A052547 -id:A052547 - cp's OEIS frontend

A187066 Let i be in {1,2,3} and let r >= 0 be an integer. Let p = {p_1, p_2, p_3} = {-2,0,1}, n=2r+p_i, and define a(-2)=0. Then, a(n)=a(2r+p_i) gives the quantity of H_(7,2,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt(2*cos(Pi/7)).

1, 0, 0, 1, 2, 1, 1, 3, 5, 4, 5, 9, 14, 14, 19, 28, 42, 47, 66, 89, 131, 155, 221, 286, 417, 507, 728, 924, 1341, 1652, 2380, 2993, 4334, 5373, 7753, 9707, 14041, 17460, 25213, 31501, 45542, 56714, 81927, 102256, 147798, 184183

Offset: 0

L. Edson Jeffery, Mar 09 2011

(Start) See A187067 for supporting theory. Define the matrix
  U_1= (0 1 0)
       (1 0 1)
       (0 1 1).
Let r>=0 and M=(m_(i,j))=(U_1)^r, i,j=1,2,3. Let B_r be the r-th "block" defined by B_r={a(2*r-2),a(2*r),a(2*r+1)} with a(-2)=0. Note that B_r-B_(r-1)-2*B_(r-2)+B_(r-3)={0,0,0}, with B_0={a(-2),a(0),a(1)}={0,1,0}. Let p={p_1,p_2,p_3}=(-2,0,1) and n=2*r+p_i. Then a(n)=a(2*r+p_i)=m_(i,2), where M=(m_(i,j))=(U_1)^r was defined above. Hence the block B_r corresponds component-wise to the second column of M, and a(n)=m_(i,2) gives the quantity of H_(7,2,0) tiles that should appear in a subdivided H_(7,i,r) tile. (End)
Combining blocks A_r, B_r and C_r, from A187065, this sequence and A187067, respectively, as matrix columns [A_r,B_r,C_r] generates the matrix (U_1)^r, and a negative index (-1)*r yields the corresponding inverse [A_(-r),B_(-r),C_(-r)]=(U_1)^(-r) of (U_1)^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 second-column entries m_(2,2) and m_(3,2) from successive matrices M=(U_1)^r.

			Suppose r=3. Then
B_r = B_3 = {a(2*r-2),a(2*r),a(2*r+1)}={a(4),a(6),a(7)} = {2,1,3},
corresponding to the entries in the third column of
M = (U_2)^3 = (0 2 1)
              (2 1 3)
              (1 3 3).
Choose i=2 and set n=2*r+p_i. Then a(n) = a(2*r+p_i) = a(6+0) = a(6) = 1, which equals the entry in row 2 and column 2 of M. Hence a subdivided H_(7,2,3) tile should contain a(6) = m_(2,2) = 1 H_(7,2,0) tiles.

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

Cf. A187065, A187067, A187068, A187069, A187070.

LinearRecurrence[{0,1,0,2,0,-1},{1,0,0,1,2,1},50] (* Harvey P. Dale, Aug 16 2012 *)
CoefficientList[Series[(1 - x^2 + x^3)/(1 - x^2 - 2*x^4 + x^6), {x, 0, 50}], x] (* G. C. Greubel, Oct 20 2017 *)

my(x='x+O('x^50)); Vec((1-x^2+x^3)/(1-x^2-2*x^4+x^6)) \\ G. C. Greubel, Oct 20 2017

Recurrence: a(n) = a(n-2)+2*a(n-4)-a(n-6).
a(2*n) = A052547(n), a(2n+1) = A006053(n+1).
G.f.: (1-x^2+x^3)/(1-x^2-2*x^4+x^6).
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[2*(-1)^(k-1)*cos(k*Pi/7)], X_k = (w_k)^5-(w_k)^3+(w_k)^2 and Y_k = -(w_k)^5+(w_k)^3+(w_k)^2, k=1,2,3.

A096975 Trace sequence of a path graph plus loop.

