A005178 -id:A005178 - cp's OEIS frontend

A049086 Number of tilings of 4 X 3n rectangle by 1 X 3 rectangles. Rotations and reflections are considered distinct tilings.

1, 3, 13, 57, 249, 1087, 4745, 20713, 90417, 394691, 1722917, 7520929, 32830585, 143313055, 625594449, 2730863665, 11920848033, 52037243619, 227154537661, 991581805481, 4328482658041, 18894822411423, 82480245888473, 360045244866137, 1571680309076689, 6860746056673507

Offset: 0

Allan C. Wechsler

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mudit Aggarwal and Samrith Ram, Generating functions for straight polyomino tilings of narrow rectangles, arXiv:2206.04437 [math.CO], 2022.
R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 19.
R. J. Mathar, Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices, arXiv:1406.7788 (2014), eq. (10)
Index entries for linear recurrences with constant coefficients, signature (5,-3,1).

Cf. A000930, A005178.

a[0]:=1:a[1]:=3:a[2]:=13: for n from 3 to 25 do a[n]:=5*a[n-1]-3*a[n-2]+a[n-3] od: seq(a[n],n=0..25); # Emeric Deutsch, Feb 15 2005
a := n -> hypergeom([(n+1)/2, n/2+1, -n], [1/3, 2/3], -8/27):
seq(simplify(a(n)), n=0..25); # Peter Luschny, Dec 09 2020

LinearRecurrence[{5,-3,1},{1,3,13},50] (* Vincenzo Librandi, Feb 18 2012 *)
CoefficientList[Series[(1-x)^2/(1-5x+3x^2-x^3), {x, 0, 40}], x] (* M. Poyraz Torcuk, Nov 06 2021 *)

a(n) = 5*a(n-1) - 3*a(n-2) + a(n-3).
a(n)/a(n-1) tends to 4.3652300134..., an eigenvalue of the matrix M and an inverse root of the polynomial x^3 - 3x^2 + 5x - 1. [a(n-2), a(n-1), a(n)] = M^n * [1 1 1], where M = the 3 X 3 matrix [ 5 -3 1 / 1 0 0 / 0 1 0]. E.g., a(3), a(4), a(5) = 57, 249, 1087. M^5 * [1 1 1] = [57, 249, 1087] - Gary W. Adamson, Apr 25 2004
G.f.: (1-x)^2/(1-5*x+3*x^2-x^3). - Colin Barker, Feb 03 2012
a(n) = Sum_{k=0..n} A109955(n,k)*2^k. - Philippe Deléham, Feb 18 2012
a(n) = hypergeom([(n+1)/2, n/2+1, -n], [1/3, 2/3], -8/27). - Peter Luschny, Dec 09 2020

More terms from Emeric Deutsch, Feb 15 2005

A028471 Number of perfect matchings (or domino tilings) in the graph P_9 X P_2n.

Original entry on oeis.org

1, 55, 6336, 817991, 108435745, 14479521761, 1937528668711, 259423766712000, 34741645659770711, 4652799879944138561, 623139489426439754945, 83456125990631342400791, 11177167872295392172767936, 1496943834332592837945956455, 200483802581126644843760725601

Offset: 0

Per H. Lundow

nonn

Sean A. Irvine, Table of n, a(n) for n = 0..150
J. L. Hock and R. B. McQuistan, A note on the occupational degeneracy for dimers on a saturated two-dimensional lattice space, Discrete Applied Mathematics, 1984, v.8, 101-104.
Per Hakan Lundow, Computation of matching polynomials and the number of 1-factors in polygraphs, Research report, No 12, 1996, Department of Math., Umea University, Sweden.
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
Index entries for linear recurrences with constant coefficients, signature (209, -11936, 274208, -3112032, 19456019, -70651107, 152325888, -196664896, 152325888, -70651107, 19456019, -3112032, 274208, -11936, 209, -1).

Cf. A000045, A001835, A005178, A003775, A028468, A028469, A028470.

