A071101 -id:A071101 - cp's OEIS frontend

A192232 Constant term of the reduction of n-th Fibonacci polynomial by x^2 -> x+1. (See Comments.)

1, 0, 2, 1, 6, 7, 22, 36, 89, 168, 377, 756, 1630, 3353, 7110, 14783, 31130, 65016, 136513, 285648, 599041, 1254456, 2629418, 5508097, 11542854, 24183271, 50674318, 106173180, 222470009, 466131960, 976694489, 2046447180, 4287928678, 8984443769, 18825088134

Offset: 1

Clark Kimberling, Jun 26 2011

Polynomial reduction: an introduction
...
We begin with an example.  Suppose that p(x) is a polynomial, so that p(x)=(x^2)t(x)+r(x) for some polynomials t(x) and r(x), where r(x) has degree 0 or 1.  Replace x^2 by x+1 to get (x+1)t(x)+r(x), which is (x^2)u(x)+v(x) for some u(x) and v(x), where v(x) has degree 0 or 1.  Continuing in this manner results in a fixed polynomial w(x) of degree 0 or 1.  If p(x)=x^n, then w(x)=x*F(n)+F(n-1), where F=A000045, the sequence of Fibonacci numbers.
In order to generalize, write d(g) for the degree of an arbitrary polynomial g(x), and suppose that p, q, s are polynomials satisfying d(s)s in this manner until reaching w such that d(w)s.
The coefficients of (reduction of p by q->s) comprise a vector of length d(q)-1, so that a sequence p(n,x) of polynomials begets a sequence of vectors, such as (F(n), F(n-1)) in the above example.  We are interested in the component sequences (e.g., F(n-1) and F(n)) for various choices of p(n,x).
Following are examples of reduction by x^2->x+1:
n-th Fibonacci p(x) -> A192232+x*A112576
n-th cyclotomic p(x) -> A192233+x*A051258
n-th 1st-kind Chebyshev p(x) -> A192234+x*A071101
n-th 2nd-kind Chebyshev p(x) -> A192235+x*A192236
x(x+1)(x+2)...(x+n-1) -> A192238+x*A192239
(x+1)^n -> A001519+x*A001906
(x^2+x+1)^n -> A154626+x*A087635
(x+2)^n -> A020876+x*A030191
(x+3)^n -> A192240+x*A099453
...
Suppose that b=(b(0), b(1),...) is a sequence, and let p(n,x)=b(0)+b(1)x+b(2)x^2+...+b(n)x^n.  We define (reduction of sequence b by q->s) to be the vector given by (reduction of p(n,x) by q->s), with components in the order of powers, from 0 up to d(q)-1.  For k=0,1,...,d(q)-1, we then have the "k-sequence of (reduction of sequence b by q->s)".  Continuing the example, if b is the sequence given by b(k)=1 if k=n and b(k)=0 otherwise, then the 0-sequence of (reduction of b by x^2->x+1) is (F(n-1)), and the 1-sequence is (F(n)).
...
For selected sequences b, here are the 0-sequences and 1-sequences of (reduction of b by x^2->x+1):
b=A000045, Fibonacci sequence (1,1,2,3,5,8,...) yields
   0-sequence A166536 and 1-sequence A064831.
b=(1,A000045)=(1,1,1,2,3,5,8,...) yields
   0-sequence A166516 and 1-sequence A001654.
b=A000027, natural number sequence (1,2,3,4,...) yields
  0-sequence A190062 and 1-sequence A122491.
b=A000032, Lucas sequence (1,3,4,7,11,...) yields
  0-sequence A192243 and 1-sequence A192068.
b=A000217, triangular sequence (1,3,6,10,...) yields
  0-sequence A192244 and 1-sequence A192245.
b=A000290, squares sequence (1,4,9,16,...) yields
  0-sequence A192254 and 1-sequence A192255.
More examples:  A192245-A192257.
...
More comments:
(1) If s(n,x)=(reduction of x^n by q->s) and
    p(x)=p(0)x^n+p(1)x^(n-1)+...+p(n)x^0, then
    (reduction of p by q->s)=p(0)s(n,x)+p(1)s(n-1,x)
    +...+p(n-1)s(1,x)+p(n)s(0,x).  See A192744.
(2) For any polynomial p(x), let P(x)=(reduction of p(x)
    by q->s).  Then P(r)=p(r) for each zero r of
    q(x)-s(x).  In particular, if q(x)=x^2 and s(x)=x+1,
    then P(r)=p(r) if r=(1+sqrt(5))/2 (golden ratio) or
    r=(1-sqrt(5))/2.

