A080145 a(n) = Sum_{m=1..n} Sum_{i=1..m} F(i)*F(i+1) where F(n)=Fibonacci numbers A000045.

0, 1, 4, 13, 37, 101, 269, 710, 1865, 4890, 12810, 33546, 87834, 229963, 602062, 1576231, 4126639, 10803695, 28284455, 74049680, 193864595, 507544116, 1328767764, 3478759188, 9107509812, 23843770261, 62423800984, 163427632705

Offset

0,3

Comments

This is the 2-fold convolution of A001654 with the sequence 1,1,1,....

Equivalently, partial sums of A064831 which is the partial sums of A001654. - Joerg Arndt, Oct 01 2021

a(n) is the number of permutations p in Sn(321) such that p^(-1) has exactly one left peak. See Troyka and Zhuang. - Michel Marcus, Oct 01 2021

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Kenneth Edwards and Michael A. Allen, New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile, J. Int. Seq. 24 (2021) Article 21.3.8.

Justin M. Troyka and Yan Zhuang, Fibonacci numbers, consecutive patterns, and inverse peaks, arXiv:2109.14774 [math.CO], 2021.

Yan Zhuang, A lifting of the Goulden-Jackson cluster method to the Malvenuto-Reutenauer algebra, arXiv:2108.10309 [math.CO], 2021.

Index entries for linear recurrences with constant coefficients, signature (4,-3,-3,4,-1).

Formula

a(n) = F(n+1)*F(n+2) - floor((n+2)/2).

G.f.: x/((1 - 2*x - 2*x^2 + x^3)*(1-x)^2).

a(n) = (4*Lucas(2*n + 3) + (-1)^(n+1) - 10*n - 15)/20. - Ehren Metcalfe, Aug 21 2017

a(n) = (4*Fibonacci(n+1)*Fibonacci(n+2) - 2*n - 3 - (-1)^n)/4. - G. C. Greubel, Jul 23 2019

a(n) = Sum_{j=1..n} j*F(n+1-j)*F(n+2-j). - Michael A. Allen, Jan 07 2022

Mathematica

CoefficientList[Series[x/((1-2x-2x^2+x^3)(1-x)^2), {x, 0, 30}], x] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2009 *)
With[{F=Fibonacci}, Table[(4*F[n+1]*F[n+2]-2*n-3-(-1)^n)/4, {n, 0, 30}]] (* G. C. Greubel, Jul 23 2019 *)

Prog

(Magma) [(4*Lucas(2*n+3)+(-1)^(n+1)-10*n-15)/20: n in [0..30]]; // Vincenzo Librandi, Aug 22 2017
(PARI) L(n)=fibonacci(n-1)+fibonacci(n+1)
a(n)=(4*L(2*n+3)-(-1)^n-10*n-15)/20 \\ Charles R Greathouse IV, Aug 26 2017
(Sage) f=fibonacci; [(4*f(n+1)*f(n+2)-2*n-3-(-1)^n)/4 for n in (0..30)] # G. C. Greubel, Jul 23 2019
(GAP) F:=Fibonacci;; List([0..30], n-> (4*F(n+1)*F(n+2)-2*n-3-(-1)^n)/4); # G. C. Greubel, Jul 23 2019

Crossrefs

Cf. A000032, A000045, A001654, A064831.

Sequence in context: A054761 A244197 A300985 * A097551 A224033 A095126

Adjacent sequences: A080142 A080143 A080144 * A080146 A080147 A080148

Keyword

easy,nonn

Author

Mario Catalani (mario.catalani(AT)unito.it), Jan 31 2003

Status

approved