A007502 Les Marvin sequence: a(n) = F(n)+(n-1)*F(n-1), F() = Fibonacci numbers. (Formerly M1170)

1, 2, 4, 9, 17, 33, 61, 112, 202, 361, 639, 1123, 1961, 3406, 5888, 10137, 17389, 29733, 50693, 86204, 146246, 247577, 418299, 705479, 1187857, 1997018, 3352636, 5621097, 9412937, 15744681, 26307469, 43912648

Offset

1,2

Comments

Denominators of convergents of the continued fraction with the n partial quotients: [1;1,1,...(n-1 1's)...,1,n], starting with [1], [1;2], [1;1,3], [1;1,1,4], ... Numerators are A088209(n-1). - Paul D. Hanna, Sep 23 2003

References

Les Marvin, Problem, J. Rec. Math., Vol. 10 (No. 3, 1976-1977), p. 213.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Formula

G.f.: (1-x^2+x^3)/(1-x-x^2)^2. - Paul D. Hanna, Sep 23 2003

a(n+1) = A109754(n, n+1) = A101220(n, 0, n+1). - N. J. A. Sloane, May 19 2006

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) for n>3, a(0)=1, a(1)=2, a(2)=4, a(3)=9. - Harvey P. Dale, Jul 13 2011

Example

a(7) = F(7)+6*F(6) = 13+6*8 = 61.

Mathematica

Table[Fibonacci[n]+(n-1)*Fibonacci[n-1], {n, 40}] (* or *) LinearRecurrence[ {2, 1, -2, -1}, {1, 2, 4, 9}, 40](* Harvey P. Dale, Jul 13 2011 *)
f[n_] := Denominator@  FromContinuedFraction@ Join[ Table[1, {n}], {n + 1}]; Array[f, 30, 0] (* Robert G. Wilson v, Mar 04 2012 *)

Prog

(Haskell)
a007502 n = a007502_list !! (n-1)
a007502_list = zipWith (+) a045925_list $ tail a000045_list
-- Reinhard Zumkeller, Oct 01 2012, Mar 04 2012
(PARI) Vec((1-x^2+x^3)/(1-x-x^2)^2+O(x^99)) \\ Charles R Greathouse IV, Mar 04, 2012
(Julia) # The function 'fibrec' is defined in A354044.
function A007502(n)
    n == 0 && return BigInt(1)
    a, b = fibrec(n-1)
    (n-1)*a + b
end
println([A007502(n) for n in 1:32]) # Peter Luschny, May 18 2022

Crossrefs

Cf. A088209 (numerators), A000045, A109754, A101220, A045925.

Sequence in context: A192967 A182806 A302832 * A088039 A266108 A360303

Adjacent sequences: A007499 A007500 A007501 * A007503 A007504 A007505

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, Robert G. Wilson v

Status

approved