A052547 -id:A052547 - cp's OEIS frontend

A006054 a(n) = 2*a(n-1) + a(n-2) - a(n-3), with a(0) = a(1) = 0, a(2) = 1.

0, 0, 1, 2, 5, 11, 25, 56, 126, 283, 636, 1429, 3211, 7215, 16212, 36428, 81853, 183922, 413269, 928607, 2086561, 4688460, 10534874, 23671647, 53189708, 119516189, 268550439, 603427359, 1355888968, 3046654856, 6845771321, 15382308530, 34563733525

Offset: 0

N. J. A. Sloane

Let u(k), v(k), w(k) be defined by u(1)=1, v(1)=0, w(1)=0 and u(k+1)=u(k)+v(k)+w(k), v(k+1)=u(k)+v(k), w(k+1)=u(k); then {u(n)} = 1,1,3,6,14,31,... (A006356 with an extra initial 1), {v(n)} = 0,1,2,5,11,25,... (this sequence with its initial 0 deleted) and {w(n)} = {u(n)} prefixed by an extra 0 = A077998 with an extra initial 0. - Benoit Cloitre, Apr 05 2002. Also u(k)^2+v(k)^2+w(k)^2 = u(2k). - Gary W. Adamson, Dec 23 2003
Form the graph with matrix A=[1, 1, 1; 1, 0, 0; 1, 0, 1]. Then A006054 counts walks of length n between the vertex of degree 1 and the vertex of degree 3. - Paul Barry, Oct 02 2004
Form the digraph with matrix [1,1,0; 1,0,1; 1,1,1]. A006054(n) counts walks of length n between the vertices with loops. - Paul Barry, Oct 15 2004
Nonzero terms = INVERT transform of (1, 1, 2, 2, 3, 3, ...). Example: 56 = (1, 1, 2, 5, 11, 25) dot (3, 3, 2, 2, 1, 1) = (3 + 3 + 4 + 10 + 11 + 25). - Gary W. Adamson, Apr 20 2009
-a(n+1) appears in the formula for the nonpositive 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+1)*sigma, n >= 0, with C(n)=A077998(n), C(-1):=0. See the Steinbach reference, and a comment under A052547.
If, with the above notations, the power basis of the field Q(rho) is taken one has for nonpositive powers of rho, rho^(-n) = a(n+2)*1 + A077998(n-1)*rho - a(n+1)*rho^2. For nonnegative powers see A006053. See also the Steinbach reference. - Wolfdieter Lang, May 06 2011
a(n) appears also in the nonnegative powers of sigma,(defined in the above comment, where also the basis is given). See a comment in A106803.
The sequence b(n):=(-1)^(n+1)*a(n) forms the negative part (i.e., with nonpositive indices) of the sequence (-1)^n*A006053(n+1). In this way we obtain what we shall call the Ramanujan-type sequence number 2a for the argument 2*Pi/7 (see the comment to Witula's formula in A006053). We have b(n) = -2*b(n-1) + b(n-2) + b(n-3) and b(n) * 49^(1/3) = (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) = (c(2)/c(1))^(1/3) * (c(1))^(-n+1) + (c(4)/c(2))^(1/3) * (c(2))^(-n+1) + (c(1)/c(4))^(1/3) * (c(4))^(-n+1), where c(j) := 2*cos(2*Pi*j/7) (for the proof, see the comments to A215112). - Roman Witula, Aug 06 2012
(1, 1, 2, 5, 11, 25, 56, ...) * (1, 0, 1, 0, 1, ...) = the variant of A006356: (1, 1, 3, 6, 14, 31, ...). - Gary W. Adamson, May 15 2013
The limit of a(n+1)/a(n) for n -> infinity is, for all generic sequences with this recurrence of signature (2,1,-1), sigma = rho^2-1, approximately 2.246979603, the length ratio (largest diagonal)/side in the regular heptagon (7-gon). For rho = 2*cos(Pi/7) and sigma see a comment above, and the P. Steinbach reference. Proof: a(n+1)/a(n) = 2 + 1/(a(n)/a(n-1)) - 1/((a(n)/a(n-1))*(a(n-1)/a(n-2))), leading in the limit to sigma^3 -2*sigma^2 - sigma + 1, which is solved by sigma = rho^2-1, due to C(7, rho) = 0 , with the minimal polynomial C(7, x) = x^3 - x^2 - 2*x + 1 of rho (see A187360). - Wolfdieter Lang, Nov 07 2013
Numbers of straight-chain aliphatic amino acids involving single, double or triple bonds (allowing adjacent double bonds) when cis/trans isomerism is neglected. - Stefan Schuster, Apr 19 2018
Let A(r,n) be the total number of ordered arrangements of an n+r tiling of r red squares and white tiles of total length n, where the individual tile lengths can range from 1 to n. A(r,0) corresponds to a tiling of r red squares only, and so A(r,0) = 1. Also, A(r,n)=0 for n<0. Let A_1(r,n) = Sum_{j=0..n} A(r,j). Then the expansion of 1/(1 - 2*x - x^2 + x^3) is A_1(0,n) + A_1(1,n-2) + A_1(n-4) + ... = a(n) without the initial two 0's. In general, the expansion of 1/(1 - 2*x -x^k + x^(k+1)) is equal to Sum_{j>=0} A_1(j, n-j*k). - Gregory L. Simay, May 25 2018
For n>1, a(n) is the number of ways to tile a strip of length n-1 with one color of squares and dominos, two colors of trominos and quadrominos, 3 colors of 5-minos and 6-minos, and so on. - Greg Dresden and Zhiyu Zhang, Jun 26 2025