Row 9 of array A099390.

T[?OddQ, ?OddQ] = 0;
T[m_, n_] := Product[2(2+Cos[2 j Pi/(m+1)]+Cos[2 k Pi/(n+1)]), {k, 1, n/2}, {j, 1, m/2}];
a[n_] := T[2n, 9] // Round;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 28 2022 *)

{a(n) = sqrtint(polresultant(polchebyshev(2*n, 2, x/2), polchebyshev(9, 2, I*x/2)))} \\ Seiichi Manyama, Apr 13 2020

a(n) = 209*a(n-1) - 11936*a(n-2) + 274208*a(n-3) - 3112032*a(n-4) + 19456019*a(n-5) - 70651107*a(n-6) + 152325888*a(n-7) - 196664896*a(n-8) + 152325888*a(n-9) - 70651107*a(n-10) + 19456019*a(n-11) - 3112032*a(n-12) + 274208*a(n-13) - 11936*a(n-14) + 209*a(n-15) - a(n-16). - Jay Anderson (horndude77(AT)gmail.com), Apr 07 2007

G.f.: (1 - 154x + 6777x^2 - 123961x^3 + 1132714x^4 - 5684515x^5 + 16401668x^6 - 27757938x^7 + 27757938*x^8 - 16401668x^9 + 5684515x^10 - 1132714x^11 + 123961x^12 -6777x^13 + 154x^14 - x^15)/(1 - 209x + 11936x^2 - 274208x^3 + 3112032x^4 - 19456019x^5 + 70651107x^6 - 152325888x^7 + 196664896x^8 - 152325888x^9 + 70651107x^10 -19456019x^11 + 3112032x^12 - 274208x^13 + 11936x^14 - 209x^15 + x^16). - Sergey Perepechko, Nov 23 2012

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A033507 Number of matchings in graph P_{4} X P_{n}.

Original entry on oeis.org

1, 5, 71, 823, 10012, 120465, 1453535, 17525619, 211351945, 2548684656, 30734932553, 370635224561, 4469527322891, 53898461609719, 649966808093412, 7838012982224913, 94519361817920403, 1139818186429110279, 13745178487929574337, 165754445655292452448

Offset: 0

Per H. Lundow

nonn

			a(1) = 5: the graph is
. o-o-o-o
and the five matchings are
. o o o o
. o-o o o
. o o-o o
. o o o-o
. o-o o-o

H. Hosoya and A. Motoyama, An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two- and three-dimensional rectangular and torus lattices, J. Math. Phys., 26(1985), 157-167.

Alois P. Heinz, Table of n, a(n) for n = 0..500
David Friedhelm Kind, The Gunport Problem: An Evolutionary Approach, De Montfort University (Leicester, UK, 2020).
Per Hakan Lundow, Computation of matching polynomials and the number of 1-factors in polygraphs, Research report, No 12, 1996, Department of Math., Umea University, Sweden.
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
Index entries for linear recurrences with constant coefficients, signature (9,41,-41,-111,91,29,-23,-1,1).

Column 4 of triangle A210662. Row sums of A100265.
For perfect matchings see A005178.
Cf. A033508-A033511.
Bisection (even part) gives A260034.

a:=[1,5,71,823,10012,120465, 1453535,17525619,211351945];; for n in [10..30] do a[n]:=9*a[n-1]+41*a[n-2]-41*a[n-3]-111*a[n-4]+91*a[n-5] +29*a[n-6]-23*a[n-7]-a[n-8]+a[n-9]; od; a; # G. C. Greubel, Oct 26 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)*(1 -3*x-18*x^2+2*x^3+12*x^4+x^5-x^6)/(1-9*x-41*x^2+41*x^3+111*x^4-91*x^5 -29*x^6+23*x^7+x^8-x^9) )); // G. C. Greubel, Oct 26 2019

a:=array(0..20,[1, 5, 71, 823, 10012, 120465, 1453535, 17525619, 211351945]):
for j from 9 to 20 do
  a[j]:=9*a[j-1]+41*a[j-2]-41*a[j-3]-111*a[j-4]+91*a[j-5]+
        29*a[j-6]-23*a[j-7]-a[j-8]+a[j-9]
od:
convert(a,list);
# Sergey Perepechko, Apr 24 2013

