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A134410
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Second-order Lucas numbers; a(n) := (2n+3)*Lucas(n) - n*Lucas(n-1).
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1
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6, 3, 19, 27, 61, 108, 204, 367, 661, 1173, 2069, 3622, 6306, 10923, 18839, 32367, 55421, 94608, 161064, 273527, 463481, 783753, 1322869, 2229002, 3749886, 6299283, 10567579, 17705667, 29630461, 49532148, 82715844, 137997247
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| This sequence is defined by analogy with the sequence of second-order Fibonacci numbers A010049.
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (2,1,-2,-1).
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FORMULA
| Defining equation a(n) := (2n+3)*Lucas(n) - n*Lucas(n-1).
Recurrence: a(0) = 6, a(1) = 3, a(n+2) = a(n+1) + a(n) + 5*Lucas(n).
O.g.f.: (2-x)*(3-3x+2x^2)/(1-x-x^2)^2.
Set A(n) = (a(n-1) + a(n+1))/5, B(n) = a(n+1) - a(n-1). Then A(n+2) = A(n+1) + A(n) + 5*Fibonacci(n) and B(n+2) = B(n+1) + B(n) + 5*Lucas(n). The polynomials L_2(n,-x) = sum {k = 0..n} C(n,k)*a(n-k)*(-x)^k appear to satisfy a Riemann hypothesis; their zeros appear to lie on the vertical line Re x = 1/2 in the complex plane. Compare with the polynomials L(n,-x) defined in A132148.
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CROSSREFS
| Cf. A000032, A010049, A132148.
Sequence in context: A050008 A166450 A019069 * A123153 A185783 A088697
Adjacent sequences: A134407 A134408 A134409 * A134411 A134412 A134413
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KEYWORD
| easy,nonn
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AUTHOR
| Peter Bala (pbala(AT)toucansurf.com), Oct 24 2007
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