			The first four Fibonacci polynomials and their reductions by x^2->x+1 are shown here:
F1(x)=1 -> 1 + 0x
F2(x)=x -> 0 + 1x
F3(x)=x^2+1 -> 2+1x
F4(x)=x^3+2x -> 1+4x
F5(x)=x^4+3x^2+1 -> (x+1)^2+3(x+1)+1 -> 6+6x.
From these, read A192232=(1,0,1,1,6,...) and A112576=(0,1,1,4,6,...).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-1).

Cf. A168561, A192233 - A192240, A192744.

Cf. A206296, A265398, A265399, A265752, A265753.

q[x_] := x + 1;
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),  x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[FixedPoint[Expand[#1 /. reductionRules] &, Fibonacci[n, x]], {n, 1, 40}];
Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}]
  (* A192232 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}]
(* A112576 *)
(* Peter J. C. Moses, Jun 25 2011 *)
LinearRecurrence[{1, 3, -1, -1}, {1, 0, 2, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)

Vec((1-x-x^2)/(1-x-3*x^2+x^3+x^4)+O(x^99)) \\ Charles R Greathouse IV, Jan 08 2013

Empirical G.f.: -x*(x^2+x-1)/(x^4+x^3-3*x^2-x+1). - Colin Barker, Sep 11 2012
The above formula is correct. - Charles R Greathouse IV, Jan 08 2013
a(n) = A265752(A206296(n)). - Antti Karttunen, Dec 15 2015
a(n) = A112576(n) -A112576(n-1) -A112576(n-2). - R. J. Mathar, Dec 16 2015

Example corrected by Clark Kimberling, Dec 18 2017

A112833 Number of domino tilings of a 3-pillow of order n.

Original entry on oeis.org

1, 2, 5, 20, 117, 1024, 13357, 259920, 7539421, 326177280, 21040987113, 2024032315968, 290333133984905, 62102074862600192, 19808204598680574457, 9421371079480456587520, 6682097668647718038428569, 7067102111711681259234263040, 11145503882824383823706372042925

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

nonn

A 3-pillow is also called an Aztec pillow. The 3-pillow of order n is a rotationally-symmetric region. It has a 2 X 2n central band of squares and then steps up from this band with steps of 3 horizontal squares to every 1 vertical square and steps down with steps of 1 horizontal square to every 1 vertical square.

a(n)^(1/n^2) tends to 1.2211384384439007690866503099... - Vaclav Kotesovec, May 19 2020

			The number of domino tilings of the 3-pillow of order 4 is 117=3^2*13.

Alois P. Heinz, Table of n, a(n) for n = 0..100
C. Hanusa, A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows, PhD Thesis, 2005, University of Washington, Seattle, USA.

This sequence breaks down as A112834^2 times A112835, where A112835 is not necessarily squarefree.
5-pillows: A112836-A112838; 7-pillows: A112839-A112841; 9-pillows: A112842-A112844.
Related to A071101 and A071100.

with(LinearAlgebra):
b:= proc(x, y, k) option remember;
      `if`(y>x or y Matrix(n, (i, j)-> b(i-1, i-1, j-1)):
R:= n-> Matrix(n, (i, j)-> `if`(i+j=n+1, 1, 0)):
a:= n-> Determinant(P(n)+R(n).(P(n)^(-1)).R(n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 26 2013

b[x_, y_, k_] := b[x, y, k] = If[y>x || yJean-François Alcover, Nov 08 2015, after Alois P. Heinz *)

A112835 Small-number statistic from the enumeration of domino tilings of a 3-pillow of order n.

Original entry on oeis.org

1, 2, 5, 5, 13, 16, 37, 45, 109, 130, 313, 377, 905, 1088, 2617, 3145, 7561, 9090, 21853, 26269, 63157, 75920, 182525, 219413, 527509, 634114, 1524529, 1832625, 4405969, 5296384, 12733489, 15306833, 36800465, 44237570, 106355317

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

Plotting A112835(n+2)/A112835(n) gives an intriguing damped sine curve.

			1 + 2*x + 5*x^2 + 5*x^3 + 13*x^4 + 16*x^5 + 37*x^6 + 45*x^7 + 109*x^8 + ...
The number of domino tilings of the 3-pillow of order 4 is 117=3^2*13. A112835(4)=13.

C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.

