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A082762
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Trinomial transform of Lucas numbers (A000032).
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4
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1, 8, 44, 232, 1216, 6368, 33344, 174592, 914176, 4786688, 25063424, 131233792, 687149056, 3597959168, 18839158784, 98643116032, 516502061056, 2704439902208, 14160631169024, 74146027405312, 388233639755776
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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FORMULA
| a(n) = Sum[ Trinomial[n, k] Lucas[k+1], {k, 0, 2n} ] where Trinomial[n, k] = trinomial coefficients (A027907)
a(n) = 2^n Lucas[2n+1] where Lucas[n] = Lucas numbers (A000032).
a(n) = 2^n*A002878(n) = 2^(-n)*Sum_{k>=0} binomial(2*n+1, 2*k)*5^k; see A091042 . a(0) = 1, a(1) = 8, a(n+1) = 6*a(n) - 4*a(n-1) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 01 2004
a(n)=(1+sqrt5)(3+sqrt5)^n+(1-sqrt5)(3-sqrt5)^n)/2 offset 0. a(n)=third binomial transform of 1,5,5,25,25,125 [From Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009]
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MATHEMATICA
| a[n_]:=(MatrixPower[{{2, 2}, {2, 4}}, n].{{2}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 20 2010]
f[n_] := Block[{s = Sqrt@ 5}, Simplify[((1 + s)(3 + s)^n + (1 - s)(3 - s)^n)/2]]; Array[f, 21, 0] (* RGWv *)
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CROSSREFS
| Sequence in context: A176688 A197213 A198768 * A147828 A155604 A126476
Adjacent sequences: A082759 A082760 A082761 * A082763 A082764 A082765
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KEYWORD
| easy,nonn
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AUTHOR
| Emanuele Munarini (munarini(AT)mate.polimi.it), May 21 2003
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