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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 (list; graph; refs; listen; history; text; internal format)
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

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,-1).

FORMULA

a(n) = F(2n-2) + F(2n-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*(x-1)^2*(x+1) / (x^2-3*x+1). - Colin Barker, Mar 26 2015

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

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

CROSSREFS

Cf. A088305.

Sequence in context: A148139 A336871 A061860 * A267912 A118311 A132836

Adjacent sequences:  A093957 A093958 A093959 * A093961 A093962 A093963

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

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Last modified September 25 12:42 EDT 2020. Contains 337344 sequences. (Running on oeis4.)