A D Mednykh, I A Mednykh, The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic, arXiv preprint arXiv:1711.00175, 2017. See Section 4.

A112833 breaks down as A112834^2 times A112835, where A112835 is not necessarily squarefree.
5-pillows: A112836-A112838; 7-pillows: A112839-A112841; 9-pillows: A112842-A112844.
Cf. A071100, A071101.

{a(n) = local(m = abs(n+3)); polcoeff( (x + x^2 - x^3 + x^5 - x^6 - x^7) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8)  + x * O(x^m), m)} /* Michael Somos, Dec 15 2011 */

a(2*n + 2) = A071100(n). a(2*n + 3) = A071101(n).
G.f.: (1 + x - x^2 + x^4 - x^5 - x^6) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8) = (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8). - Michael Somos, Dec 15 2011
a(-n) = a(-6 + n). a(-1) = a(-2) = 1, a(-3) = 0. a(n) = 2*a(n-2) + 2*a(n-4) + 2*a(n-6) - a(n-8). - Michael Somos, Dec 15 2011

A138573 a(n) = 2a(n-1) + 2a(n-2) + 2*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=2, a(3)=5.

Original entry on oeis.org

0, 1, 2, 5, 16, 45, 130, 377, 1088, 3145, 9090, 26269, 75920, 219413, 634114, 1832625, 5296384, 15306833, 44237570, 127848949, 369490320, 1067846845, 3086134658, 8919094697, 25776662080, 74495936025, 215297250946, 622220603405

Offset: 0

Benoit Cloitre, May 12 2008

nonn

This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m). - T. D. Noe, Dec 23 2008

Case P1 = 2, P2 = -4, Q = 1 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 04 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kunle Adegoke, Robert Frontczak, and Taras Goy, Binomial Fibonacci sums from Chebyshev polynomials, J. Integer Seq., Vol. 26 (2023), Art. 23.9.6. See page 4; Preprint, arXiv:2308.04567 [math.CO], 2023.
Peter Bala, Linear divisibility sequences and Chebyshev polynomials
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

Cf. A071101, A000045, A100047, A138574, A143056, A272665.

a:=[0,1,2,5];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]+2*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Sep 12 2018

seq(coeff(series((x*(1-x)*(x+1))/(1-2*x-2*x^2-2*x^3+x^4),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Sep 12 2018

Round@Table[(((GoldenRatio + Sqrt[GoldenRatio])^n + (GoldenRatio - Sqrt[GoldenRatio])^n)/2 - (-1)^n Cos[n ArcTan[Sqrt[GoldenRatio]]])/Sqrt[5], {n, 0, 20}] (* or *) LinearRecurrence[{2, 2, 2, -1}, {0, 1, 2, 5}, 20] (* Vladimir Reshetnikov, May 11 2016 *)
Table[Abs[Fibonacci[n, 1 + I]]^2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 05 2016 *)
CoefficientList[Series[-x*(x-1)*(1+x)/(1-2*x-2*x^2-2*x^3+x^4), {x, 0, 20}], x] (* Stefano Spezia, Sep 12 2018 *)

my(x='x+O('x^50)); concat([0], Vec(x*(1-x)*(1+x)/(1 -2*x -2*x^2 -2*x^3 +x^4))) \\ G. C. Greubel, Aug 08 2017

