 / (1 - 2*x - x^2 + x^3) + x * O(x^n), n), polcoeff( x^2 / (1 - x - 2*x^2 + x^3) + x * O(x^n), n))}; /* Michael Somos, Nov 30 2014 */

@CachedFunction
def a(n): # a = A006053
    if (n<3): return (n//2)
    else: return a(n-1) + 2*a(n-2) - a(n-3)
[a(n) for n in range(41)] # G. C. Greubel, Feb 12 2023

G.f.: x^2/(1 - x - 2*x^2 + x^3). - Emeric Deutsch, Dec 14 2004
a(n) = c^(n-2) - a(n-1)*(c-1) + (1/c)*a(n-2) for n > 3 where c = 2*cos(Pi/7). Example: a(7) = 14 = c^5 - 9*(c-1) + 4/c = 18.997607... - 7.21743962... + 2.219832528... - Gary W. Adamson, Jan 24 2010
G.f.: -1 + 1/(1 - Sum_{j>=1} floor(j/2)*x^j). - Joerg Arndt, Jul 06 2011
a(n+2) = A094790(n/2+1)*(1+(-1)^n)/2 + A094789((n+1)/2)*(1-(-1)^n)/2. - Paul Barry, Oct 30 2004
First differences of A028495. - Floor van Lamoen, Nov 02 2005
a(n) = A187065(2*n+1); a(n+1) = A187066(2*n+1) = A187067(2*n). - L. Edson Jeffery, Mar 16 2011
a(n) = 2^n*(c(1)^(n-1)*(c(1)+c(2)) + c(3)^(n-1)*(c(3)+c(6)) + c(5)^(n-1)*(c(5)+c(4)) )/7, with c(j):=cos(Pi*j/7). - Herbert Kociemba, Dec 18 2011
a(n+1)*(-1)^n*49^(1/3) = (c(1)/c(4))^(1/3)*(2*c(1))^n + (c(2)/c(1))^(1/3)*(2*c(2))^n + (c(4)/c(2))^(1/3)*(2c(4))^n = (c(2)/c(1))^(1/3)*(2*c(1))^(n+1) + (c(4)/c(2))^(1/3)*(c(2))^(n+1) + (c(1)/c(4))^(1/3)*(2*c(4))^(n+1), where c(j) := cos(2Pi*j/7); for the proof, see Witula et al.'s papers. - Roman Witula, Jul 21 2012
The previous formula connects the sequence a(n) with A214683, A215076, A215100, A120757. We may call a(n) the Ramanujan-type sequence number 2 for the argument 2*Pi/7. - Roman Witula, Aug 02 2012
a(n) = -A006054(1-n) for all n in Z. - Michael Somos, Nov 30 2014
G.f.: x^2 / (1 - x / (1 - 2*x / (1 + 5*x / (2 - x / (5 - 2*x))))). - Michael Somos, Jan 20 2017
a(n) ~ r*c^n, where r=0.241717... is one of the roots of 49*x^3-7*x+1, and c=2*cos(Pi/7) (as in Gary W. Adamson's formula). - Daniel Checa, Nov 04 2022
a(2n-1) = 2*a(n+1)*a(n) - a(n)^2 - a(n-1)^2. - Richard Peterson, May 25 2023

More terms from Emeric Deutsch, Dec 14 2004

Typo in definition fixed by Reinhard Zumkeller, Oct 14 2011

A005021 Random walks (binomial transform of A006054).

Original entry on oeis.org

1, 5, 19, 66, 221, 728, 2380, 7753, 25213, 81927, 266110, 864201, 2806272, 9112264, 29587889, 96072133, 311945595, 1012883066, 3288813893, 10678716664, 34673583028, 112584429049, 365559363741, 1186963827439, 3854047383798, 12514013318097, 40632746115136

Offset: 0

N. J. A. Sloane

Number of walks of length 2n+5 in the path graph P_6 from one end to the other one. Example: a(1)=5 because in the path ABCDEF we have ABABCDEF, ABCBCDEF, ABCDCDEF, ABCDEDEF and ABCDEFEF. - Emeric Deutsch, Apr 02 2004
Since a(n) is the binomial transform of A006054 from formula (3.63) in the Witula-Slota-Warzynski paper, it follows that a(n)=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)), where A(n;1)=A077998(n), B(n;1)=A006054(n+1), C(n;1)=A006054(n), A(n;-1)=A121449(n), B(n+1;-1)=-A085810(n+1), C(n;-1)=A215404(n) and A(n;d), B(n;d), C(n;d), n in N, d in C, denote the quasi-Fibonacci numbers defined and discussed in comments in A121449 and in the cited paper. - Roman Witula, Aug 09 2012
From Wolfdieter Lang, Mar 30 2020: (Start)
With offset -4 this sequence 6, 1, 0, 0, 1, 5, ... appears in the formula for 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: (M_3)^n = a(n-2)*(M_3)^2 - (6*a(n-3) - a(n-4))*M_3 + a(n-3)*1_3, with the 3 X 3 unit matrix 1_3, for n >= 0. Proof from Cayley-Hamilton: (M_3)^n = 5*(M_3)^3 - 6*M_3 + 1_3 (see A332602 for the characteristic polynomial Phi(3, x)), and the recurrence (M_3)^n = M_3*(M_3)^(n-1). For (M_3)^n[1,1] = 2*a(n-2) - 5*a(n-3) + a(n-4), for n >= 0, see A080937(n).
The formula for a(n) in terms of r = rho(7) = A160389 given below shows that a(n)/a(n-1) converges to rho(7)^2 = A116425 = 3.2469796... for n -> infinity. This is because r - 2/r = 0.692..., and r - 1 - 1/r = 0.137... .
(End)

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).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Andrei Asinowski and Michaela A. Polley, Patterns in rectangulations. Part I: T-like patterns, inversion sequence classes I(010, 101, 120, 201) and I(011, 201), and rushed Dyck paths, arXiv:2501.11781 [math.CO], 2025. See p. 26.
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.
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
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]
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
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. - From _N. J. A. Sloane_, Dec 26 2012
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
N. J. A. Sloane, Transforms
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).