Original entry on oeis.org

3, 1, 5, 4, 13, 16, 38, 57, 117, 193, 370, 639, 1186, 2094, 3827, 6829, 12389, 22220, 40169, 72220, 130338, 234609, 423065, 761945, 1373466, 2474291, 4459278, 8034394, 14478659, 26088169, 47011093, 84708772, 152642789, 275049240

Offset: 0

Paul Barry, Jul 16 2004

Let A be the adjacency matrix of the graph P_3 with a loop added at the end. Then a(n) = trace(A^n). A is a 'reverse Jordan matrix' [0,0,1;0,1,1;1,1,0]. a(n) = abs(A094648(n)).
From L. Edson Jeffery, Mar 22 2011: (Start)
Let A be the unit-primitive matrix (see [Jeffery])
A = A_(7,1) =
(0 1 0)
(1 0 1)
(0 1 1).
Then a(n) = Trace(A^n). (End)

Michael De Vlieger, Table of n, a(n) for n = 0..3910
A. Akbary, Q. Wang, A generalized Lucas sequence and permutations binomials, Proc. Am. Math. Soc. 134 (2006) 15-22, sequence a(n) with l=7.
Robin Chapman and Nicholas C. Singer, Eigenvalues of a bidiagonal matrix, Amer. Math. Monthly, 111 (2004), p. 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 matrix
Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.
Q. Wang, On generalized Lucas sequences, Contemp. Math. 531 (2010) 127-141, Table 1 (k=3).
Index entries for linear recurrences with constant coefficients, signature (1,2,-1).

Cf. A006053, A052547, A096976.

A033304(n) = a(-1-n). - Michael Somos, Aug 03 2006

CoefficientList[Series[(3 - 2 x - 2 x^2)/(1 - x - 2 x^2 + x^3), {x, 0, 33}], x] (* Michael De Vlieger, Aug 21 2019 *)

{a(n)=if(n>=0, n+=1; polsym(x^3-x^2-2*x+1,n-1)[n], n=1-n; polsym(1-x-2*x^2+x^3,n-1)[n])} /* Michael Somos, Aug 03 2006 */

a(n)=trace([0,1,0;1,0,1;0,1,1]^n); /* Joerg Arndt, Apr 30 2011 */

G.f.: (3-2*x-2*x^2)/(1-x-2*x^2+x^3);
a(n) = a(n-1) + 2*a(n-2) - a(n-3);
a(n) = (2*sqrt(7)*sin(atan(sqrt(3)/9)/3)/3+1/3)^n + (1/3-2*sqrt(7)*sin(atan(sqrt(3)/9)/3+Pi/3)/3)^n + (2*sqrt(7)*cos(acot(-sqrt(3)/9)/3)/3+1/3)^n.
a(n) = 2^n*((cos(Pi/7))^n+(cos(3*Pi/7))^n+(cos(5*Pi/7))^n). - Vladimir Shevelev, Aug 25 2010
a(n) = (-1)^n*A094648(n). - R. J. Mathar, Nov 05 2024

A116423 Binomial transform of A006053.

Original entry on oeis.org

0, 1, 3, 9, 26, 74, 209, 588, 1651, 4631, 12983, 36388, 101972, 285741, 800660, 2243445, 6286059, 17613241, 49351342, 138279586, 387451077, 1085614208, 3041824015, 8523002359, 23880923183, 66912861640, 187485674652, 525323190505, 1471922876424, 4124236259529

Offset: 1

Gary W. Adamson, Feb 14 2006

nonn

a(n)/a(n-1) tends to 2.801... = 1 + 2*cos(Pi/7).
A(n) := a(n+1)*(-1)^(n+1) 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 C(n)= A085810(n)*(-1)^n, and B(n)= A181880(n-2)*(-1)^n. For the nonnegative powers see A120757(n), |A122600(n-1)| and A181879(n), respectively. See also a comment under A052547.
This sequence is constructible as a spiral tiling of similar trapezoids, as follows: start with an isosceles trapezoid with side lengths 3,1,4,1. Each new trapezoid is rotated and scaled so one leg fills all unoccupied space on the short base of the previous trapezoid. a(n) is given by the length of the n-th trapezoid's legs. This process is identical to the recursion relation added by R. J. Mathar in the Formula section. See the Links section for an illustration. - Andrew B. Hudson, Jun 19 2019

			a(5) = 26 = 1*0 + 1*4 + 4*1 + 4*3 + 6*1 = 4 + 4 + 12 + 6 = 26.