			G.f. = x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 25*x^6 + 56*x^7 + 126*x^8 + 283*x^9 + ... - _Michael Somos_, Jun 25 2018

Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..150
C. P. de Andrade, J. P. de Oliveira Santos, E. V. P. da Silva and K. C. P. Silva, Polynomial Generalizations and Combinatorial Interpretations for Sequences Including the Fibonacci and Pell Numbers, Open Journal of Discrete Mathematics, 2013, 3, 25-32 doi:10.4236/ojdm.2013.31006. - From _N. J. A. Sloane_, Feb 20 2013
Maximilian Fichtner, K. Voigt, and S. Schuster, The tip and hidden part of the iceberg: Proteinogenic and non-proteinogenic aliphatic amino acids, Biochimica et Biophysica Acta (BBA)-General, 2016, Volume 1861, Issue 1, Part A, January 2017, Pages 3258-3269.
Brian Hopkins and Hua Wang, Restricted Color n-color Compositions, arXiv:2003.05291 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 434
S. Morier-Genoud, V. Ovsienko, and S. Tabachnikov, Introducing supersymmetric frieze patterns and linear difference operators, Math. Z. 281 (2015) 1061.
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.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Alexey Ustinov, Supercontinuants, arXiv:1503.04497 [math.NT], 2015.
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
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 (2,1,-1).

Cf. A005578, A006053, A006356, A007583, A080937, A094790, A214683, A214699, A214779, A215112, A306334.

Row sums of A144159 and A180264.

a006054 n = a006053_list !! n
a006054_list = 0 : 0 : 1 : zipWith (+) (map (2 *) $ drop 2 a006054_list)
   (zipWith (-) (tail a006054_list) a006054_list)
-- Reinhard Zumkeller, Oct 14 2011

A006054:=z**2/(1-2*z-z**2+z**3); # Simon Plouffe in his 1992 dissertation

LinearRecurrence[{2, 1, -1}, {0, 0, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)

a(n):=if n<2 then 0 else if n=2 then 1 else b(n-2);
b(n):=sum(sum(binomial(j,n-3*k+2*j)*(-1)^(j-k)*binomial(k,j)*2^(-n+3*k-j),j,0,k),k,1,n); /* Vladimir Kruchinin, May 05 2011 */

x='x+O('x^66);
concat([0, 0], Vec(x^2/(1-2*x-x^2+x^3))) \\ Joerg Arndt, May 05 2011

G.f.: x^2/(1-2*x-x^2+x^3).
Sum_{k=0..n+2} a(k) = A077850(n). - Philippe Deléham, Sep 07 2006
Let M = the 3 X 3 matrix [1,1,0; 1,2,1; 0,1,2], then M^n*[1,0,0] = [A080937(n-1), A094790(n), A006054(n-1)]. E.g., M^3*[1,0,0] = [5,9,5] = [A080937(2), A094790(3), A006054(2)]. - Gary W. Adamson, Feb 15 2006
a(n) = round(k*A006356(n-1)), for n>1, where k = 0.3568958678... = 1/(1+2*cos(Pi/7)). - Gary W. Adamson, Jun 06 2008
a(n+1) = A187070(2n+1) = A187068(2n+3). - L. Edson Jeffery, Mar 10 2011
a(n+3) = Sum_{k=1..n} Sum_{j=0..k} binomial(j,n-3*k+2*j)*(-1)^(j-k)*binomial(k,j)*2^(-n+3*k-j); a(0)=0, a(1)=0, a(2)=1. - Vladimir Kruchinin, May 05 2011
7*a(n) = (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) - for the proof see Witula et al. papers. - Roman Witula, Aug 07 2012
a(n) = -A006053(1-n) for all n in Z. - Michael Somos, Jun 25 2018

