A046741 -id:A046741 - cp's OEIS frontend

A062127 Seventh column of A046741.

13, 223, 1577, 7018, 23431, 64316, 153190, 327718, 644573, 1185025, 2061259, 3423422, 5467399, 8443318, 12664784, 18518842, 26476669, 37104995, 51078253, 69191458, 92373815, 121703056, 158420506, 203946878, 259898797, 328106053

Offset: 0

Vladeta Jovovic, Jun 04 2001

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. dumbbells: A002940, A002941, A002889, A046741, A055608, A062123-A062127.

List([0..40], n -> (81*n^6 +567*n^5 +2205*n^4 +4545*n^3 +5674*n^2 +3728*n +1040)/80); # G. C. Greubel, Jan 31 2019

[(81*n^6 +567*n^5 +2205*n^4 +4545*n^3 +5674*n^2 +3728*n +1040)/80: n in [0..40]]; // G. C. Greubel, Jan 31 2019

Table[(81*n^6 +567*n^5 +2205*n^4 +4545*n^3 +5674*n^2 +3728*n +1040)/80, {n, 0, 40}] (* G. C. Greubel, Jan 31 2019 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{13,223,1577,7018,23431,64316,153190},30] (* Harvey P. Dale, Jun 07 2022 *)

vector(40, n, n--; (81*n^6 +567*n^5 +2205*n^4 +4545*n^3 +5674*n^2 +3728*n +1040)/80) \\ G. C. Greubel, Jan 31 2019

[(81*n^6 +567*n^5 +2205*n^4 +4545*n^3 +5674*n^2 +3728*n +1040)/80 for n in range(40)] # G. C. Greubel, Jan 31 2019

G.f.: (2*x^6 + 14*x^5 + 72*x^4 + 207*x^3 + 289*x^2 + 132*x + 13)/(1-x)^7. Generally, g.f. for k-th column of A046741 is coefficient of y^k in expansion of (1-y)/((1-y-y^2)*(1-y)-(1+y)*x).
From G. C. Greubel, Jan 31 2019: (Start)
a(n) = (81*n^6 + 567*n^5 + 2205*n^4 + 4545*n^3 + 5674*n^2 + 3728*n + 1040)/80.
E.g.f.: (1040 + 16800*x + 45760*x^2 + 39240*x^3 + 13140*x^4 + 1782*x^5 + 81*x^6)*exp(x)/80. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001

A062124 Fourth column of A046741.

Original entry on oeis.org

3, 26, 94, 234, 473, 838, 1356, 2054, 2959, 4098, 5498, 7186, 9189, 11534, 14248, 17358, 20891, 24874, 29334, 34298, 39793, 45846, 52484, 59734, 67623, 76178, 85426, 95394, 106109, 117598, 129888, 143006, 156979, 171834, 187598, 204298

Offset: 0

Vladeta Jovovic, Jun 04 2001

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. dumbbells: A002940, A002941, A002889, A046741, A055608, A062123-A062127.

List([0..40], n -> (6+19*n+18*n^2+9*n^3)/2); # G. C. Greubel, Jan 31 2019

[(6+19*n+18*n^2+9*n^3)/2: n in [0..40]]; // G. C. Greubel, Jan 31 2019

Table[(6+19*n+18*n^2+9*n^3)/2, {n,0,40}] (* G. C. Greubel, Jan 31 2019 *)
LinearRecurrence[{4,-6,4,-1},{3,26,94,234},40] (* Harvey P. Dale, Feb 20 2022 *)

vector(40, n, n--; (6+19*n+18*n^2+9*n^3)/2) \\ G. C. Greubel, Jan 31 2019

[(6+19*n+18*n^2+9*n^3)/2 for n in range(40)] # G. C. Greubel, Jan 31 2019

G.f.: (3 + 14*x + 8*x^2 + 2*x^3)/(1-x)^4. Generally, g.f. for k-th column of A046741 is coefficient of y^k in expansion of (1-y)/((1-y-y^2)*(1-y) - (1+y)*x).
From G. C. Greubel, Jan 31 2019: (Start)
a(n) = (6 + 19*n + 18*n^2 + 9*n^3)/2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
E.g.f.: (6 + 46*x + 45*x^2 + 9*x^3)*exp(x)/2. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001

A062125 Fifth column of A046741.

