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%I M1170 #43 May 18 2022 05:58:27
%S 1,2,4,9,17,33,61,112,202,361,639,1123,1961,3406,5888,10137,17389,
%T 29733,50693,86204,146246,247577,418299,705479,1187857,1997018,
%U 3352636,5621097,9412937,15744681,26307469,43912648
%N Les Marvin sequence: a(n) = F(n)+(n-1)*F(n-1), F() = Fibonacci numbers.
%C 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
%D Les Marvin, Problem, J. Rec. Math., Vol. 10 (No. 3, 1976-1977), p. 213.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A007502/b007502.txt">Table of n, a(n) for n = 1..500</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F G.f.: (1-x^2+x^3)/(1-x-x^2)^2. - _Paul D. Hanna_, Sep 23 2003
%F a(n+1) = A109754(n, n+1) = A101220(n, 0, n+1). - _N. J. A. Sloane_, May 19 2006
%F 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
%e a(7) = F(7)+6*F(6) = 13+6*8 = 61.
%t 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 *)
%t f[n_] := Denominator@ FromContinuedFraction@ Join[ Table[1, {n}], {n + 1}]; Array[f, 30, 0] (* _Robert G. Wilson v_, Mar 04 2012 *)
%o (Haskell)
%o a007502 n = a007502_list !! (n-1)
%o a007502_list = zipWith (+) a045925_list $ tail a000045_list
%o -- _Reinhard Zumkeller_, Oct 01 2012, Mar 04 2012
%o (PARI) Vec((1-x^2+x^3)/(1-x-x^2)^2+O(x^99)) \\ _Charles R Greathouse IV_, Mar 04, 2012
%o (Julia) # The function 'fibrec' is defined in A354044.
%o function A007502(n)
%o n == 0 && return BigInt(1)
%o a, b = fibrec(n-1)
%o (n-1)*a + b
%o end
%o println([A007502(n) for n in 1:32]) # _Peter Luschny_, May 18 2022
%Y Cf. A088209 (numerators), A000045, A109754, A101220, A045925.
%K nonn,nice,easy
%O 1,2
%A _N. J. A. Sloane_, _Robert G. Wilson v_
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