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

Original entry on oeis.org

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

A077998 Expansion of (1-x)/(1-2*x-x^2+x^3).

Original entry on oeis.org

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, 19181424995, 43100270734, 96845429254

Offset: 0

N. J. A. Sloane, Nov 17 2002

Let u(k), v(k), w(k) be defined by u(1)=1, v(1)=0, w(1)=0 and u(k+1)=u(k)+v(k)+w(k), v(k+1)=u(k)+v(k), w(k+1)=u(k); then {u(n)} = 1,1,3,6,14,31,... (A006356 with an extra initial 1), {v(n)} = 0,1,2,5,11,25,... (A006054 with its initial 0 deleted) and {w(n)} = {u(n)} prefixed by an extra 0 = this sequence with an extra initial 0. - Benoit Cloitre, Apr 05 2002 [Also u(k)^2+v(k)^2+w(k)^2 = u(2k). - Gary W. Adamson, Dec 23 2003]
Form the graph with matrix A=[1, 1, 1; 1, 0, 0; 1, 0, 1]. Then A077998 counts closed walks of length n at the vertex of degree 4. - Paul Barry, Oct 02 2004
a(n) is the number of Motzkin (n+2)-sequences with no flatsteps at ground level and whose height is <=2. For example, a(3)=6 counts UDUFD, UFDUD, UFFFD, UFUDD, UUDFD, UUFDD. - David Callan, Dec 09 2004
Number of compositions of n if there are two kinds of part 2. Example: a(3)=6 because we have (3),(1,2),(1,2'),(2,1),(2',1) and (1,1,1). Row sums of A105477. - Emeric Deutsch, Apr 09 2005
Diagonal sums of A056242. - Paul Barry, Dec 26 2007
Diagonal sums of triangle in A105306. - Philippe Deléham, Nov 16 2008
a(n) appears in the formula for the nonpositive 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) = a(n)*1 + a(n-1)*rho - C(n)*sigma, n>=0, with C(n)=A006054(n+1). Put a(-1):=0. See the Steinbach reference, and a comment under A052547.
The limit a(n+1)/a(n) for n -> infinity is sigma = rho^2-1, approximately 2.246979603. See a Nov 07 2013 comment on A006054 for the proof, and the preceding comment for rho and sigma and the P. Steinbach reference. - Wolfdieter Lang, Nov 07 2013
From Greg Dresden and Aaron Zhou, Jun 15 2023: (Start)
a(n) is the number of ways to tile a skew double-strip of 3*n cells using all possible "trominos". Here is the skew double-strip corresponding to n=4, with 12 cells:
   _ ___ _ ___ _ ___
  |   |   |   |   |   |   |
 |__|___|_|___| |___|
|   |   |   |   |   |   |
|_|___|_|___|_|___|,
and here are the three possible "tromino" tiles, which can be rotated or reflected as needed:
   _              _
  |   |            |   |
 |__|_      ___|___|     _________
|   |   |    |   |   |      |   |   |   |
|_|___|,   |_|___|  ,   |_|___|_|.
As an example, here is one of the a(4) = 14 ways to tile the skew double-strip of 12 cells:
   _ ___ _____ _______
  |   |   |       |       |
 |   |  |___  |     |
|       |       |   |   |
|_____|_______|_|___|. (End)

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 14*x^4 + 31*x^5 + 70*x^6 + 157*x^7 + 353*x^8 + ... - _Michael Somos_, Dec 12 2023

Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.
Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.

Vincenzo Librandi, 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.
S. Morier-Genoud, V. Ovsienko, and S. Tabachnikov, Introducing supersymmetric frieze patterns and linear difference operators, Math. Z. 281 (2015) 1061.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Alexey Ustinov, Supercontinuants, arXiv:1503.04497 [math.NT], 2015.
Floor van Lamoen, Wave sequences
R. 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 (2,1,-1).

