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

%I A007502 M1170 #55 Aug 28 2025 09:21:05
%S A007502 1,2,4,9,17,33,61,112,202,361,639,1123,1961,3406,5888,10137,17389,
%T A007502 29733,50693,86204,146246,247577,418299,705479,1187857,1997018,
%U A007502 3352636,5621097,9412937,15744681,26307469,43912648
%N A007502 Les Marvin sequence: a(n) = F(n) + (n-1)*F(n-1), F() = Fibonacci numbers.
%C A007502 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 A007502 Les Marvin, Problem, J. Rec. Math., Vol. 10 (No. 3, 1976-1977), p. 213.
%D A007502 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007502 T. D. Noe, <a href="/A007502/b007502.txt">Table of n, a(n) for n = 1..500</a>
%H A007502 Ignas Gasparavičius, Andrius Grigutis, and Juozas Petkelis, <a href="https://arxiv.org/abs/2507.23619">Picturesque convolution-like recurrences and partial sums' generation</a>, arXiv:2507.23619 [math.NT], 2025. See p. 27.
%H A007502 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F A007502 G.f.: (1-x^2+x^3)/(1-x-x^2)^2. - _Paul D. Hanna_, Sep 23 2003
%F A007502 a(n+1) = A109754(n, n+1) = A101220(n, 0, n+1). - _N. J. A. Sloane_, May 19 2006
%F A007502 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
%F A007502 E.g.f.: exp(x/2)*( ((3 + 2*x)/sqrt(5))*sinh(sqrt(5)*x/2) - cosh(sqrt(5)*x/2) ) + 1. - _G. C. Greubel_, Aug 26 2025
%e A007502 a(7) = F(7) + 6*F(6) = 13 + 6*8 = 61.
%t A007502 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 A007502 f[n_] := Denominator@  FromContinuedFraction@ Join[ Table[1, {n}], {n + 1}]; Array[f, 30, 0] (* _Robert G. Wilson v_, Mar 04 2012 *)
%o A007502 (Haskell)
%o A007502 a007502 n = a007502_list !! (n-1)
%o A007502 a007502_list = zipWith (+) a045925_list $ tail a000045_list
%o A007502 -- _Reinhard Zumkeller_, Oct 01 2012, Mar 04 2012
%o A007502 (PARI) Vec((1-x^2+x^3)/(1-x-x^2)^2+O(x^99)) \\ _Charles R Greathouse IV_, Mar 04 2012
%o A007502 (Julia) # The function 'fibrec' is defined in A354044.
%o A007502 function A007502(n)
%o A007502     n == 0 && return BigInt(1)
%o A007502     a, b = fibrec(n-1)
%o A007502     (n-1)*a + b
%o A007502 end
%o A007502 println([A007502(n) for n in 1:32]) # _Peter Luschny_, May 18 2022
%o A007502 (Magma)
%o A007502 A007502:= func< n | Fibonacci(n) +(n-1)*Fibonacci(n-1) >;
%o A007502 [A007502(n): n in [1..40]]; // _G. C. Greubel_, Aug 26 2025
%o A007502 (SageMath)
%o A007502 def A007502(n): return fibonacci(n) +(n-1)*fibonacci(n-1)
%o A007502 print([A007502(n) for n in range(1,41)]) # _G. C. Greubel_, Aug 26 2025
%Y A007502 Cf. A000045, A045925, A088209 (numerators), A101220, A109754.
%K A007502 nonn,nice,easy,changed
%O A007502 1,2
%A A007502 _N. J. A. Sloane_, _Robert G. Wilson v_