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A108851
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a(n) = 4*a(n-1) + 3*a(n-2), a(0) = 1, a(1) = 2.
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8
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1, 2, 11, 50, 233, 1082, 5027, 23354, 108497, 504050, 2341691, 10878914, 50540729, 234799658, 1090820819, 5067682250, 23543191457, 109375812578, 508132824683, 2360658736466, 10967033419913, 50950109889050
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Binomial transform of A083098, second binomial transform of (1, 0, 7, 0, 49, 0, 243, 0, ..).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to sequences with linear recurrences with constant coefficients, signature (4,3).
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FORMULA
| a(n) = ((2 + sqrt(7))^n + (2 - sqrt(7))^n) / 2.
G.f.: (1 - 2*x) / (1 - 4*x - 3*x^2).
E.g.f.: exp(2*x)*cosh(sqrt(7)*x).
a(n+1)/a(n) converges to 2 + sqrt(7) = 4.645751311064...
Limit(a(n+k)/a(k), k=infinity) = A108851(n)+A015530(n)*sqrt(7); also Limit(A108851(n)/A015530(n), n=infinity) = sqrt(7). [Johannes W. Meijer, Aug 01 2010]
a(n) = Sum_{k, 0<=k<=n} A201730(n,k)*6^k . - DELEHAM Philippe, Dec 06 2011
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PROG
| (Sage) [lucas_number2(n, 4, -3)/2 for n in xrange(0, 22)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 14 2009]
(MAGMA) [Floor(((2 + Sqrt(7))^n + (2 - Sqrt(7))^n) / 2): n in [0..30]]; // Vincenzo Librandi, Jul 18 2011
(PARI) a(n)=round(((2+sqrt(7))^n+(2-sqrt(7))^n)/2) \\ Charles R Greathouse IV, Dec 06 2011
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CROSSREFS
| Cf. A080042 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 14 2009]
Appears in A179596, A179597 and A126473. [Johannes W. Meijer, Aug 01 2010]
Sequence in context: A151314 A187000 A154415 * A105486 A137960 A018933
Adjacent sequences: A108848 A108849 A108850 * A108852 A108853 A108854
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KEYWORD
| easy,nonn
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AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 11 2005
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