LinearRecurrence[{9,41,-41,-111,91,29,-23,-1,1},{1,5,71,823,10012,120465, 1453535,17525619,211351945},30] (* Harvey P. Dale, Mar 27 2015 *)

my(x='x+O('x^30)); Vec((1-x)*(1 -3*x-18*x^2+2*x^3+12*x^4+x^5-x^6)/(1-9*x-41*x^2+41*x^3+111*x^4-91*x^5 -29*x^6+23*x^7+x^8-x^9)) \\ G. C. Greubel, Oct 26 2019

def A033507_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)*(1 -3*x-18*x^2+2*x^3+12*x^4+x^5-x^6)/(1-9*x-41*x^2+41*x^3+111*x^4-91*x^5 -29*x^6+23*x^7+x^8-x^9) ).list()
A033507_list(30) # G. C. Greubel, Oct 26 2019

From Sergey Perepechko, Apr 24 2013: (Start)
a(n) = 9*a(n-1) +41*a(n-2) -41*a(n-3) -111*a(n-4) +91*a(n-5) +29*a(n-6) -23*a(n-7) -a(n-8) +a(n-9).
G.f.: (1-x) * (1 -3*x -18*x^2 +2*x^3 +12*x^4 +x^5 -x^6) / (1 -9*x -41*x^2 +41*x^3 +111*x^4 -91*x^5 -29*x^6 +23*x^7 +x^8 -x^9). (End)

Edited by N. J. A. Sloane, Nov 15 2009

A188899 Third row of array in A187617.

Original entry on oeis.org

1, 5, 36, 281, 2245, 18061, 145601, 1174500, 9475901, 76455961, 616891945, 4977472781, 40161441636, 324048393905, 2614631600701, 21096536145301, 170220478472105, 1373448758774436, 11081871650713781, 89415697915538545, 721463601671126161, 5821234309893001301, 46969478172465070500, 378980086070257592201, 3057856106268358639861

Offset: 0

N. J. A. Sloane, Apr 13 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400
N. Allegra, Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics, arXiv:1410.4131 [cond-mat.stat-mech], 2014. See Table 1.
Index entries for linear recurrences with constant coefficients, signature (11,-25,11,-1).

Bisection (odd part) of A005178. - Alois P. Heinz, Oct 28 2012

ft:=(m,n)->
2^(m*n/2)*mul( mul(
(cos(Pi*i/(n+1))^2+cos(Pi*j/(m+1))^2), j=1..m/2), i=1..n/2);
gt:=(m,n)->round(evalf(ft(m,n),300));
tt:=[seq(gt(4,2*n),n=0..10)];
# second Maple program:
a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|11|-25|11>>^n.
        <<1, 5, 36, 281>>)[1, 1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 28 2012

LinearRecurrence[{11, -25, 11, -1}, {1, 5, 36, 281}, 25] (* Jean-François Alcover, Jun 17 2018 *)

x='x+O('x^200); Vec((1-x)*(x^2-5*x+1)/(x^4-11*x^3+25*x^2-11*x+1)) \\ Altug Alkan, Mar 23 2016

G.f.: (1-x)*(x^2-5*x+1)/(x^4-11*x^3+25*x^2-11*x+1). - Alois P. Heinz, Oct 28 2012