Original entry on oeis.org

5, 56, 263, 815, 1982, 4115, 7646, 13088, 21035, 32162, 47225, 67061, 92588, 124805, 164792, 213710, 272801, 343388, 426875, 524747, 638570, 769991, 920738, 1092620, 1287527, 1507430, 1754381, 2030513, 2338040, 2679257, 3056540

Offset: 0

Vladeta Jovovic, Jun 04 2001

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. dumbbells: A002940, A002941, A002889, A046741, A055608, A062123-A062127.

List([0..40], n -> (40+126*n+165*n^2+90*n^3+27*n^4)/8); # G. C. Greubel, Jan 31 2019

[(40+126*n+165*n^2+90*n^3+27*n^4)/8: n in [0..40]]; // G. C. Greubel, Jan 31 2019

LinearRecurrence[{5, -10, 10, -5, 1}, {5, 56, 263, 815, 1982}, 31] (* or *) CoefficientList[Series[(5+33x^2+10x^3+31x+2x^4)/(1-x)^5,{x,0,30}],x] (* Harvey P. Dale, Dec 21 2011 *)
Table[(40+126*n+165*n^2+90*n^3+27*n^4)/8, {n,0,40}] (* G. C. Greubel, Jan 31 2019 *)

vector(40, n, n--; (40+126*n+165*n^2+90*n^3+27*n^4)/8) \\ G. C. Greubel, Jan 31 2019

[(40+126*n+165*n^2+90*n^3+27*n^4)/8 for n in range(40)] # G. C. Greubel, Jan 31 2019

G.f.: (5 + 33*x^2 + 10*x^3 + 31*x + 2*x^4)/(1-x)^5. Generally, g.f. for k-th column of A046741 is coefficient of y^k in expansion of (1-y)/((1-y-y^2)*(1-y)-(1+y)*x).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), where a(0)=5, a(1)=56, a(2)=263, a(3)=815, a(4)=1982. - Harvey P. Dale, Dec 21 2011
From G. C. Greubel, Jan 31 2019: (Start)
a(n) = (40 + 126*n + 165*n^2 + 90*n^3 + 27*n^4)/8.
E.g.f.: (40 + 408*x + 624*x^2 + 252*x^3 + 27*x^4)*exp(x)/8. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001

A062126 Sixth column of A046741.

Original entry on oeis.org

8, 114, 667, 2504, 7191, 17266, 36482, 70050, 124882, 209834, 335949, 516700, 768233, 1109610, 1563052, 2154182, 2912268, 3870466, 5066063, 6540720, 8340715, 10517186, 13126374, 16229866, 19894838, 24194298, 29207329

Offset: 0

Vladeta Jovovic, Jun 04 2001

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.3.14).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. dumbbells: A002940, A002941, A002889, A046741, A055608, A062123-A062127.

List([0..40], n -> (320+1114*n+1515*n^2+1125*n^3+405*n^4+81*n^5 )/40); # G. C. Greubel, Jan 31 2019

[(320+1114*n+1515*n^2+1125*n^3+405*n^4+81*n^5)/40: n in [0..40]]; // G. C. Greubel, Jan 31 2019

Table[(320+1114*n+1515*n^2+1125*n^3+405*n^4+81*n^5)/40, {n, 0, 40}] (* G. C. Greubel, Jan 31 2019 *)

vector(40, n, n--; (320+1114*n+1515*n^2+1125*n^3+405*n^4 + 81*n^5)/40) \\ G. C. Greubel, Jan 31 2019

[(320+1114*n+1515*n^2+1125*n^3+405*n^4+81*n^5)/40 for n in range(40)] # G. C. Greubel, Jan 31 2019

G.f.: (x+2)*(2*x^4 + 8*x^3 + 36*x^2 + 31*x + 4)/(1-x)^6. Generally, g.f. for k-th column of A046741 is coefficient of y^k in expansion of (1-y)/((1 - y - y^2)*(1-y) - (1+y)*x).
From G. C. Greubel, Jan 31 2019: (Start)
a(n) = (320 + 1114*n + 1515*n^2 + 1125*n^3 + 405*n^4 + 81*n^5)/40.
E.g.f.: (320 + 4240*x + 8940*x^2 + 5580*x^3 + 1215*x^4 + 81*x^5)*exp(x)/40. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001

A039962 Duplicate of A046741.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 7, 11, 3, 1, 10, 29, 26, 5, 1, 13, 56, 94, 56, 8, 1, 16, 92, 234, 263, 114

Offset: 0

dead

A030186 a(n) = 3*a(n-1) + a(n-2) - a(n-3) for n >= 3, a(0)=1, a(1)=2, a(2)=7.