Apart from initial term, same as A006356, which is the main entry for this sequence. A106803 is yet another version.

Cf. A096976, A105477.

a:=[1,1,3];; for n in [4..40] do a[n]:=2*a[n-1]+a[n-2]-a[n-3]; od; a; # G. C. Greubel, Jun 27 2019

I:=[1,1,3]; [n le 3 select I[n] else 2*Self(n-1)+Self(n-2)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jun 01 2017

CoefficientList[Series[(1-x)/(1-2*x-x^2+x^3), {x, 0, 40}], x] (* Stefan Steinerberger, Sep 11 2006 *)
LinearRecurrence[{2,1,-1},{1,1,3},40] (* Roman Witula, Aug 07 2012 *)
a[ n_] := {1, 0, 0} . MatrixPower[{{0, 1, 0}, {0, 0, 1}, {-1, 1, 2}}, n] . {1, 1, 3}; (* Michael Somos, Dec 12 2023 *)

a(n)=([0,1,0; 0,0,1; -1,1,2]^n*[1;1;3])[1,1] \\ Charles R Greathouse IV, May 10 2016

((1-x)/(1-2*x-x^2+x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 27 2019

a(0)=a(1)=1, a(2)=3, a(n+1) = 2*a(n) + a(n-1) - a(n-2) for n>=2. - Philippe Deléham, Sep 07 2006
7*a(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(2Pi*j/7) and s(j) = 2*Sin(2Pi*j/7) - for the proof of this one and many other relations for the sequences u(k), v(k) and w(k) defined on the top of the comments by Benoit Cloitre - see Witula et al.'s paper. - Roman Witula, Aug 07 2012
a(n) = b(n+2)- b(n+1), first differences of b(n) = A006054(n). - Wolfdieter Lang, Nov 07 2013; corrected by Kai Wang, May 31 2017
a(n) = A096976(-n) for all n in Z. - Michael Somos, Dec 12 2023

Edited by N. J. A. Sloane, Aug 08 2008 at the suggestion of R. J. Mathar

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

Original entry on oeis.org

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

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

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

A120757 Expansion of x^2(2+x)/(1-3x-4*x^2-x^3).

Original entry on oeis.org

0, 2, 7, 29, 117, 474, 1919, 7770, 31460, 127379, 515747, 2088217, 8455018, 34233669, 138609296, 561217582, 2272323599, 9200450421, 37251863241, 150829715006, 610697048403, 2472661868474, 10011603514040, 40536155064419

Offset: 1

Gary W. Adamson & Roger L. Bagula, Jul 01 2006

The (1,1)-entry of the matrix M^n, where M is the 3 X 3 matrix [0,1,1; 1,1,2; 1,2,2].
a(n)/a(n-1) tends to 4.0489173...an eigenvalue of M and a root to the characteristic polynomial x^3 - 3x^2 - 4x - 1.
C(n):=a(n), with a(0):=1 (hence the o.g.f. for C(n) is (1-3*x-2*x^2)/(1-3*x-4*x^2-x^3)), appears in the following formula for the nonnegative 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)= |A122600(n-1)|, B(0)=0, and A(n)= A181879(n). For the nonpositive powers see A085810(n)*(-1)^n, A181880(n-2)*(-1)^n and A116423(n+1)*(-1)^(n+1), respectively. See also a comment under A052547.
We have a(n)=cs(3n+1), where the sequence cs(n) and its two conjugate sequences as(n) and bs(n) are defined in the comments to the sequence A214683 (see also A215076, A215100, A006053). We call the sequence a(n) the Ramanujan-type sequence number 5 for the argument 2Pi/7. Since as(3n+1)=bs(3n+1)=0, we obtain the following relation: 49^(1/3)*a(n) = (c(1)/c(4))^(n + 1/3) + (c(4)/c(2))^(n + 1/3) + (c(2)/c(1))^(n + 1/3), where c(j) := Cos(2Pi/7) (for more details and proofs see Witula et al.'s papers). - Roman Witula, Aug 02 2012

			a(7)=1919 because M^7= [1919,3458,4312;3458,6231,7770;4312,7770,9689].

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 Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.

G. C. Greubel, Table of n, a(n) for n = 1..1000
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31, MR 1439165
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
Roman Witula, Full Description of Ramanujan Cubic Polynomials, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.
Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.
Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
Index entries for linear recurrences with constant coefficients, signature (3,4,1).

