THE DYING RABBIT PROBLEM. V. E. HOGGATT, JR., and D. A. LIIMD San Jose State College, San Jose, Calif., and University of Cambridge, England

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1 THE DYING RABBIT PROBLEM V. E. HOGGATT, JR., ad D. A. LIIMD Sa Jose State College, Sa Jose, Calif., ad Uiversity of Cambridge, Eglad 1. INTRODUCTION Fiboacci umbers origially arose i the aswer to the followig problem posed by Leoardo de Pisa i Suppose there is oe pair of rabbits i a eclosure at the 0 moth, ad that this pair breeds aother pair i each of the succeedig moths. Also suppose that pairs of rabbits breed i the secod moth followig birth, ad thereafter produce oe pair mothly. What is the umber of pairs of rabbits at the ed of the moth? It is ot diffi th cult to establish by J iductio that the aswer is F l 0 +2, where F is the Fiboacci umber. I [l] Brother Alfred asked for a solutio to this problem if, like Socrates, our rabbits are motral, say each pair dies oe year after birth. His aswer [2], however, cotaied a error. The mistake was oted by Coh [ 3], who also supplied the correct solutio. I this paper we geeralize the dyig rabbit problem to arbitrary breedig patters ad death times. 2. SOLUTION TO THE GENERALIZED DYING RABBIT PROBLEM Suppose that there is oe pair of rabbits at the 0 time poit, that this pair produces Bf pairs at the first time poit, B 2 pairs at the secod time poit, ad so forth, ad that each offsprig pair breeds i the same maer. We shall let B 0 = 0, ad put B(x) = B x 11 =0 so that B(x) is the birth polyomial associated with the birth sequece ( B j =0 The degree of B(x), deg B(x), may be fiite or ifiite. Now suppose a pair of rabbits dies at the m time poit after birth (after possible breedig), ad let D(x) = x be the associated death polyomial. If our rabbits are immortal, 482

2 Dec THE DYING RABBIT PROBLEM 483 put D(x) = 0. Clearly deg D(x) > 0 implies deg D(x) > deg B(x), uless the rabbits have strage matig habits. Let T be the total umber of live pairs of rabbits at the time poit, ad put T(x) = Tx 11 =0 where T 0 = 1. Our problem is the to determie T(x), where B(x) ad D(x) are kow. Let R be the umber of pairs of rabbits bor at the time poit assumig o deaths. With the covetio that the origial pair was bor at the 0 time poit, ad recallig that B 0 = 0, we have R 0-1, Rj := BQR^ + BJRQ, R 2 = B0R2 + B-^i + B 2 RQ ad i geeral that (1) R = B.R. ( > 1). po J "J Note that for = 0 this expressio yields the icorrect R 0 = 0. The if RW = E R x, =0 equatio (1) is equivalet to R(x) = R(x)B(x) + 1, so that

3 484 THE DYING RABBIT PROBLEM [Dec The total tota] umber T* of pairs at the give by time poit assumig o deaths is T* = Y, R., 3 ad we fid (2) (1 - x) [1 - B(x7) 1 - A k = Q *».l±±){i^) 00 / \ = E E H. x = D T x = T*(x) =0\j=0 7 =0 Hoggatt [4] used slightly differet methods to show both (1) ad (2). We must ow allow for deaths. Sice each pair dies m time poits afterbirth, the umber of deaths D at the time poit equals the umber of births R at the (-m) time poit. Therefore W =»«JV' = ^B=0 =0 Lettig the total umber of dead pairs of rabbits at the time poit be c = y\ D., -* 3 we have D(x) (1 - x) [1 ^=(lo*%fov Ho(N X " = E Cx 11 = C(x) =0

