|
| |
|
|
A001946
|
|
a(n) = 11*a(n-1) + a(n-2).
(Formerly M2009 N0794)
|
|
7
| |
|
|
2, 11, 123, 1364, 15127, 167761, 1860498, 20633239, 228826127, 2537720636, 28143753123, 312119004989, 3461452808002, 38388099893011, 425730551631123, 4721424167835364, 52361396397820127, 580696784543856761
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,1
|
|
|
COMMENTS
| For odd n there is the Aurifeuillian factorization a(n) = Lucas[5n] = Lucas[n]*A[n]*B[n] = A000032[n]*A124296[n]*A124297[n], where A[n] = A124296[n] = 5*F(n)^2 - 5*F(n) + 1 and B[n] = A124297[n] = 5*F(n)^2 + 5*F(n) + 1, where F(n) = Fibonacci[n]. The largest prime divisors of a(n) for n>0 are listed in A121171[n] = {11, 41, 31, 2161, 151, 2521, 911, ...}. - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 25 2006
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 12 2010: (Start)
For more information about this type of recurrence follow the Khovanova link and see A086902 and A054413.
(End)
|
|
|
REFERENCES
| J. Riordan, Combinatorial Identities, Wiley, 1968, p. 139.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
| Tanya Khovanova, Recursive Sequences
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
|
|
|
FORMULA
| a(n) = Lucas(5n) = Fibonacci(5n-1) + Fibonacci(5n+1). - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 25 2006
a(n) = ((11 + 5sqrt(5))/2)^n + ((11 - 5sqrt(5))/2)^n. - Tanya Khovanova (tanyakh(AT)yahoo.com), Feb 06 2007
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 12 2010: (Start)
a(2n+1) = 11*A097842(n), a(2n) = A065705(n).
a(3n+1) = A041226(5n), a(3n+2) = A041226(5n+3), a(3n+3) = 2* A041226(5n+4).
Limit(a(n+k)/a(k), k=infinity) = (A001946(n) + A049666(n)*sqrt(125))/2.
Limit(A001946(n)/A049666(n), n=infinity) = sqrt(125).
(End)
|
|
|
MAPLE
| A001946:=(-2+11*z)/(-1+11*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]
|
|
|
MATHEMATICA
| Table[Fibonacci[5n-1]+Fibonacci[5n+1], {n, 0, 30}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 25 2006
|
|
|
PROG
| (MAGMA) [ Lucas(5*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011
|
|
|
CROSSREFS
| Cf. A000032, A000045, A121171, A124296, A124297.
Sequence in context: A057076 A118794 A155928 * A206401 A193207 A112864
Adjacent sequences: A001943 A001944 A001945 * A001947 A001948 A001949
|
|
|
KEYWORD
| easy,nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
|
| |
|
|
