A159616
Expansion of (1-x)/(1-5*x-2*x^2+8*x^3).
Original entry on oeis.org
1, 4, 22, 110, 562, 2854, 14514, 73782, 375106, 1906982, 9694866, 49287446, 250571106, 1273871494, 6476200114, 32924174710, 167382301826, 850950257638, 4326122494162, 21993454571478, 111811915784610, 568437508112710
Offset: 0
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a:=[1,4,22];; for n in [4..40] do a[n]:=5*a[n-1]+2*a[n-2]-8*a[n-3]; od; a; # G. C. Greubel, Oct 27 2019
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R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x)/(1-5*x-2*x^2+8*x^3) )); // G. C. Greubel, Oct 27 2019
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seq(coeff(series((1-x)/(1-5*x-2*x^2+8*x^3), x, n+1), x, n), n=0..40); # G. C. Greubel, Oct 27 2019~
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CoefficientList[Series[(1-x)/(1-5*x-2*x^2+8*x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 11 2012 *)
LinearRecurrence[{5,2,-8}, {1,4,22}, 30] (* Harvey P. Dale, Dec 22 2013 *)
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my(x='x+O('x^40)); Vec((1-x)/(1-5*x-2*x^2+8*x^3)) \\ G. C. Greubel, Oct 27 2019
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def A159616_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P((1-x)/(1-5*x-2*x^2+8*x^3)).list()
A159616_list(40) # G. C. Greubel, Oct 27 2019
A159617
G.f.: (1-x)/(1-8*x-8*x^2+8*x^3).
Original entry on oeis.org
1, 7, 64, 560, 4936, 43456, 382656, 3369408, 29668864, 261244928, 2300355072, 20255449088, 178356473856, 1570492542976, 13828748541952, 121767076888576, 1072202663100416, 9441127931576320, 83132508142305280, 732011467286249472
Offset: 0
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CoefficientList[Series[(1 - x)/(1 - 8 x - 8 x^2 + 8 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 11 2012 *)
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Vec((1 - x) / (1 - 8*x - 8*x^2 + 8*x^3) + O(x^25)) \\ Colin Barker, Jul 05 2020
A253265
The number of tilings of 2 X n boards with squares of 2 colors and dominoes of 3 colors.
Original entry on oeis.org
1, 7, 82, 877, 9565, 103960, 1130701, 12296275, 133724242, 1454268793, 15815379409, 171994465072, 1870463946217, 20341557798991, 221217294787570, 2405769114915733, 26163076626035413, 284527128680078536, 3094272440210485525, 33650646877362841531, 365955505581792121138
Offset: 0
- G. C. Greubel, Table of n, a(n) for n = 0..950
- M. Katz, C. Stenson, Tiling a (2 x n)-board with squares and dominoes, JIS 12 (2009) 09.2.2, Table 1, a=2, b=3.
- Index entries for linear recurrences with constant coefficients, signature (10,12,-27).
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a:=[1,7,82];; for n in [4..30] do a[n]:=10*a[n-1]+12*a[n-2] -27*a[n-3]; od; a; # G. C. Greubel, Oct 28 2019
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R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-3*x)/(1-10*x-12*x^2+27*x^3) )); // G. C. Greubel, Oct 28 2019
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seq(coeff(series((1-3*x)/(1-10*x-12*x^2+27*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 28 2019
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CoefficientList[Series[(1-3x)/(1-10x-12x^2+27x^3), {x, 0, 20}], x] (* Michael De Vlieger, Sep 30 2015 *)
LinearRecurrence[{10,12,-27},{1,7,82},30] (* Harvey P. Dale, Dec 30 2015 *)
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my(x='x+O('x^30)); Vec((1-3*x)/(1-10*x-12*x^2+27*x^3)) \\ G. C. Greubel, Oct 28 2019
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def A253265_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P((1-3*x)/(1-10*x-12*x^2+27*x^3)).list()
A253265_list(30) # G. C. Greubel, Oct 28 2019
A102435
Triangle read by rows: T(n,k) is the number of k-matchings of the corona L'(n) of the ladder graph L(n)=P_2 X P_n. and the complete graph K(1); in other words, L'(n) is the graph constructed from L(n) by adding for each vertex v a new vertex v' and the edge vv'.
Original entry on oeis.org
1, 1, 3, 1, 1, 8, 16, 8, 1, 1, 13, 54, 87, 54, 13, 1, 1, 18, 117, 348, 501, 348, 117, 18, 1, 1, 23, 205, 914, 2210, 2966, 2210, 914, 205, 23, 1, 1, 28, 318, 1910, 6658, 13980, 17895, 13980, 6658, 1910, 318, 28, 1, 1, 33, 456, 3461, 15945, 46648, 88425, 109391, 88425, 46648, 15945, 3461, 456, 33, 1
Offset: 0
T(2,2)=16 because in the graph L'(2) with vertex set {A,B,C,D,a,b,c,d} and edge set {AB,BC,CD,AD,Aa,Bb,Cc,Dd} we have sixteen 2-matchings. Indeed, each of the edges Aa,Bb,Cc,Dd can be matched with five edges and each of the edges AB,BC,CD,AD can be matched with three edges. Thus we have (4*5+4*3)/2=16 matchings.
Triangle begins:
1;
1,3,1;
1,8,16,8,1;
1,13,54,87,54,13,1;
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P[0]:=1: P[1]:=1+3*t+t^2: P[2]:=1+8*t+16*t^2+8*t^3+t^4: for n from 3 to 8 do P[n]:=sort(expand((1+4*t+t^2)*P[n-1]+t*(1+t)^2*P[n-2]-t^3*P[n-3])) 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
Showing 1-4 of 4 results.
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