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A114696
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Expansion of (1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.
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4
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1, 6, 15, 40, 97, 238, 575, 1392, 3361, 8118, 19599, 47320, 114241, 275806, 665855, 1607520, 3880897, 9369318, 22619535, 54608392, 131836321, 318281038, 768398399, 1855077840, 4478554081, 10812186006, 26102926095, 63018038200, 152139002497, 367296043198
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OFFSET
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0,2
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COMMENTS
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Elements of odd index give match to A065113: Sum of the squares of the n-th and the (n+1)st triangular numbers (A000217) is a perfect square.
Generating floretion: - 1.5'i + 'j + 'k - .5i' + j' + k' + .5'ii' - .5'jj' - .5'kk' - 'ij' + 'ik' - 'ji' + .5'jk' + 2'ki' - .5'kj' + .5e
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LINKS
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Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).
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FORMULA
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G.f.: (1 +4*x +x^2)/((1-x)*(1+x)*(1-2*x-x^2)).
a(0)=1, a(1)=6, a(2)=15, a(3)=40, a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4). - Harvey P. Dale, Jan 23 2014
a(n) = (-3 - (-1)^n + (3-2*sqrt(2))*(1-sqrt(2))^n + (1+sqrt(2))^n*(3+2*sqrt(2)))/2. - Colin Barker, May 26 2016
From G. C. Greubel, May 24 2021: (Start)
a(n) = 3*A000129(n+1) + A000129(n) - (3 + (-1)^n)/2.
a(n) = (1/2)*(A002203(n+2) - 3 - (-1)^n). (End)
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MAPLE
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Q:= proc(n) option remember; # Q=A002203
if n<2 then 2
else 2*Q(n-1) + Q(n-2)
fi; end:
seq((Q(n+2) -3 -(-1)^n)/2, n=0..40); # G. C. Greubel, May 24 2021
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MATHEMATICA
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CoefficientList[Series[(1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)), {x, 0, 30}], x] (* or *) LinearRecurrence[{2, 2, -2, -1}, {1, 6, 15, 40}, 30] (* Harvey P. Dale, Jan 23 2014 *)
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PROG
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(PARI) Vec((1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^30)) \\ Colin Barker, May 26 2016
(Sage) [(lucas_number2(n+2, 2, -1) -3 -(-1)^n)/2 for n in (0..30)] # G. C. Greubel, May 24 2021
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CROSSREFS
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Cf. A000129, A002203, A005409, A100828, A111954, A113224.
Cf. A114647, A114688, A114689, A114695, A114697.
Sequence in context: A273748 A272847 A273829 * A106368 A100491 A220030
Adjacent sequences: A114693 A114694 A114695 * A114697 A114698 A114699
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KEYWORD
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easy,nonn
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AUTHOR
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Creighton Dement, Feb 18 2006
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STATUS
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approved
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