Pellian sequence relationships among π, e, 2

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1 otes o umber Theory ad Discrete Mathematics Vol. 8, 0, o., 58 6 Pellia sequece relatioships amog π, e, J. V. Leyedekkers ad A. G. Shao Faculty of Sciece, The Uiversity of Sydey Sydey, SW 006, Australia Faculty of Egieerig & IT, Uiversity of Techology Sydey, SW 00, Australia s: tshao8@gmail.co athoy.shao@uts.edu.au Abstract: The umerators ad deomiators of the covergets of the cotiued fractios of π, e ad are show to be elemets of secod order recurrece sequeces of the Pellia or Fiboacci variety which are related to Pythagorea triples (c = b + a, b > a). π ad have surprisigly similar structures except that has primitive Pythagorea triples with c b = or b a =, whereas π has c b eve ad ot costat ad b a ot costat, although the right-ed-digits are costat. Keywords: Iteger structure aalysis, Modular rigs, Prime umbers, Fiboacci umbers, Ifiite series, Pell sequece, Cotiued fractios, Primitive Pythagorea triples, Right-ed-digits. AMS Classificatio: A4, A55, A0. Itroductio We have recetly show that π = Q (.) where Q is the ratio of the quarter circumferece of a circle to the side of the iscribed square []. Here we exted the study to the structure of the irratioals e ad ad compare with π. The first six covergets of their cotiued fractios are set out i Table. umber Covergets π e Table. First six covergets for, π ad e 58

2 The covergets for The covergets D from the first row of Table satisfy the secod order liear recurrece relatios [8]:, D D D. (.) with iitial terms ad i the umerators { }, ad ad i the deomiators {D } (the stadard Pell sequece [8]). From the relatioship betwee the Pell ad Pell-Lucas sequeces [8], it has bee show [] that Pellia sequeces ca be geerated from the z j grid [] set up to characterise Pythagorea triples (Table ) c = b + a, b > a Two questios arise: Are the sequeces, { } ad {D } related to primitive Pythagorea triples (ppts)?, ad Are there similar structures for π ad e? odd eve z = c b, b > a c b a y = b a (K ) j + (j + z ½ ) j(j + z ½ ) z ½ (j + z ½ ) j z d + f df f d K [( z) ½ + j ] + z [( z) ½ + j ] z [( z) ½ +j ] ( z) ½ d + f f d df Table. z j grid for Pythagorea triples: j is the iteger couter; criterio for geeratig ppts is (j, z ½ ) = whe z > ; if z = oly ppts are obtaied. 59 (j ) z The elemets of the umerator sequece, { }, are all odd ad it is foud that they equal d (= j) for ppts with z = (Table ). d = j f = j + z ½ ppts y = j z z = j y 5, 4, 4 5, 4, 8,, , 6, , 444, , 9800, , 40, , 660, Table. umerators ad ppts The sequeces {f} ad {y} are ot preset i [8]. However, {f} {u } where u u u. a Pellia o-homogeeous secod order recurrece relatio with iitial terms, ad []. That is, {f} satisifies

3 f f f The other iteral parameters are z =, ad y (c b) which also satisfies a Pellia ohomogeeous recurrece relatio: y 6 y y a (.) i which 4 eve, a odd. The elemets of the sequece of deomiators, {D }, equal d, f pairs for ppts with y (b a) = (Table 4).. d = j f = j + z ½ ppts z z y, 5, 4,, , 0, , , 4060, , , 904, Table 4. Deomiators ad ppts Agai we ca fid Pellia-type recurrece relatios; for istace, { z } satisfies (.) with a = 0. Covergece of e ad π The covergets of e i the third row of Table oscillate betwee Pellia ad Fiboacci sequeces (Table 5).,,,,4,5 4,5,6 Recurrece relatio 60 4 Type Pellia Fiboacci Pellia Table 5. Recurrece relatios for covergets of e The covergets of π i the secod row of Table also oscillate betwee Pellia ad Fiboacci sequeces (Table 6).,,,,4,4,5 4,5,6 Recurrece 5 relatio 9 Type Pellia Fiboacci Pellia Fiboacci Table 6. Recurrece relatios (Rr) for covergets of π The Pellia-type sequeces are agai associated with ppts. For example, for π the first two are the d ad f of the triple: {49, 45, } with z = 8 ad y = 4. If () is

4 take as d, the the triple is {505, 555, 55464} with z = 8 (8 ) ad y = 6. Whe D =, 06 for d ad f, this yields the triple {85, 8, 484} with z = 98 ad y = 90. The value of z is eve ad has a right-ed-digit (RED) of 8 while y has a RED of. The REDs remai costat while the values of z ad y vary. This is i cotrast to the system where z = for { } ad y = for {D }. The cotiued fractio for π is:... (.) 5 9 The occurrece of 5, ad 9, ad, i the partial quotiets ad the coefficiets i the Pellia-type recurrece relatios ivites further ivestigatio. These partial quotiets, ulike those for e ad, have ot bee foud to obey ay simple laws []. It is somewhat surprisig the that the Pellia relatios for e ad π have similar patters which are a mix of the secod order recurrece relatios [4]. The recurrece relatios associated with differet patters i their liks with ppts, i cotrast to those of e ad π. follow 4 Cocludig commets Topics for further research readily emerge. For istace, if we take the recurrece relatio (.) ad geeralize it to the homogeeous form w w w (4.) 6 with iitial coditios w, w, m =,,,, we get the tableau i Table : m m Referece [5] 64 4 [4] [9] [] [5] [6] [5] [6] Table. Examples of recursive sequece defied by (4.); w w w m, We otice that w wm, w6,, (4.) ad if w w w, the w6 w6, 5m ad w 6, w5,. To what extet ca these results be geeralized?

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