A-Probabilistic-View-of-Certain-Weighted-Fibonacci-Sums

1 , THE OFFICIAL JOURNAL OF THE FIBONACCI ASSOCIATION TABLE OF' , b ~ Double Indexed Fibonacci Sequences and the Distribution Heights of Happy Numbers and Cubic Happy Numbers H.G Grupdman qnd8.A Tee~le - * - - - : :=&% - On Modular Fibonacci Sets Gihai Caragiu and William Webb 307; A - +'2:-.,< &' ;ZI ;i A Nim-Type Game and Continued Fractions ; Tamcis Lengyel 10 ; g: a ' ' &lH Generating Functions, Weighted and Non-Weighted Sums for Powers of Second-Order Recurrence Sequences Pantelimon Stinica' 321 Some Comments on Baillie-PSW Pseudoprimes Zhuo Chen and John Greene 334 On the kh-OrderF-L Identity Chizhong Zhou and Fredric ?: Howard 345 352 ' Unexpected Pell an uasi Morgan-Voyce Summation Connections A E Horadam .2 ** >- A Probabilistic mew ot Certain Weighted Fibonacci Sums .Arthur 7: Benjamin, Judson D Nee6 Daniel E Otero and James A Sellers -*-&- On Positive Numbers n for Which Q(n)Divides F,, Florian Luca - A -6-'I - !.idEdited by Russ Euler and Jawad Sadek Elementary Problems and Solutions Advanced Problems and Solutions - ,', lAd.= I :.' lI'hinternational ConferGce on Fibona& ,, ,,w ,A m ' - - , -L - 365 , - 'd :ZG.m .a!& + Sums ani~ifferencesof Values of a Quadratic Polynomial Alan E Beardon 372 L + -+ r -* J + ,-.i ' - - -3 4, - -;!;- - v - Y r *- A PROBABILISTIC VIEW OF CERTAIN WEIGHTED FIBONACCI SUMS Arthur T Benjamin Dept of Mathematics, Harvey Mudd College, Claremont, CA 91711 beqjaminQhmc.edu Judson D Neer Dept of Science and Mathematics, Cedarville University, 251 N Main St., Cedarville, OH 453140601 jud@poboxes.com Daniel E Otero Dept of Mathematics and Computer Science, Xavier University, Cincinnati, OH 45207-4441 oteroQxu.edu James A Sellers Dept of Mathematics, Penn State University, University Park, PA 16802 sellersjQmath.psu.edu (Submitted May 2001) INTRODUCTION In this paper we investigate sums of the form For any given n, such a sum can be determined [3] by applying the x& operator n times to the generating function then evaluating the resulting expression at x = 112 This leads to a0 = 1,a1 = 5,a2 = 47, and so on Thesc sums may be used to determine the expected value and higher moments of the number of flips needed of a fair coin until two consecutive heads appear [3] In this article, we pursue the reverse strategy of using probability to derive a, and develop an exponential generating function for a, in Section In Section 4, we present a method for finding an exact, non-recursive, formula for a, PROBABILISTIC INTERPRETATION Consider an infinitely long binary sequence of independent random variables bl, b2, bs, where P(bi = 0) = P(bi = 1) = 112 Let Y denote the random variable denoting the beginning of the first 00 substring That is, by = by+l = and no 00 occurs before then Thus P(Y = 1) = 114 For k 2, we have P ( Y = k) is equal to the probability that our sequence begins bl , b2, ,b k - z , l , 0,O, where no 00 occurs among the first k - terms Since 360 [AUG A PROBABILISTIC VIEW OF CERTAIN WEIGHTED FIBONACCI SUMS the probability of occurence of each such string is (1/2)"', and it is well known [I]that there are exactly Fk binary strings of length k - with no consecutive O's, we have for k 1, Fk P ( Y = k) = F Since Y is finite with probability 1, it follows that For n > 0, the expected value of Yn is > Thus a0 = For n 1, we use conditional expectation to find a recursive formula for a, We illustrate our argument with n = and n = before proceeding with the general case For a random sequence bl, b2, , we compute E(Y) by conditioning on bl and b2 If bl = b2 = 0, then Y = If bl = 1, then we have wasted a flip, and we are back to the drawing board; let Y' denote the number of remaining flips needed If bl = and b2 = 1, then we have wasted two flips, and we are back to the drawing board; let Y" denote the number of remaining flips needed in this case Now by conditional expectation we have E(Y) = -(I) + ?1E ( l + Y') + -E(2 + Y") since E(Y') = E(Y1') = E(Y) Solving for E(Y) gives us E(Y) = Hence, Conditioning on the first two outcomes again allows us to compute Since E(Y) = 5, it follcnrs that E(Y*)= 47 Thus, A PROBABILISTIC VIEW O F CERTAIN WEIGHTED FIBONACCI SUMS Following the same logic for higher moments, we derive for n 1, E(Yn) = -(ln) + -E + [(I Y)n] + -E [(2 + Y)"] Consequently, we have the following recursive equation: Thus for all n 1, Using equation (3), one can easily derive as = 665, a4 = 12,551, and so on GENERATING FUNCTION AND ASYMPTOTICS For n 0, define the exponential generating function It follows from equation (3) that Consequently, For the asymptotic growth of a,, one need only look a t the leading term of the Laurent series expansion [4] of a(x) This leads to [AUG A PROBABILISTIC VIEW OF CERTAIN WEIGHTED FIBONACCI SUMS CLOSED FORM While the recurrence (3), generating function (4), and asymptotic result (5) are satisfying, a closed form for a, might also be desired For the sake of completeness, we demonstrate such a closed form here To calculate we first recall the Binet formula for Fk [3]: Then ( ) implies that (1) can be rewritten as Next, we remember the formula for the geometric series: This holds for all real numbers x such that 1x1 < We now apply the x& operator n times to (8) It is clcar that the left-hand side of (8) will then become knsk C k> The right-hand side of (8) is transformed into the rational function where the coefficients e(n, j ) are the Eulerian numbers [2, Sequence A0082921, defined by e(n, j ) = j - e(n - 1,j) + ( n - j + 1) e(n - 1,j - 1) with e ( , l ) = (Thc fact that these are indeed the coefficients of the polynomial in the numerator of (9) can bc proved quickly by induction.) Ftom the information found in [2, Sequence A0082921, we know A PROBABILISTIC VIEW OF CERTAIN WEIGHTED FIBONACCI SUMS Therefore, Thus the two sums that appear in (7) can be determined explicity using (10) since 1'+-4 f i < and q < Hence, an exact, non-recursive, formula for a, can be developed REFERENCES [I] A.T Benjamin and J.J Quinn "Recounting Fibonacci and Lucas Identities." College Mathematics Journal 30.5 (1999): 359366 [2] N.J.A Sloane The On-Line Encyclopedia of Integer Sequences, published electronically at http://www.research.att.com/~njas/sequences/,2000 [3] S Vajda Fibonacci and Lucas Numbers, and the Golden Section, John Wiley and Sons, New York, 1989 [4] H.S Wilf Generatingfunctionology, Academic Press, Boston, 1994 AMS Classification Numbers: llB39 [AUG ~