a(n) = round(w^n/2/sqrt(5)) where w = (1+r)/(1-r) = 2.89005363826396... and r = sqrt(sqrt(5)-2) = 0.485868271756...; for n >= 3, a(n) = A071101(n+3).
G.f.: -x*(x-1)*(1+x)/(1 - 2*x - 2*x^2 - 2*x^3 + x^4). - R. J. Mathar, Jun 03 2009
From Peter Bala, Mar 04 2014: (Start)
Define a Lucas sequence {U(n)} in the ring of Gaussian integers by the recurrence U(n) = (1 + i)*U(n-1) + U(n-2) with U(0) = 0 and U(1) = 1. Then a(n) = |U(n)|^2.
Let a, b denote the zeros of x^2 - (1 + i)*x - 1 and c, d denote the zeros of x^2 - (1 - i)*x - 1.
Then a(n) = (a^n - b^n)*(c^n - d^n)/((a - b)*(c - d)).
a(n) = (alpha(1)^n + beta(1)^n - alpha(2)^n - beta(2)^n)/(2*sqrt(5)), where alpha(1), beta(1) are the roots of x^2 - ( 1 + sqrt(5))*x + 1 = 0, and alpha(2), beta(2) are the roots of x^2 - (1 - sqrt(5))*x + 1 = 0.
The o.g.f. is the Hadamard product of the rational functions x/(1 - (1 + i)x - x^2) and x/(1 - (1 - i)x - x^2). (End)
From Peter Bala, Mar 24 2014: (Start)
a(n) = (1/sqrt(5))*(T(n,phi) - T(n,-1/phi)), where phi = 1/2*(1 + sqrt(5)) is the golden ratio and T(n,x) denotes the Chebyshev polynomial of the first kind. Compare with the Fibonacci numbers, A000045, whose terms are given by the Binet formula 1/sqrt(5)*( phi^n - (-1/phi)^n ).
a(n) = top right (or bottom left) entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 1; 1, 1]; the off-diagonal elements of M^n give the sequence of Fibonacci numbers. Bottom right entry of the matrix T(n, M) gives A138574. See the remarks in A100047 for the general connection between Chebyshev polynomials and linear divisibility sequences of the fourth order. (End)
a(n) = (((phi + sqrt(phi))^n + (phi - sqrt(phi))^n)/2 - (-1)^n * cos(n*arctan(sqrt(phi))))/sqrt(5), where phi=(1+sqrt(5))/2. - Vladimir Reshetnikov, May 11 2016
a(n) = A143056(n+1)^2 + A272665(n+1)^2. - Vladimir Reshetnikov, Oct 05 2016
Limit_{n -> oo} a(n)/a(n-1) = A318605. - A.H.M. Smeets, Sep 12 2018

A192234 a(n) = 2*(a(n-1) + a(n-2) + a(n-3)) - a(n-4) for n >= 4, with initial terms 0,1,0,1.

Original entry on oeis.org

0, 1, 0, 1, 4, 9, 28, 81, 232, 673, 1944, 5617, 16236, 46921, 135604, 391905, 1132624, 3273345, 9460144, 27340321, 79014996, 228357577, 659965644, 1907336113, 5512303672, 15930853281, 46041020488, 133061018769, 384553481404, 1111380188041

Offset: 0

Clark Kimberling, Jun 26 2011

With a different offset, constant term of the reduction of the n-th 1st-kind Chebyshev polynomial by x^2->x+1. See A192232.

Colin Barker, Table of n, a(n) for n = 0..1000
H. S. M. Coxeter, Loxodromic sequences of tangent spheres, Aequationes Mathematicae, 1.1-2 (1968): 104-121. See p. 112.
Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

Cf. A192232.

a:=[0,1,0,1];; for n in [5..40] do a[n]:=2*a[n-1]+2*a[n-2]+2*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Jul 29 2019

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1-2*x-x^2)/(1-2*x-2*x^2-2*x^3+x^4) )); // G. C. Greubel, Jul 29 2019

q[x_]:= x + 1;
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, ChebyshevT[n, x]]]], {n, 1, 40}];
Table[Coefficient[Part[t, n], x, 0], {n, 1, 40}]  (* A192234 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 40}]  (* A071101 *)
(* Peter J. C. Moses, Jun 25 2011 *)

a(n)=my(t=polchebyshev(n));while(poldegree(t)>1, t=substpol(t, x^2,x+1));subst(t,x,0) \\ Charles R Greathouse IV, Feb 09 2012

concat(0, Vec(x*(1-2*x-x^2)/(1-2*x-2*x^2-2*x^3+x^4) + O(x^40))) \\ Colin Barker, Sep 09 2018

(x*(1-2*x-x^2)/(1-2*x-2*x^2-2*x^3+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 29 2019

G.f.: x*(1 - 2*x - x^2) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4). - Colin Barker, Feb 09 2012 and Sep 09 2018

Entry revised (with new offset and initial terms) by N. J. A. Sloane, Sep 03 2018

A138574 a(n) = 2a(n-1) + 2a(n-2) + 2*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=3, a(3)=9, a(4)=25.

Original entry on oeis.org

0, 1, 3, 9, 25, 73, 211, 609, 1761, 5089, 14707, 42505, 122841, 355017, 1026019, 2965249, 8569729, 24766977, 71577891, 206863945, 597847897, 1727812489, 4993470771, 14431398369, 41707515361, 120536956513, 348358269715, 1006774084809, 2909631106713, 8408989965961

Offset: 0

Benoit Cloitre, May 12 2008

nonn

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

Cf. A071101.

