A105635 a(n) = (2*Pell(n+1) - (1+(-1)^n))/4.
0, 1, 2, 6, 14, 35, 84, 204, 492, 1189, 2870, 6930, 16730, 40391, 97512, 235416, 568344, 1372105, 3312554, 7997214, 19306982, 46611179, 112529340, 271669860, 655869060, 1583407981, 3822685022, 9228778026, 22280241074, 53789260175, 129858761424, 313506783024
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Eric Weisstein's World of Mathematics, Matching Number
- Eric Weisstein's World of Mathematics, Pell Graph
- Eric Weisstein's World of Mathematics, Vertex Cover Number
- Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).
Crossrefs
Cf. A000129.
Cf. A157901. - Gary W. Adamson, Mar 08 2009
Programs
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GAP
a:=[0,1,2,6];; 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; # G. C. Greubel, Oct 27 2019
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Magma
R
:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x/((1-x^2)*(1-2*x-x^2)) )); // G. C. Greubel, Oct 27 2019 -
Maple
with(combinat): seq(iquo(fibonacci(n+1,2),2),n=0..30); # Zerinvary Lajos, Apr 20 2008 # second Maple program: a:= n-> (<<1|1|0|0>, <3|0|1|0>, <1|0|0|0>, <1|0|0|1>>^n)[4, 1]: seq(a(n), n=0..50); # Alois P. Heinz, Jul 24 2008
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Mathematica
Table[(Fibonacci[n + 1, 2] - Fibonacci[n + 1, 0])/2, {n, 0, 30}] (* G. C. Greubel, Oct 27 2019 *) Floor[Fibonacci[Range[20], 2]/2] (* Eric W. Weisstein, Aug 01 2023 *) Table[(2 Fibonacci[n + 1, 2] - (-1)^n - 1)/4, {n, 0, 10}] (* Eric W. Weisstein, Aug 01 2023 *) CoefficientList[Series[x/(1 - 2 x - 2 x^2 + 2 x^3 + x^4), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 01 2023 *) LinearRecurrence[{2, 2, -2, -1}, {0, 1, 2, 6, 14}, 20] (* Eric W. Weisstein, Aug 01 2023 *) -
PARI
my(x='x+O('x^30)); concat([0], Vec(x/((1-x^2)*(1-2*x-x^2)))) \\ G. C. Greubel, Oct 27 2019 -
Sage
def A105635_list(prec): P.
= PowerSeriesRing(ZZ, prec) return P(x/((1-x^2)*(1-2*x-x^2))).list() A105635_list(30) # G. C. Greubel, Oct 27 2019
Formula
G.f.: x/((1-x^2)*(1-2*x-x^2)).
a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4).
a(n) = Sum_{k=0..floor((n-1)/2)} Pell(n-2k).
a(n) = Sum_{k=0..n} Pell(k)*(1-(-1)^(n+k-1))/2.
a(n) = term (4,1) in the 4 X 4 matrix [1,1,0,0; 3,0,1,0; 1,0,0,0; 1,0,0,1]^n. - Alois P. Heinz, Jul 24 2008
a(n) = floor(Pell(n)/2). - Eric W. Weisstein, Aug 01 2023
Comments