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A093960
a(1) = 1, a(2) = 2, a(n+1) = n*a(1) + (n-1)*a(2) + ... + (n-r)*a(r+1) + ... + a(n).
3
1, 2, 4, 11, 29, 76, 199, 521, 1364, 3571, 9349, 24476, 64079, 167761, 439204, 1149851, 3010349, 7881196, 20633239, 54018521, 141422324, 370248451, 969323029, 2537720636, 6643838879, 17393796001, 45537549124, 119218851371, 312119004989, 817138163596
OFFSET
1,2
COMMENTS
a(1) = a(2) = 1 gives A088305, i.e., Fibonacci numbers with even indices. This can be called 'fake Fibonacci sequence'. 4 = 3+1, 11 = 8+3, 29 = 21+8, 76 = 55+21, etc. a(n) = F(2n-2) + F(2n-4).
Except for the initial terms, this is the same as the bisection of the Lucas sequence (A002878). - Franklin T. Adams-Watters, Jul 17 2006
FORMULA
a(n) = F(2*n-2) + F(2*n-4), where F(k) is k-th Fibonacci number, n > 2.
a(n) = 3*a(n-1) - a(n-2) for n>4. - Colin Barker, Mar 26 2015
G.f.: x*(1-x)^2*(1+x) / (1-3*x+x^2). - Colin Barker, Mar 26 2015
a(n) = 2^(2-n)*[n<3] + LucasL(2*n-3). - G. C. Greubel, Dec 30 2021
MAPLE
a[1]:=1: a[2]:=2: for n from 2 to 33 do a[n+1]:=sum((n-r)*a[r+1], r=0..n-1) od: seq(a[n], n=1..33); # Emeric Deutsch, Aug 01 2005
A093960List := proc(m) local A, P, n; A := [1, 2]; P := [1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(A), P[-1]]);
A := [op(A), P[-1]] od; A end: A093960List(30); # Peter Luschny, Mar 24 2022
MATHEMATICA
Print[1]; Print[2]; Do[Print[Fibonacci[2*n - 2] + Fibonacci[2*n - 4]], {n, 3, 20}] (* Ryan Propper, Jun 19 2005 *)
LinearRecurrence[{3, -1}, {1, 2, 4, 11}, 30] (* Harvey P. Dale, Nov 17 2018 *)
PROG
(PARI) Vec(x*(x-1)^2*(x+1)/(x^2-3*x+1) + O(x^100)) \\ Colin Barker, Mar 26 2015
(Magma) [1, 2] cat [Lucas(2*n-3): n in [3..30]]; // G. C. Greubel, Dec 30 2021
(Sage) [2^(2-n)*bool(n<3) + lucas_number2(2*n-3, 1, -1) for n in (1..30)] # G. C. Greubel, Dec 30 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amarnath Murthy, May 22 2004
EXTENSIONS
More terms from Ryan Propper, Jun 19 2005
More terms from Emeric Deutsch, Aug 01 2005
STATUS
approved