I:=[0,1,3,9,25]; [n le 5 select I[n] else 2*Self(n-1)+2*Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Sep 14 2018

seq(coeff(series((x*(1+x+x^2-x^3))/(1-2*x-2*x^2-2*x^3+x^4),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Sep 12 2018

LinearRecurrence[{2, 2, 2, -1}, {0, 1, 3, 9, 25}, 50] (* G. C. Greubel, Aug 08 2017 *)
CoefficientList[Series[x*( 1+x+x^2-x^3 )/(1-2*x-2*x^2-2*x^3+x^4), {x, 0, 20}], x] (* Stefano Spezia, Sep 12 2018 *)

x='x+O('x^50); concat([0], Vec(x*(1 +x +x^2 -x^3)/(1 -2*x -2*x^2 -2*x^3 +x^4))) \\ G. C. Greubel, Aug 08 2017

a(n) = round(w^n*(1 + 1/sqrt(5))/4) where w = (1+r)/(1-r) = 2.89005363826396... and r = sqrt(sqrt(5)-2) = 0.485868271756... .
G.f.: x*( 1 + x + x^2 - x^3 ) / ( 1 - 2*x - 2*x^2 - 2*x^3 + x^4 ). - R. J. Mathar, Jun 29 2013
Lim_{n -> inf} a(n)/a(n-1) = A318605. - A.H.M. Smeets, Sep 12 2018

Terms corrected by Colin Barker, Jun 28 2013

A206625 Expansion of x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2x^2 - 2x^4 - 2*x^6 + x^8) in powers of x.

Original entry on oeis.org

0, 1, 1, 1, 2, 5, 5, 13, 16, 37, 45, 109, 130, 313, 377, 905, 1088, 2617, 3145, 7561, 9090, 21853, 26269, 63157, 75920, 182525, 219413, 527509, 634114, 1524529, 1832625, 4405969, 5296384, 12733489, 15306833, 36800465, 44237570, 106355317

Offset: 0

Michael Somos, Feb 10 2012

This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m).

			G.f. = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 5*x^6 + 13*x^7 + 16*x^8 + 37*x^9 + ...

J. A. Sjogren, Cycles and spanning trees. Math. Comput. Modelling 15, No.9, 87-102 (1991).

G. C. Greubel, Table of n, a(n) for n = 0..2500
J. A. Sjogren, Cycles and Spanning Trees, see equation (3.5)
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,2,0,2,0,2,0,-1).

Cf. A071100 (bisection), A071101 (bisection), A112835, A138573.

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x)*(1-x^2)*(1+x^3)/(1-2*x^2-2*x^4-2*x^6+x^8 ))); // G. C. Greubel, Aug 12 2018

CoefficientList[Series[x*(1+x)*(1-x^2)*(1+x^3)/(1-2*x^2-2*x^4-2*x^6+x^8 ), {x, 0, 50}], x] (* G. C. Greubel, Aug 12 2018 *)

{a(n) = my(m = abs(n)); polcoeff( x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8) + x * O(x^m), m)};

{a(n) = my(m = abs(n), v); v = polroots( Pol([ 1, 2, 4, 2, 1])); sqrtint( round( prod( k=1, 4, v[k]^m - 1, 2^(m%2) / 20)))};

G.f.: x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8).
a(n) = a(-n) = 2*a(n-2) + 2*a(n-4) + 2*a(n-6) - a(n-8) for all n in Z.
a(2*n + 5) = A071100(n).  a(2*n + 6) = A071101(n). a(n + 3) = A112835(n). a(2*n) = A138573(n).

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A192232 Constant term of the reduction of n-th Fibonacci polynomial by x^2 -> x+1. (See Comments.)

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A112833 Number of domino tilings of a 3-pillow of order n.

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A112835 Small-number statistic from the enumeration of domino tilings of a 3-pillow of order n.

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A138573 a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=2, a(3)=5.

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A192234 a(n) = 2*(a(n-1) + a(n-2) + a(n-3)) - a(n-4) for n >= 4, with initial terms 0,1,0,1.

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A138574 a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=3, a(3)=9, a(4)=25.

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A206625 Expansion of x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8) in powers of x.

A138573 a(n) = 2a(n-1) + 2a(n-2) + 2*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=2, a(3)=5.

A138574 a(n) = 2a(n-1) + 2a(n-2) + 2*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=3, a(3)=9, a(4)=25.

A206625 Expansion of x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2x^2 - 2x^4 - 2*x^6 + x^8) in powers of x.