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A122491 a(n) = n * Fibonacci(n) - Sum_{i=0..n} Fibonacci(i).

%I A122491 #46 Feb 16 2025 08:33:02
%S A122491 0,0,0,2,5,13,28,58,114,218,407,747,1352,2420,4292,7554,13209,22969,
%T A122491 39748,68494,117590,201210,343275,584087,991440,1679208,2838408,
%U A122491 4789058,8066669,13566373,22782892,38209762,64003002,107083610,178967807,298803459,498404504
%N A122491 a(n) = n * Fibonacci(n) - Sum_{i=0..n} Fibonacci(i).
%C A122491 Similar to A190062.
%C A122491 Also the circuit rank and corank of the n-Lucas cube graph. - _Eric W. Weisstein_, Jul 28 2023
%H A122491 Bruno Berselli, <a href="/A122491/b122491.txt">Table of n, a(n) for n = 0..1000</a>
%H A122491 Carlos Alirio Rico Acevedo, Ana Paula Chaves, <a href="https://arxiv.org/abs/1903.07490">Double-Recurrence Fibonacci Numbers and Generalizations</a>, arXiv:1903.07490 [math.NT], 2019.
%H A122491 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CircuitRank.html">Circuit Rank</a>
%H A122491 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Corank.html">Corank</a>
%H A122491 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasCubeGraph.html">Lucas Cube Graph</a>
%H A122491 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,1,1).
%F A122491 a(n) = n * Fibonacci(n) - Fibonacci(n+2) + 1. - _Stefan Steinerberger_, Feb 22 2008
%F A122491 G.f.: x^3*(2-x)/((1-x)*(1-x-x^2)^2). - _Colin Barker_, Feb 10 2012
%F A122491 a(n+2) = Sum_{k=0..n} A099920(k). - _J. M. Bergot_, Apr 13 2013
%F A122491 a(n) = 2*A006478(n)-A006478(n-1). - _R. J. Mathar_, May 04 2014
%e A122491 a(5) = 13 because Fib(5) = 5, times 5 = 25 and subtract sum(Fib(5)) = 12 to get 13.
%p A122491 with(combinat, fibonacci): for i from 1 to 30 do i*fibonacci(i) - sum(fibonacci(k), k=0..i); end do;
%t A122491 Table[n Fibonacci[n] - Fibonacci[n + 2] + 1, {n, 0, 40}] (* _Stefan Steinerberger_, Feb 22 2008 *)
%t A122491 LinearRecurrence[{3, -1, -3, 1, 1}, {0, 0, 0, 2, 5}, 40] (* _Harvey P. Dale_, May 17 2016 *)
%o A122491 (PARI) a(n)=n*fibonacci(n) - fibonacci(n+2) + 1 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A122491 Cf. A000045.
%K A122491 nonn,easy
%O A122491 0,4
%A A122491 _Ben Paul Thurston_, Sep 16 2006
%E A122491 Edited by _N. J. A. Sloane_, Sep 17 2006