Original entry on oeis.org

1, 2, 7, 22, 71, 228, 733, 2356, 7573, 24342, 78243, 251498, 808395, 2598440, 8352217, 26846696, 86293865, 277376074, 891575391, 2865808382, 9211624463, 29609106380, 95173135221, 305916887580, 983314691581, 3160687827102, 10159461285307, 32655756991442

Offset: 0

Ottavio D'Antona (dantona(AT)dsi.unimi.it)

Number of matchings in ladder graph L_n = P_2 X P_n.
Cycle-path coverings of a family of digraphs.
a(n+1) = Fibonacci(n+1)^2 + Sum_{k=0..n} Fibonacci(k)^2*a(n-k) (with the offset convention Fibonacci(2)=2). - Barry Cipra, Jun 11 2003
Equivalently, ways of paving a 2 X n grid cells using only singletons and dominoes. - Lekraj Beedassy, Mar 25 2005
It is easy to see that the g.f. for indecomposable tilings (pavings) i.e. those that cannot be split vertically into smaller tilings, is g=2x+3x^2+2x^3+2x^4+2x^5+...=x(2+x-x^2)/(1-x); then G.f.=1/(1-g)=(1-x)/(1-3x-x^2+x^3). - Emeric Deutsch, Oct 16 2006
Row sums of A046741. - Emeric Deutsch, Oct 16 2006
Equals binomial transform of A156096. - Gary W. Adamson, Feb 03 2009
a(n) = Lucas(2n) + Sum_{k=2..n-1} Fibonacci(2k-1)*a(n-k). This relationship can be proven by a visual proof using the idea of tiling a 2 X n board with only singletons and dominoes while conditioning on where the vertical dominoes first appear. If there are no vertical dominoes positioned at lengths 2 through n-1, there will be Lucas(2n) ways to tile the board since a complete tour around the board will be made possible. If the first vertical domino appears at length k (where 2 <= k <= n-1) there will be Fibonacci(2k-1)*a(n-k) ways to tile the board. - Rana Ürek, Jun 25 2018

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 25.
J. D. E. Konhauser et al., Which Way Did The Bicycle Go? Problem 140 "Count The Tilings" pp. 42; 180-1 Dolciani Math. Exp. No. 18 MAA Washington DC 1996.

Indranil Ghosh, Table of n, a(n) for n = 0..1968 (terms 0..200 from T. D. Noe)
Ottavio M. D'Antona and Emanuele Munarini, The Cycle-path Indicator Polynomial of a Digraph, Advances in Applied Mathematics 25 (2000), 41-56.
Roger C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15 (1974), 214-216.
Svenja Huntemann and Neil A. McKay, Counting Domineering Positions, arXiv:1909.12419 [math.CO], 2019.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 473
Matt Katz and Catherine Stenson, Tiling a (2xn)-Board with Squares and Dominoes, JIS 12 (2009) 09.2.2, Table 1.
David Friedhelm Kind, The Gunport Problem: An Evolutionary Approach, De Montfort University (Leicester, UK, 2020).
Richard M. Low and Ardak Kapbasov, Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 2.
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.
Richmond B. McQuistan and S. J. Lichtman, Exact recursion relation for (2 × N) arrays of dumbbells, J. Math Phys. 11 (1970), 3095-3099.
László Németh, Tilings of (2 X 2 X n)-board with colored cubes and bricks, arXiv:1909.11729 [math.CO], 2019.
László Németh, Walks on tiled boards, arXiv:2403.12159 [math.CO], 2024. See pp. 1, 7.
N. J. A. Sloane Notes on A030186 and A033505
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Ladder Graph
Eric Weisstein's World of Mathematics, Matching
Yifan Zhang and George Grossman, A Combinatorial Proof for the Generating Function of Powers of a Second-Order Recurrence Sequence, J. Int. Seq. 21 (2018), #18.3.3.
Index entries for linear recurrences with constant coefficients, signature (3,1,-1).