Cf. A006053, A214683, A215076, A215100.

a:=[0,2,7]; [ n le 3 select a[n] else 3*Self(n-1) + 4*Self(n-2) + Self(n-3): n in [1..25]]; // Marius A. Burtea, Oct 03 2019

with(linalg): M[1]:=matrix(3,3,[0,1,1,1,1,2,1,2,2]): for n from 2 to 25 do M[n]:=multiply(M[1],M[n-1]) od: seq(M[n][1,1],n=1..25);

LinearRecurrence[{3,4,1},{0,2,7},40] (* Roman Witula, Aug 02 2012 *)

a(n)=([0,1,0; 0,0,1; 1,4,3]^(n-1)*[0;2;7])[1,1] \\ Charles R Greathouse IV, Jun 22 2016

@CachedFunction
def a(n): # a = A120757
    if (n<3): return (0,2,7)[n]
    else: return 3*a(n-1) + 4*a(n-2) + a(n-3)
[a(n) for n in range(40)] # G. C. Greubel, Nov 25 2022

a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3) (follows from the minimal polynomial of the matrix M). See also the o.g.f. given in the name.

Edited by N. J. A. Sloane, Dec 03 2006

New name, old name as comment; o.g.f.; reference.

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

A122600 Expansion of 1/(1 + 3x - 4x^2 + x^3).

Original entry on oeis.org

1, -3, 13, -52, 211, -854, 3458, -14001, 56689, -229529, 929344, -3762837, 15235416, -61686940, 249765321, -1011279139, 4094585641, -16578638800, 67125538103, -271785755150, 1100438056662, -4455582728689, 18040286167865, -73043627475013, 295747609825188, -1197457625543481

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 20 2006

Suggested by the Steinbach heptagon polynomial p^3 - 2*p^2*(1 - p) - p(1 - p)^2 + (1 - p)^3 = (1 - 4 p + 3 p^2 + p^3).

B(n):=|a(n-1)| = a(n-1)*(-1)^(n-1) with B(0):=0 (hence the o.g.f. for B(n) is x/(1 + 3*x - 4*x^2 + x^3)) appears in the following formula for the nonnegative 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 C(n)= A120757(n) with C(0):=1, and A(n)= A181879(n). For the nonpositive powers see A085810*(-1)^n, A181880(n) and A116423(n)*(-1)^n, respectively. See also a comment under A052547.

Toufik Mansour and Mark Shattuck, Avoidance of vincular patterns by flattened derangements, arXiv:2504.14713 [math.CO], 2025. See p. 6.
Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Index entries for linear recurrences with constant coefficients, signature (-3,4,-1).

Cf. A065941.

p[x_] := 1 - 4 x + 3x^2 + x^3; q[x_] := ExpandAll[x^3*p[1/x]]; Table[ SeriesCoefficient[ Series[x/q[x], {x, 0, 30}], n], {n, 0, 30}]
CoefficientList[Series[1/(1 + 3*x - 4*x^2 + x^3), {x, 0, 50}], x] (* or *) LinearRecurrence[{-3, 4, -1}, {1, -3, 13}, 40] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2012 *)

a(n)= -3*a(n-1) + 4*a(n-2) - a(n-3), n>=2, a(-1):=0, a(1)=0, a(1)=-3 (from the o.g.f. given in the name).
a(n) = (-1)^n*Sum_{k=0..n} binomial(n+k+2,3*k+2)*7^k. - Emanuele Munarini, Aug 27 2017
From Kai Wang, Jul 05 2020: (Start)
a(n) =  Sum_{i+2j+3k=n} (-1)^(i+k)*3^i*4^j*((i+j+k)!)/(i!*j!*k!).
a(n) = (-1)^n*(6*A215076(n+4) - 21*A215076(n+3) - 13*A215076(n+2))/7. (End)

Edited by N. J. A. Sloane, Feb 01 2007

cp's OEIS Frontend

A362379 Convolution triangle of A052547(n).

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A006054 a(n) = 2*a(n-1) + a(n-2) - a(n-3), with a(0) = a(1) = 0, a(2) = 1.

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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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A077998 Expansion of (1-x)/(1-2*x-x^2+x^3).

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

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A005021 Random walks (binomial transform of A006054).

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A120757 Expansion of x^2(2+x)/(1-3x-4*x^2-x^3).

A122600 Expansion of 1/(1 + 3x - 4x^2 + x^3).