A220614 T(n,k)=Number of ways to reciprocally link elements of an nXk array either to themselves or to exactly two horizontal and vertical neighbors, without consecutive collinear links.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 5, 1, 1, 8, 11, 11, 8, 1, 1, 13, 21, 36, 21, 13, 1, 1, 21, 43, 95, 95, 43, 21, 1, 1, 34, 85, 281, 324, 281, 85, 34, 1, 1, 55, 171, 781, 1277, 1277, 781, 171, 55, 1, 1, 89, 341, 2245, 4673, 7041, 4673, 2245, 341, 89, 1, 1, 144, 683, 6336

Offset: 1

R. H. Hardin Dec 17 2012

Table starts
.1...1....1......1.......1.........1..........1............1.............1
.1...2....3......5.......8........13.........21...........34............55
.1...3....5.....11......21........43.........85..........171...........341
.1...5...11.....36......95.......281........781.........2245..........6336
.1...8...21.....95.....324......1277.......4673........17768.........66245
.1..13...43....281....1277......7041......34981.......183920........937663
.1..21...85....781....4673.....34981.....233265......1653229......11354857
.1..34..171...2245...17768....183920....1653229.....16038658.....149429377
.1..55..341...6336...66245....937663...11354857....149429377....1884307184
.1..89..683..18061..249499...4861869...79290645...1423339849...24310532213
.1.144.1365..51205..934816..24977087..548972737..13397669320..309983079825
.1.233.2731.145601.3511945.128971605.3817993565.126938044289.3977585437731

			Some solutions for n=3 k=4 0=self 2=n 4=w 6=e 8=s (reciprocal directions total 10)
.00.00.68.48...68.48.68.48...68.48.00.00...00.00.00.00...00.68.48.00
.00.00.26.24...26.24.26.24...26.24.00.00...00.00.00.00...00.26.24.00
.00.00.00.00...00.00.00.00...00.00.00.00...00.00.00.00...00.00.00.00

R. H. Hardin, Table of n, a(n) for n = 1..544

Column 2 is A000045(n+1)
Column 3 is A001045(n+1)
Column 4 is A005178(n+1)
Diagonal is A086832

A100265 Triangle read by rows: T(n,k) is the number of k-matchings in the P_4 X P_n lattice graph.

Original entry on oeis.org

1, 1, 3, 1, 1, 10, 29, 26, 5, 1, 17, 102, 267, 302, 123, 11, 1, 24, 224, 1044, 2593, 3388, 2150, 552, 36, 1, 31, 395, 2696, 10769, 25835, 36771, 29580, 12181, 2111, 95, 1, 38, 615, 5566, 31106, 111882, 261965, 395184, 372109, 206206, 60730, 7852, 281, 1, 45

Offset: 0

Emeric Deutsch, Dec 28 2004

Row sums yield A033507. T(n,2n) yields A005178.

			T(2,4)=5 because in the graph P_4 X P_2 with vertices a(0,0), b(0,1), c(0,2),
d(0,3),a'(1,0),b'(1,1),c'(1,2),d'(1,3), we have the following 4-matchings
{aa',bb',cc',dd'},{aa',bb',cd,c'd'},{ab,a'b',cc',dd'},{ab,a'b',cd,c'd'} and {aa',bc,b'c',dd'} (perfect matchings, of course).
Triangle starts:
1;
1, 3, 1;
1, 10, 29, 26, 5;
1, 17, 102, 267, 302, 123, 11;
1, 24, 224, 1044, 2593, 3388, 2150, 552, 36;

H. Hosoya and A. Motoyama, An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two- and three-dimensional rectangular and torus lattices, J. Math. Physics 26 (1985) 157-167 (eq. (46) and Table VI).

Cf. A033507, A005178.