Partial sums give A033505.
Column 2 of triangle A210662.
Cf. A054894, A055518, A055519, A088016.
Cf. A156096. - Gary W. Adamson, Feb 03 2009
Bisection (even part) gives A260033.

a:=[1,2,7];; for n in [4..30] do a[n]:=3*a[n-1]+a[n-2]-a[n-3]; od; a; # G. C. Greubel, Sep 27 2019

a030186 n = a030186_list !! n
a030186_list = 1 : 2 : 7 : zipWith (-) (tail $
   zipWith (+) a030186_list $ tail $ map (* 3) a030186_list) a030186_list
-- Reinhard Zumkeller, Oct 20 2011

a[0]:=1: a[1]:=2: a[2]:=7: for n from 3 to 30 do a[n]:=3*a[n-1]+a[n-2]-a[n-3] od: seq(a[n],n=0..30); # Emeric Deutsch, Oct 16 2006
# second Maple program:
a:= n-> (<<0|1|0>, <0|0|1>, <-1|1|3>>^(n+1))[3,2]:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 07 2024

LinearRecurrence[{3,1,-1}, {1,2,7}, 26] (* Robert G. Wilson v, Nov 20 2012 *)
Table[RootSum[1 -# -3#^2 +#^3 &, 9#^n -10#^(n+1) +7#^(n+2) &]/74, {n, 0, 30}] (* Eric W. Weisstein, Oct 03 2017 *)
CoefficientList[Series[(1-x)/(1-3x-x^2+x^3), {x,0,30}], x] (* Eric W. Weisstein, Oct 03 2017 *)

{a(n)=if(n==0,1,if(n==1,2,(sum(k=0, n-1, a(k))^2+sum(k=0, n-1, a(k)^2))/a(n-1)))} \\ Paul D. Hanna, Nov 20 2012

def A030186_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1-3*x-x^2+x^3)).list()
A030186_list(30) # G. C. Greubel, Sep 27 2019

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

a(n) = ( (Sum_{k=0..n-1} a(k))^2 + (Sum_{k=0..n-1} a(k)^2) ) / a(n-1) for n>1 with a(0)=1, a(1)=2 (similar to A088016). - Paul D. Hanna, Nov 20 2012

More terms from James Sellers

Entry revised by N. J. A. Sloane, Feb 13 2002

A002940 Arrays of dumbbells.

Original entry on oeis.org

1, 4, 11, 26, 56, 114, 223, 424, 789, 1444, 2608, 4660, 8253, 14508, 25343, 44030, 76136, 131110, 224955, 384720, 656041, 1115784, 1893216, 3205416, 5416441, 9136084, 15384563, 25866914, 43429784, 72821274, 121953943, 204002680, 340886973, 569047468, 949022608

Offset: 1

N. J. A. Sloane

Whitney transform of n. The Whitney transform maps the sequence with g.f. g(x) to that with g.f. (1/(1-x))g(x(1+x)). - Paul Barry, Feb 16 2005
a(n-1) is the permanent of the n X n 0-1 matrix with 1 in (i,j) position iff (i=1 and j1). For example, with n=5, a(4) = per([[1, 1, 1, 1, 0], [1, 1, 1, 1, 1], [1, 1, 1, 1, 1], [0, 1, 1, 1, 1], [0, 0, 1, 1, 1]]) = 26. - David Callan, Jun 07 2006
a(n) is the internal path length of the Fibonacci tree of order n+2. A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node. The internal path length of a tree is the sum of the levels of all of its internal (i.e. non-leaf) nodes. - Emeric Deutsch, Jun 15 2010
Partial Sums of A023610 - John Molokach, Jul 03 2013

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(2.3.14).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Carlos Alirio Rico Acevedo and Ana Paula Chaves, Double-Recurrence Fibonacci Numbers and Generalizations, arXiv:1903.07490 [math.NT], 2019.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See pp. 23-24.
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15 (1974), 214-216.
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15.2 (1974), 214-216. (Annotated scanned copy)
Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
R. B. McQuistan and S. J. Lichtman, Exact recursion relation for 2 x N arrays of dumbbells, J. Math. Phys., 11 (1970), 3095-3099.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,1).

Cf. A002941, A002889, A055608, A062123, A062124, A062125, A062126, A062127, A046741.