Double partial sums of A060557. Bisection of A052547.

Cf. A094789, A094790, A080937, A160389, A116425, A322602.

I:=[1,5,19]; [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

a:=k->sum(binomial(5+2*k,7*j+k-2),j=ceil((2-k)/7)..floor((7+k)/7))-sum(binomial(5+2*k,7*j+k-1),j=ceil((1-k)/7)..floor((6+k)/7)): seq(a(k),k=0..25);
A005021:=-(z-1)*(z-5)/(-1+5*z-6*z**2+z**3); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence apart from the initial 1

LinearRecurrence[{5,-6,1}, {1,5,19}, 50] (* Roman Witula, Aug 09 2012 *)
CoefficientList[Series[1/(1 - 5 x + 6 x^2 - x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 18 2015 *)

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

G.f.: 1/(1-5x+6x^2-x^3). - Emeric Deutsch, Apr 02 2004
a(n) = 5*a(n-1) -6*a(n-2) +a(n-3). - Emeric Deutsch, Apr 02 2004
a(n) = Sum_{j=-infinity..infinity} (binomial(5+2*k, 7*j+k-2) - binomial(5+2*k, 7*j+k-1)) (a finite sum).
a(n-2) = 2^n*C(n;1/2)=(1/7)*((c(2)-c(4))*(c(4))^(2n) + (c(4)-c(1))*(c(1))^(2n) + (c(1)-c(2))*(c(2))^(2n)), where a(-2)=a(-1):=0, c(j):=2*cos(2Pi*j/7). This formula follows from the Binet formula for C(n;d)--one of the quasi-Fibonacci numbers (see comments in A121449 and the formula (3.17) in the Witula-Slota-Warzynski paper). - Roman Witula, Aug 09 2012
In terms of the algebraic number r = rho(7) = 2*cos(Pi/7) = A160389 of degree 3 the preceding formula gives a(n) = r^(2*(n+2))*(A1(r) + A2(r)*(r - 2/r)^(2*(n+1)) = A3(r)*(r - 1 - 1/r)^(2*(n+1)))/7, for n >= -4 (see a comment above for this offset), with A1(r) = -r^2 + 2*r + 1, A2(r) = -r^2 - r + 2, and A3(r) = 2*r^2 - r - 3. - Wolfdieter Lang, Mar 30 2020

a(25)-a(26) from Vincenzo Librandi, Sep 18 2015

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

Original entry on oeis.org

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)

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

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

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

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).

A216054 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 1 or if k-n >= 6, T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 5, 0, 0, 0, 0, 5, 9, 5, 0, 0, 0, 0, 5, 14, 14, 0, 0, 0, 0, 0, 0, 19, 28, 14, 0, 0, 0, 0, 0, 0, 19, 47, 42, 0, 0, 0, 0, 0, 0, 0, 0, 66, 89, 42, 0, 0, 0, 0, 0, 0, 0, 0, 66, 155, 131, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 221, 286, 131, 0, 0

Offset: 0

Philippe Deléham, Mar 16 2013

A hexagon arithmetic of E. Lucas.

			Square array begins:
1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=0
0, 1, 2, 3, 4, 5, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=1
0, 0, 2, 5, 9, 14, 19, 19, 0, 0, 0, 0, 0, 0, 0, ... row n=2
0, 0, 0, 5, 14, 28, 47, 66, 66, 0, 0, 0, 0, 0, 0, ... row n=3
0, 0, 0, 0, 14, 42, 89, 155, 221, 221, 0, 0, 0, 0, ... row n=4
0, 0, 0, 0, 0, 0, 42, 131, 286, 507, 728, 728, 0, 0, ... row n=5
0, 0, 0, 0, 0, 0, 131, 417, 924, 1652, 2380, 2380, 0, ... row n=6
...

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

Cf. A006053, A052547, A096976, A187066,

Cf. Similar sequences A216230, A216228, A216226, A216238

Clear[t]; t[0, k_ /; k <= 5] = 1; t[n_, k_] /; k < n || k > n+5 = 0; t[n_, k_] := t[n, k] = t[n-1, k] + t[n, k-1]; Table[t[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 18 2013 *)

T(n,n) = A080937(n).
T(n,n+1) = A080937(n+1).
T(n,n+2) = A094790(n+1).
T(n,n+3) = A094789(n+1).
T(n,n4) = T(n,n+5) = A005021(n).
Sum_{k, 0<=k<=n} T(n-k,k) = A028495(n).

cp's OEIS Frontend

A006053 a(n) = a(n-1) + 2*a(n-2) - a(n-3), with a(0) = a(1) = 0, a(2) = 1.

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A005021 Random walks (binomial transform of A006054).

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

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

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