G:= - (1 + 3*z^3*t^4 + 11*z^3*t^5 + 6*z^3*t^6 - 2*z*t - 2*z*t^2 - 3*z^2*t^2 - 9*z^2*t^3 - 3*z^2*t^4 + z^7*t^14 + 3*z^4*t^6 + 5*z^4*t^7 + 2*z^4*t^8 - 3*z^5*t^8 - 3*z^5*t^9 - 5*z^5*t^10 - 2*z^6*t^11)/( - 1 + z + t^18*z^9 + z^3*t^2 + 4*z^3*t^3 - 4*z^3*t^4 - 27*z^3*t^5 - 15*z^3*t^6 + 5*z*t + 3*z*t^2 + 2*z^2*t + 13*z^2*t^2 + 21*z^2*t^3 + 5*z^2*t^4 - 2*z^7*t^11 - 3*z^7*t^12 - 9*z^7*t^13 - 9*z^7*t^14 - 3*z^4*t^4 - 18*z^4*t^5 - 41*z^4*t^6 - 40*z^4*t^7 - 9*z^4*t^8 - z^8*t^14 - z^8*t^16 + z^8*t^15 + 3*z^5*t^6 + 14*z^5*t^7 + 29*z^5*t^8 + 24*z^5*t^9 + 21*z^5*t^10 - z^6*t^8 + 6*z^6*t^10 + 19*z^6*t^11 + 5*z^6*t^12):
Gser:=simplify(series(G,z=0,11)): P[0]:=1: for n from 1 to 8 do P[n]:=coeff(Gser,z^n) od:for n from 0 to 8 do seq(coeff(t*P[n],t^k),k=1..2*n + 1) od; # yields sequence in triangular form

G.f.= (1 - zt^2)(z^6*t^12 + z^5*t^10 - 2z^5*t^9 - 4z^4*t^8 - 5z^4*t^7 - 3z^4*t^6 - 2z^3*t^6 + 4z^2*t^4 + 11z^2*t^3 + 3z^2*t^2 + zt^2 + 2zt - 1)/( - 1 + z + t^18*z^9 + z^3*t^2 + 4z^3*t^3 - 4z^3*t^4 - 27z^3*t^5 - 15z^3*t^6 + 5z*t + 3zt^2 + 2tz^2 + 13z^2*t^2 + 21z^2*t^3 + 5z^2*t^4 - 2z^7*t^11 - 3z^7*t^12 - 9z^7*t^13 - 9z^7*t^14 - 3z^4*t^4 - 18z^4*t^5 - 41z^4*t^6 - 40z^4*t^7 - 9z^4*t^8 - z^8*t^14 - z^8*t^16 + z^8*t^15 + 3z^5*t^6 + 14z^5*t^7 + 29z^5*t^8 + 24z^5*t^9 + 21z^5*t^10 - z^6*t^8 + 6z^6*t^10 + 19z^6*t^11 + 5z^6*t^12).

The row generating polynomials A[n] satisfy A[n] = (5t + 1 + 3t^2)A[n - 1] + (13t^2 + 21t^3 + 5t^4 + 2t)A[n - 2] + ( - 27t^5 - 15t^6 + t^2 - 4t^4 + 4t^3)A[n - 3] + ( - 40t^7 - 9t^8 - 41t^6 - 18t^5 - 3t^4)A[n - 4] + (29t^8 + 21t^10 + 3t^6 + 24t^9 + 14t^7)A[n - 5] + (6t^10 + 5t^12 - t^8 + 19t^11)A[n - 6] + ( - 9t^13 - 2t^11 - 3t^12 - 9t^14)A[n - 7] + ( - t^16 - t^14 + t^15)A[n - 8] + t^18*A[n - 9]

A220123 Number of tilings of a 4 X n rectangle using integer-sided rectangular tiles of area 4.

Original entry on oeis.org

1, 1, 2, 3, 9, 16, 35, 65, 143, 281, 590, 1174, 2440, 4925, 10142, 20563, 42178, 85819, 175632, 357875, 731536, 1491966, 3047879, 6218844, 12699982, 25919176, 52922491, 108022099, 220541999, 450186874, 919074255, 1876149465, 3830134125, 7818778884, 15961716918

Offset: 0

Alois P. Heinz, Dec 05 2012

			a(3) = 3, because there are 3 tilings of a 4 X 3 rectangle using integer-sided rectangular tiles of area 4:
._._._.   ._.___.   .___._.
| | | |   | |   |   |   | |
| | | |   | |___|   |___| |
| | | |   | |   |   |   | |
|_|_|_|   |_|___|   |___|_|

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Caleb Wagner, Number of tilings of a 4 X n rectangle using integer sided rectangular tiles of area 4, Nov 2013
Index entries for linear recurrences with constant coefficients, signature (1,1,0,5,-1,1,0,-1).