Cf. A000045, A001925, A023610, A054454, A006478, A067331, A178523.

a002940 n = a002940_list !! (n-1)
a002940_list = 1 : 4 : 11 : zipWith (+)
   (zipWith (-) (map (* 2) $ drop 2 a002940_list) a002940_list)
   (drop 5 a000045_list)
-- Reinhard Zumkeller, Jan 18 2014

m:=35; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)/((1-x)*(1-x-x^2)^2) )); // G. C. Greubel, Jan 31 2019

a[n_]:= a[n]= If[n<3, n^2, 2a[n-1] -a[n-3] +Fibonacci[n+1]]; Array[a, 32] (* Jean-François Alcover, Jul 31 2018 *)

my(x='x+O('x^35)); Vec((1+x)/((1-x)*(1-x-x^2)^2)) \\ G. C. Greubel, Jan 31 2019

((1+x)/((1-x)*(1-x-x^2)^2)).series(x, 35).coefficients(x, sparse=False) # G. C. Greubel, Jan 31 2019

a(n) = 2*a(n-1) - a(n-3) + A000045(n+1).
G.f.: x*(1+x)/((1-x)*(1-x-x^2)^2).
a(n) = Sum_{k=0..n} ( Sum_{i=0..n} k*C(k, i-k) ). - Paul Barry, Feb 16 2005
E.g.f.: 2*exp(x) + exp(x/2)*((55*x - 50)*cosh(sqrt(5)*x/2) + sqrt(5)*(25*x - 22)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Dec 03 2023

More terms from Henry Bottomley, Jun 02 2000

A002941 Arrays of dumbbells.

Original entry on oeis.org

1, 7, 29, 94, 263, 667, 1577, 3538, 7622, 15900, 32314, 64274, 125561, 241569, 458715, 861242, 1601081, 2950693, 5396209, 9801012, 17692092, 31759800, 56727588, 100861716, 178585489, 314995915, 553650761, 969967510, 1694235803

Offset: 1

N. J. A. Sloane

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(2.3.14).
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15 (1974), 214-216.
R. B. McQuistan and S. J. Lichtman, Exact recursion relation for 2 x N arrays of dumbbells, J. Math. Phys., 11 (1970), 3095-3099.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15.2 (1974), 214-216. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (5,-7,-2,10,-2,-5,1,1).

Cf. A046741, A002940, A002889, A055608, A062123-A062127.

a002941 n = a002941_list !! (n-1)
a002941_list = 1 : 7 : 29 : zipWith (+)
   (zipWith (-) (map (* 2) $ drop 2 a002941_list) a002941_list)
   (drop 2 $ zipWith (+) (tail a002940_list) a002940_list)
-- Reinhard Zumkeller, Jan 18 2014

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)^2/((1-x-x^2)^3*(1-x)^2) )); // G. C. Greubel, Jan 31 2019

CoefficientList[(1+x)^2/((1-x-x^2)^3*(1-x)^2) + O[x]^30, x] (* Jean-François Alcover, Jul 31 2018 *)
LinearRecurrence[{5,-7,-2,10,-2,-5,1,1},{1,7,29,94,263,667,1577,3538},30] (* Harvey P. Dale, Aug 29 2021 *)

x='x+O('x^30); Vec((1+x)^2/((1-x-x^2)^3*(1-x)^2)) \\ Altug Alkan, Jul 31 2018

((1+x)^2/((1-x-x^2)^3*(1-x)^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 31 2019

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

a(n) = 2*a(n-1) - a(n-3) + A002940(n) + A002940(n-1).

More terms from Henry Bottomley, Jun 02 2000

A002889 Arrays of dumbbells.

Original entry on oeis.org

1, 10, 56, 234, 815, 2504, 7018, 18336, 45328, 107160, 244198, 539656, 1161987, 2446906, 5054440, 10266850, 20549117, 40595568, 79271188, 153190480, 293278496, 556737696, 1048772300, 1961855408, 3646420325, 6737649754

Offset: 1

N. J. A. Sloane

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(2.3.14).
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15 (1974), 214-216.
R. B. McQuistan and S. J. Lichtman, Exact recursion relation for 2 x N arrays of dumbbells, J. Math. Phys., 11 (1970), 3095-3099.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15.2 (1974), 214-216. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (7,-17,11,19,-29,-3,21,-3,-7,1,1).

Cf. A046741, A002940, A002941.