Jinyuan Wang, Table of n, a(n) for n = 1..1000
Andrew B. Hudson, Illustration of the first 7 terms as a spiral tiling of similar trapezoids.
Peter Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-1)

Cf. A006053.

I:=[0,1,3]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 11 2019

LinearRecurrence[{4, -3, -1}, {0, 1, 3}, 40] (* Vincenzo Librandi, Jul 11 2019 *)

concat(0, Vec(x^2*(1-x)/(1-4*x+3*x^2+x^3) + O(x^50))) \\ Michel Marcus, Sep 13 2014

Binomial transform of A006053 starting with A006053(1): (0, 1, 1, 3, 4, 9, 14, ...).
From R. J. Mathar, Apr 02 2008: (Start)
O.g.f.: x^2(1-x)/(1 - 4x + 3x^2 + x^3).
a(n) = 4*a(n-1) - 3*a(n-2) - a(n-3). (End)

More terms from R. J. Mathar, Apr 02 2008

More terms from Michel Marcus, Sep 13 2014

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

A181879 Expansion of x(1+x)/(1-3x-4*x^2-x^3).

Original entry on oeis.org

0, 1, 4, 16, 65, 263, 1065, 4312, 17459, 70690, 286218, 1158873, 4692181, 18998253, 76922356, 311452261, 1261044460, 5105864780, 20673224441, 83704176903, 338911293253, 1372223811812, 5556020785351, 22495868896554, 91083913642878, 368791237300201, 1493205235368669, 6045864568949689, 24479205885623944, 99114281168039257, 401305531615563236

Offset: 0

Wolfdieter Lang, Nov 26 2010

a(n) 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, called there Q(rho). (rho*sigma)^n = C(n) + B(n)*rho + a(n)*sigma,n>=0, with C(n)= A120757(n) with C(0):=1, and B(n)= |A122600(n-1)| with B(0)=1. 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.

P. 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).

CoefficientList[Series[x (1+x)/(1-3x-4x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{3,4,1},{0,1,4},40] (* Harvey P. Dale, Feb 04 2024 *)

Vec((1+x)/(1-3*x-4*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

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

A181880 Expansion of 1/(1-4x-3x^2-x^3).

Original entry on oeis.org

1, 4, 19, 89, 417, 1954, 9156, 42903, 201034, 942001, 4414009, 20683073, 96916320, 454128508, 2127946065, 9971086104, 46722311119, 218930448853, 1025859814873, 4806952917170, 22524321562152, 105544004814991, 494555936863590, 2317380083461485, 10858732149251701, 50881624784254849, 238420075668235984, 1117183909174960184, 5234877488488803537, 24529481757148330684

Offset: 0

Wolfdieter Lang, Nov 27 2010

B(n):=a(n-2)*(-1)^n, B(0):=0, B(1):=0, (o.g.f. x^2/(1 + 4*x + 3*x^2 -x^3))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 C(n)= A085810(n)*(-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.

P. 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 (4, 3, 1).

CoefficientList[Series[1/(1-4*x-3*x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{4,3,1},{1,4,19},40] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

O.g.f.: 1/(1-4*x-3*x^2-x^3).

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

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

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).

A052672 Expansion of e.g.f. (1-x)/(1-x-2*x^2+x^3).

Original entry on oeis.org

1, 0, 4, 6, 120, 600, 10080, 95760, 1693440, 23950080, 475372800, 8821612800, 199743667200, 4533271142400, 116906088499200, 3112264995840000, 90679371374592000, 2757644630028288000, 89895729202126848000

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..400
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 620

Cf. A000142, A052547.