Column k=4 of A220122. Cf. A005178.

gf:= -(x-1)*(x+1)*(x^2+1)/(x^8-x^6+x^5-5*x^4-x^2-x+1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..50);

G.f.: -(x-1)*(x+1)*(x^2+1) / (x^8 - x^6 + x^5 - 5*x^4 - x^2 - x + 1).
a(n) = a(n-1) + a(n-2) + 5*a(n-4) - a(n-5) + a(n-6) - a(n-8). - Caleb Wagner, Nov 06 2013
a(2*n+1) = Sum_{k=0..n} A005178(k+1)*a(2*n-2*k). - Shravan Haribalaraman, Aug 29 2022

A362298 Number of tilings of a 4 X n rectangle using dominos and 2 X 2 right triangles.

Original entry on oeis.org

1, 1, 19, 55, 472, 2023, 13249, 66325, 392299, 2088856, 11877025, 64803157, 362823607, 1998759703, 11123273896, 61509329983, 341492705365, 1891193243713, 10489893539203, 58127214942544, 322296397820593, 1786338231961609, 9903234373856059, 54893955008138983

Offset: 0

Gerhard Kirchner, Apr 19 2023

Triangles only occur as pairs forming 2 X 2 squares. For program code and additional details, see A362297.

			a(2) = 19.
Partitions of a 2 X 2 square (triangles or dominos):
   ___    ___    ___    ___
  |  /|  |\  |  |___|  | | |
  |/__|  |__\|  |___|  |_|_|
       2t            2d
   ___ ___    ___ ___    ___ ___    _ ___ _    _______
  |2t |2t |  |2t |2d |  |2d |2t |  | |2t | |  |only d |
  |___|___|  |___|___|  |___|___|  |_|___|_|  |_______|
    4 ways +   4 ways +  4 ways  +   2 ways +  5 ways  = 19 ways
Only dominos: A005178(3) = 5.

Index entries for linear recurrences with constant coefficients, signature (4,18,-48,-42,99).

Column k=2 of A362297.

Cf. A351322, A352432, A352433, A006130, A362299.

LinearRecurrence[{4,18,-48,-42,99},{1,1,19,55,472},24] (* Stefano Spezia, Apr 20 2023 *)

a(n) = 4*a(n-1) + 18*a(n-2) - 48*a(n-3) - 42*a(n-4) + 99*a(n-5).

G.f.: (9*x^3-3*x^2-3*x+1)/(-99*x^5+42*x^4+48*x^3-18*x^2-4*x+1).

A129113 Expansion of x^3 / (1 - x - 5*x^2 - x^3 + x^4).

Original entry on oeis.org

0, 0, 0, 1, 1, 6, 12, 42, 107, 323, 888, 2568, 7224, 20629, 58429, 166230, 471780, 1340730, 3807431, 10816631, 30722736, 87272592, 247895472, 704164537, 2000191753, 5681637318, 16138865148, 45843078954, 130218850259

Offset: 0

Roger L. Bagula, Jun 07 2007

Li Zhou et al., Tiling 4-Rowed Rectangles with Dominoes: Problem 11187, American Mathematical Monthly, vol. 114, no. 6, 2007 (pp. 554-556).
Index entries for linear recurrences with constant coefficients, signature (1,5,1,-1).