Cf. A055608, A062123-A062127.

a002889 n = a002889_list !! (n-1)
a002889_list = 1 : 10 : 56 : zipWith (+)
   (zipWith (-) (map (* 2) $ drop 2 a002889_list) a002889_list)
   (drop 2 $ zipWith (+) (tail a002941_list) a002941_list)
-- Reinhard Zumkeller, Jan 18 2014

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)^3/((1-x)^3*(1-x-x^2)^4) )); // G. C. Greubel, Jan 31 2019

CoefficientList[(1+x)^3/((1-x)^3*(1-x-x^2)^4) + O[x]^30, x] (* Jean-François Alcover, Jul 31 2018 *)
LinearRecurrence[{7,-17,11,19,-29,-3,21,-3,-7,1,1},{1,10,56,234,815,2504,7018,18336,45328,107160,244198},30] (* Harvey P. Dale, Jul 25 2021 *)

x='x+O('x^30); Vec((1+x)^3/((1-x)^3*(1-x-x^2)^4)) \\ Altug Alkan, Jul 31 2018

((1+x)^3/((1-x)^3*(1-x-x^2)^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 31 2019

a(n) = 2*a(n-1) - a(n-3) + A002941(n) + A002941(n-1).

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

More terms from Henry Bottomley, Jun 02 2000

A242861 Triangle T(n,k) by rows: number of ways k dominoes can be placed on an n X n chessboard, k>=0.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 12, 44, 56, 18, 1, 24, 224, 1044, 2593, 3388, 2150, 552, 36, 1, 40, 686, 6632, 39979, 157000, 407620, 695848, 762180, 510752, 192672, 35104, 2180, 1, 60, 1622, 26172, 281514, 2135356, 11785382, 48145820, 146702793, 333518324, 562203148

Offset: 0

Ralf Stephan, May 24 2014

Also, coefficients of the matching-generating polynomial of the n X n grid graph.

In the n-th row there are floor(n^2/2)+1 values.

			Triangle starts:
  1
  1
  1  4   2
  1 12  44   56    18
  1 24 224 1044  2593   3388   2150    552     36
  1 40 686 6632 39979 157000 407620 695848 762180 510752 192672 35104 2180
  ...

Alois P. Heinz, Rows n = 0..14, flattened
Ralf Stephan, Two dominoes on the 3x3 chessboard, illustration of T(3,2)=44.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
Wikipedia, Matching polynomial
Index entries for sequences related to dominoes
Index entries for sequences related to matchings

Cf. A046741, A096713.

b:= proc(n, l) option remember; local k;
      if n=0 then 1
    elif min(l[])>0 then b(n-1, map(h->h-1, l))
    else for k while l[k]>0 do od; expand(`if`(n>1,
         x*b(n, subsop(k=2, l)), 0) +`if`(k (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$n])):
seq(T(n), n=0..8); # Alois P. Heinz, Jun 01 2014

b[n_, l_List] := b[n, l] =  Module[{k}, Which[n == 0, 1, Min[l]>0, b[n-1, l-1], True, For[k=1, l[[k]]>0, k++]; Expand[If[n>1, x*b[n, ReplacePart[l, k -> 2]], 0] + If[k 1, k + 1 -> 1}]], 0] + b[n, ReplacePart[l, k -> 1]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, Array[0&, n]]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jun 16 2015, after Alois P. Heinz *)

def T(n,k):
   G = Graph(graphs.Grid2dGraph(n,n))
   G.relabel()
   mu = G.matching_polynomial()
   return abs(mu[n^2-2*k])

T(n,1) = A046092(n-1), T(n,2) = A242856(n).
T(n,floor(n^2/2)) = A137308(n), T(2n,2n^2) = A004003(n).
sum(k>=0, T(n,k)) = A210662(n,n) = A028420(n).
T(n,3) = A243206(n), T(n,4) = A243215(n), T(n,5) = A243217(n), T(n,floor(n^2/4)) = A243221(n). - Alois P. Heinz, Jun 01 2014

cp's OEIS Frontend

A062127 Seventh column of A046741.

Views

Author

Keywords

References

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

Extensions

A062124 Fourth column of A046741.

Views

Author

Keywords

References

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

Extensions

A062125 Fifth column of A046741.

Views

Author

Keywords

References

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

Extensions

A062126 Sixth column of A046741.

Views

Author

Keywords

References

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

Extensions

A039962 Duplicate of A046741.

Views

Author

Keywords

A030186 a(n) = 3*a(n-1) + a(n-2) - a(n-3) for n >= 3, a(0)=1, a(1)=2, a(2)=7.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Haskell

Maple

Mathematica

PARI

Sage