I:=[1,0,4]; [n le 3 select I[n] else (n-1)*(Self(n-1) +2*(n-2)*Self(n-2) -(n-2)*(n-3)*Self(n-3)): n in [1..31]]; // G. C. Greubel, Jun 13 2022

spec := [S,{S=Sequence(Prod(Z,Union(Z,Prod(Z,Sequence(Z)))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[-(-1+x)/(x^3-2x^2-x+1),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 25 2021 *)

def A052672_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (1-x)/(1-x-2*x^2+x^3) ).egf_to_ogf().list()
A052672_list(30) # G. C. Greubel, Jun 13 2022

E.g.f.: (1 - x)/(1 - x - 2*x^2 + x^3).
Recurrence: a(0)=1, a(1)=0, a(2)=4, a(n) = n*a(n-1) + 2*n*(n-1)*a(n-2) - n*(n-1)*(n-2)*a(n-3).
a(n) = (n!/7)*Sum_{alpha=RootOf(Z^3 -2*Z^2 -Z +1)} (3 - alpha)*alpha^(-n).
a(n) = n!*A052547(n). - R. J. Mathar, Nov 27 2011

A265755 a(n) = a(n-1) + a(n-2) if n is even and a(n) = a(n-3) + a(n-4) if n is odd, with a(0) = a(1) = a(2) = 0 and a(3) = 1.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 2, 3, 1, 4, 5, 9, 5, 14, 14, 28, 19, 47, 42, 89, 66, 155, 131, 286, 221, 507, 417, 924, 728, 1652, 1341, 2993, 2380, 5373, 4334, 9707, 7753, 17460, 14041, 31501, 25213, 56714, 45542, 102256, 81927, 184183, 147798, 331981, 266110, 598091, 479779, 1077870, 864201, 1942071, 1557649

Offset: 0

Nicholas Drozd, Dec 15 2015

			a(8) = a(7) + a(6)
     = a(4) + a(3) + a(5) + a(4)
     = (a(3) + a(2)) + a(3) + (a(2) + a(1)) + (a(3) + a(2))
     = 1 + 1 + 0 + 1
     = 3

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,2,0,-1).

Interleaves A006053 and A052547, with an extra leading 0.

A187066 with even values and odd values swapped and an extra leading 0.

a[0] = a[1] = a[2] = 0; a[3] = 1; a[n_] := a[n] = If[EvenQ@ n, a[n - 1] + a[n - 2], a[n - 3] + a[n - 4]]; Table[a@ n, {n, 0, 55}] (* Michael De Vlieger, Dec 15 2015 *)
nxt[{n_,a_,b_,c_,d_}]:={n+1,b,c,d,If[OddQ[n],c+d,a+b]}; NestList[nxt,{1,0,0,0,1},60][[All,2]] (* or *) LinearRecurrence[{0,1,0,2,0,-1},{0,0,0,1,1,0},60] (* Harvey P. Dale, Nov 10 2017 *)

concat(vector(3), Vec(x^3*(1+x-x^2)/(1-x^2-2*x^4+x^6) + O(x^70))) \\ Colin Barker, Dec 16 2015

From Colin Barker, Dec 16 2015: (Start)
a(n) = a(n-2) + 2*a(n-4) - a(n-6) for n>5.
G.f.: x^3*(1+x-x^2) / (1-x^2-2*x^4+x^6).
(End)

More terms from Michael De Vlieger, Dec 15 2015

cp's OEIS Frontend

A187066 Let i be in {1,2,3} and let r >= 0 be an integer. Let p = {p_1, p_2, p_3} = {-2,0,1}, n=2*r+p_i, and define a(-2)=0. Then, a(n)=a(2*r+p_i) gives the quantity of H_(7,2,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt(2*cos(Pi/7)).

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A096975 Trace sequence of a path graph plus loop.

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A116423 Binomial transform of A006053.

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

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A181879 Expansion of x*(1+x)/(1-3*x-4*x^2-x^3).

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A181880 Expansion of 1/(1-4*x-3*x^2-x^3).

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

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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).

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A187066 Let i be in {1,2,3} and let r >= 0 be an integer. Let p = {p_1, p_2, p_3} = {-2,0,1}, n=2r+p_i, and define a(-2)=0. Then, a(n)=a(2r+p_i) gives the quantity of H_(7,2,0) tiles in a subdivided H_(7,i,r) tile after linear scaling by the factor x^r, where x=sqrt(2*cos(Pi/7)).

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

A181879 Expansion of x(1+x)/(1-3x-4*x^2-x^3).

A181880 Expansion of 1/(1-4x-3x^2-x^3).