Cf. A005178.

f[1] = f[2] = f[3] = 0; f[4] = 1; f[n_] := f[n] = f[n - 1] + 5f[n - 2] + f[n - 3] - f[n - 4]; Array[f, 29] (* or *) LinearRecurrence[{1, 5, 1, -1}, {0, 0, 0, 1}, 29] (* or *) gf = x^3/(1 - x - 5 x^2 - x^3 + x^4); CoefficientList[ Series[gf, {x, 0, 28}], x]

concat(vector(3), Vec(x^3/(1-x-5*x^2-x^3+x^4) + O(x^30))) \\ Michel Marcus, Nov 19 2017

a(n) = a(n - 1) + 5*a(n - 2) + a(n - 3) - a(n - 4).

a(n) = ((-b + c - e - g + i)*(1 + s + k)^n + (b + d - f + h - j)*(1 - s + l)^n + (b - d + f - h + j)*(1 - s - l)^n + (-b - c + e + g - i)*(1 + s - k)^n)/(5800*4^n), with b = 100*s, c = 1015*k, d = 145*sqrt(10*(7 + s)), e = 245*sqrt(58*(7 + s)), f = 75*sqrt(290*(7 + s)), g = 1914*sqrt(119 + 22*s), h = 98*sqrt(145*(119 + 22*s)), i = 382*sqrt(3451 + 638*s), j = 406*sqrt(595 + 110*s), k = sqrt(2*(7 + s)), l = sqrt(2*(7 - s)), s = sqrt(29). - Tim Monahan, Sep 09 2011; modified by Robert G. Wilson v, Sep 26 2011

A171064 G.f.: -x(x-1)(1+x)/(1-x-7*x^2-x^3+x^4).

Original entry on oeis.org

0, 1, 1, 7, 15, 64, 175, 631, 1905, 6433, 20224, 66529, 212625, 692119, 2226799, 7217728, 23284815, 75343591, 243328225, 786800449, 2542156800, 8217744577, 26556314401, 85835882791, 277405671375, 896595420736, 2897714688751

Offset: 0

R. J. Mathar, at the request of R. K. Guy, Sep 03 2010

The member k=7 of a family of sequences starting 0,1,1,k with recurrence a(n) = a(n-1)+k*a(n-2)+a(n-3)-a(n-4).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hugh Williams, R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277
Index entries for linear recurrences with constant coefficients, signature (1,7,1,-1).

Cf. A116201 (k=1), A105309 (k=2), A152090 (k=3), A007598 (k=4), A005178 (k=5), A003757 (k=6).

I:=[0, 1, 1, 7]; [n le 4 select I[n] else Self(n-1) + 7*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 19 2012

CoefficientList[Series[-x*(x - 1)*(1 + x)/(1 - x - 7*x^2 - x^3 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)
LinearRecurrence[{1,7,1,-1},{0,1,1,7},30] (* Harvey P. Dale, Nov 15 2020 *)

a(n) = +a(n-1) +7*a(n-2) +a(n-3) -a(n-4).

The roots (r1..r4) of the characteristic polynomials for this "family" of sequences have the following form (not simplified) for k= 1,2,3,4,5,6.... r1=(sqrt(4*k+10+2*sqrt(4*k+9))+sqrt(4*k-6+2*sqrt(4*k+9)))/4. r2=(sqrt(4*k+10+2*sqrt(4*k+9))-sqrt(4*k-6+2*sqrt(4*k+9)))/4. r3=(-sqrt(4*k+10-2*sqrt(4*k+9))-sqrt(4*k-6-2*sqrt(4*k+9)))/4. r4=(-sqrt(4*k+10-2*sqrt(4*k+9))+sqrt(4*k-6-2*sqrt(4*k+9)))/4. For k=1,2,3, r3 and r4 are complex . Closed-form (not simplified) is as follows for all k (note:for k1-k3 set r3 and r4 =0 and round a(n) to nearest integer): a(n)=sqrt(4*k+9)/(4*k+9)*(((r1)^n+(r2)^n)-((r3)^n+(r4)^n)). [Tim Monahan, Sep 17 2011]

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