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

a(n) = number of central Stirling numbers of the second kind S(2*k,k) with 2-adic valuation 3 and 0 <= k < 2^(n+3). - N. J. A. Sloane, Dec 14 2025

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

Arnold Adelberg and Tamás Lengyel, New Results on the 2-Adic Valuation of the Central Stirling Numbers S(2k, k), The Fibonacci Quarterly 63.2 (2025): 146-162; ResearchGate link. See Table 2.

Ignas Gasparavičius, Andrius Grigutis, and Juozas Petkelis, Picturesque convolution-like recurrences and partial sums' generation, arXiv:2507.23619 [math.NT], 2025. See p. 27.

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

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

7*a(n) + (5*n-28)*a(n-1) - (5*n-9)*a(n-2) - (5*n - 11)*a(n-3) = 0. - Greg Dresden, Feb 24 2026

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
(Magma)
A007502:= func< n | Fibonacci(n) +(n-1)*Fibonacci(n-1) >;
[A007502(n): n in [1..40]]; // G. C. Greubel, Aug 26 2025
(SageMath)
def A007502(n): return fibonacci(n) +(n-1)*fibonacci(n-1)
print([A007502(n) for n in range(1, 41)]) # G. C. Greubel, Aug 26 2025

Crossrefs

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

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