4 1969] THE DYING RABBIT PROBLEM 485 Now the total umber of live pairs of rabbits T at the time poit is T " C ' s o t h a t (3) T(x) = T*(x) - C(x) = T r ^ ^ L M _ 3. SOME PARTICULAR CASES To solve Brother Alfred f s problem, we put B(x) = x 2 + x x 12 ad D(x) = x 12 i (3) to give T(x) x x 12 X x (1 - x)(l - x 2 - x x 12 ) _ 1 - x l - x x + x 13 It follows that the sequece { T } obeys T 1-0 = T_ L T J _ T ( > 0), together with the iitial coditios T = F - for = 0, 1,, 11, ad T12 = F i3-1j which agrees with the aswer give by Coh [3]. As aother example of (3), suppose each pair produce a pair at each of the two time poits followig birth, ad the die at the m time poit after birth (m> 2). I this case, B(x) = x + x 2 ad D(x) = x. From (3), we see T(x) - -i JH l " x (1 - x)(l - x - x 2 ) Makig use of the geeratig fuctio t F +1 x, - v x - X- =0 we get

5 486 THE DYING RABBIT PROBLEM [Dec.., 1 + X + + X V^ X J T(x) = = 2^ 1 - x - x 2 3=0 1 - x - x 2 m - 1 /,. \ m-1 / \ / m - 1 \ (4) = E E F +1 x ^ = E E F k+1 )x + E E F k + 1 x 1 j=0 \ =0 X J =0 \ k=0 K + i K + / =m \ k=0 1 / m-1 = E ft (F +3 - Dx 11 - E (F +3 - F _ m+3 )x =0 =m For m = 4r it is kow [5] that F +3 F -4r+3 F 2r L -2r+3 ' where L is the Lucas umber, while for m = 4r + 2, F +3 " F -4r+l = L 2r+l F -2r+2 ' which may be used to further simplify (4). I particular, for m = 2, T(x) = 1 + 2x + E F + 2 x = E F + 2 x =0 =0 while for m = 4 we have T(x) = 1 + 2x + 4x 2 + 7x 3 + T L.x 11 ~, +1 =4 =0 ^ +l Thus for proper choices of B(x) ad D(x) we are able to get both Fiboacci ad Lucas umbers as the total populatio umbers. The secod-amed author was supported i part by the Udergraduate Research Participatio Program at the Uiversity of Sata Clara through NSF Grat GY-273.

6 ] THE DYING RABBIT PROBLEM 487 REFERENCES 1. Brother U. Alfred, "Explorig Fiboacci N u m b e r s, " Fiboacci Quarterly, 1 (1963), No. 1, Brother U. Alfred, "Dyig Rabbit P r o b l e m Revived," Fiboacci Quarterly, 1 (1963), No. 4, Joh H. E. Coh, " L e t t e r to the E d i t o r, " Fiboacci Quarterly, 2 (1964), V. E. Hoggatt, J r., " Geeralized Rabbits for Geeralized Fiboacci N u m- b e r s, " F i M a c i ^ Q u r ^ r ] ^, 6 (1968), No. 3, I. Dale Ruggles, "Some F i b o a c c i Results U s i g Fiboacci - Type Sequeces," Fiboacci Quarterly, 1 (1963), No. 2, [Cotiued from page 481. ] 3. J. H. E. Coh, "Square Fiboacci N u m b e r s, Etc., " Fiboacci Quarterly, 2 (1964), J. H. E. Coh, "O Square Fiboacci N u m b e r s, " Jour. LodoMath. Soc., 39 (1964), J. H. E. Coh, " Lucas ad Fiboacci N u m b e r s ad Some Diophatie Equatios, " El2 i^^^^im^ia^2 i' 7 (1965), Fikelstei, R. P., "O the Uits i a Quartic Field with Applicatios to MordelPs Equatio," Doctoral Dissertatio, Arizoa State Uiversity, O. H e m e r, "O the Diophatie Equatio y 2 - k = x 3, "Doctoral D i s s e r t a - tio, Uppsala, O. H e m e r, "O Some Diophatie Equatios of the Type y 2 - f 2 = x 3, " Math. Scad., 4 (1956), H. Lodo ad R. Fikelstei, "O MordelPs Equatio y 2 - k = x 3, " to appear. 10. H. Lodo ad R. Fikelstei, "O MordelPs Equatio y 2 - k = x 3 : II. The Case k < 0, " to be submitted. 11. C. Siegel, "Zum Beweise des Starksche S a t z e s, " Ivetioes Math., 5 (1968), *