A Variant of Pascal s Triangle Dennis Van Hise Stetson University

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1 A Vrit of Pscl s Trigle eis V Hise tetso Uiversity I. Itroductio d Nottio It is the purpose of this pper to explore vrit of Pscl s trigle. This vrit hs the rule tht every etry, deoted s,, where,, is clculted i wy such tht, i +. This mes tht whe these umbers re put ito trigulr formtio, every umber is the sum of ll the umbers bove it i its two digols. The begiig of the trigle loos lie the followig: For exmple, The rows of the trigle will begi t d wor dow the trigle i icremets of. The colums of the trigle poit southwest (60 o from horizotl) d will begi t, d proceed i icremets of. For exmple, we would sy tht the leftmost umber would be i row 5 d i colum. The ticolums re the colums of the trigle poitig southest tht begi t, d proceed i icremets of movig digolly from right to left. o, for exmple, the rightmost umber would be i row 5 d i ticolum. Filly, hllow digols poit southwest (0 o from horizotl). For exmple, the third shllow digol cosists of the elemets, d,.

2 II. Bsic Lemms & Theorems ome iterestig discoveries foud withi this trigle will ow be discussed, d proved, lot of which is doe through iductio. We begi with perhps the most obvious result. The trigle is symmetricl (i.e.:, +, ). Proof: (By Iductio) For d,,,+,. Now ssume tht,, + for some. The we c write i + i + i, + + by + defiitio. Further, + i - i + +, + + i +, i + + +, by the iductio hypothesis. This is the sme s + + i i, i + i + +, which mes +, +, +. Hece, by iductio, we coclude tht the trigle is symmetricl. We ow develop some recursive d explicit formuls for clcultig etries i the first three colums of the trigle. The first formul clcultes the etries i the first colum. Lemm:,, for. Proof: By defiitio,,, defiitio. Hece,,, for. +, +,, sice, by

3 A explicit formul for etries i the first colum is ow give., for. Proof: ice,,, the the theory of differece equtios implies, C. olvig for C whe 4, 4. We get C ½, which mes, for. The followig lemm is give to clculte the etries i the secod colum, d the ext theorem is the correspodig explicit formul. Lemm: +,, + for. Proof: We ow +,, i +, by defiitio. Further,,, + d i, +,. Thus, +,, +, + +. But,,. o, +, by defiitio, which implies +,, +,. Therefore, by Theorem, +,, + for., (+) 4 for 4. Proof: The theory of differece equtios implies, (A + B)( 4 ). olvig for A d B whe, d 4, 5, we get AB. o,, (+) 4. Note: A differet proof c be foud i the ppedix.

4 Filly, recursive formul d correspodig explicit formul is give to clculte the etries i the third colum, fter which there does ot pper to be simple formul for clcultig etries. Lemm:,, +, +, for 4. Proof: By defiitio,, i +, +, +, +, +, +, +, +, +, +, +, +. But,, i +, +, + by defiitio. Thus,,, +, +, +,, +, +, for 4., ( ) 7 for 4. Proof: The theory of differece equtios implies, C( 7 ) + ( 7 ) + E ( 7 ). olvig for C,, d E whe 4, 5, 5, 4, d 6, 7, we get C -4, 7, d E, which gives, ( ) 7. Note: A differet proof c be foud i the ppedix. 4

5 III. Other Lemms & Theorems These lemms d theorems expli the behvior i the rows d shllow digols of the vrit. Let deote the sum of the etries i the th row, d etries i the th shllow digol. deote the sum of the Lemm: for. Proof: Choose rbitrry elemet r,. Cosider the th row, where > r. The r, will pper i the formul for fidig, becuse it is i the sme colum. Further, r, will pper i the formul for fidig, + r becuse of the fct tht r, is i its ticolum. Hece, the rbitrry elemet, is summed exctly twice i the row for r -. Ad so, the sum of r the elemets i row would be equl to twice the sum of the elemets i ll previous rows. Therefore, we c write for. Lemm: for. Proof: It c be show tht +, d sice This mes. The ext theorem provides explicit formul for clcultig the sum of the etries i y give row. 5

6 (/9) for where d. Proof: From the theory of differece equtios d lemm 4, C +. ice d 6, solvig for C d, we get C 0 d (/9). Thus, (/9) Note: A differet proof c be foud i the ppedix.. The followig theorem c be used to clculte y give etry i the trigle, d is used to prove the ext couple theorems.,, +,, for. Proof: We ow, i +, i + ; d, i +,, d,, + i +, + i -, - i. o, ; + i, i + ;, +. o,,, +. Therefore,, -, + i,,, +, + -, i -. i Further,, -,, + -,, - i -. o,,, + -,, - i -. Also, +,,, +, -,. Hece,,, +, -,. With the ext two theorems, we re ble to clculte the sum of the etries i shllow digols. 6

7 + - for 5. Proof: We ow,, +, -, for. If we let be i,, the, is i ;, is i ; d, is i. Cosider r,, rbitrry elemet i. If we let r +-, the / / + / - / +,,, +,. It is importt to ote tht i some of the summtios there my exist terms i the form r, 0 or r, r+. These terms re ctully equl to zero d hece do ot ffect the sum. Thus, for 4. Proof: (By Iductio) For 4, Now ssume + for some. Also, recll + -. o, by the iductio hypothesis. Thus, + +. Hece, by iductio, we coclude +. 7

8 IV. The Trigle Mod d After exmiig the properties of the vrit, we decided to explore the trigle mod d, sice Pscl s Trigle cotied some iterestig properties i its trigle mod d. While Pscl s Trigle mod displys frctl structure, our vrit does ot, but it does disply some itriguig ptters mod. A picture of the trigle mod, s well s the metioed frctl ptters c be see i the colored trigle mod o the ext two pges. The followig theorem tells us tht the trigle mod does ot disply y frctl structure. The oly odd umbers i the trigle re, d etries of the form, d, +. These re cosecutive umbers i the middle of every eve umbered row. Proof: (By iductio o rows) Cosider row 4. We see tht row hs oly eve etries. Also, 4, d 4, 4 re composed of , which is eve. Further, 4, d 4, re composed of ++ 5, which is odd. o, 4, d 4, re the oly odd etries i row 4, d re i the ceter of the trigle. Thus, the hypothesis is true for row d row 4. Now ssume for some row r tht ll rows before d icludig it behve i the hypothesized fshio. Cosider rows r+ d r+. Recll tht,, +,,. o,, will be eve if, 0 (mod ). Thus, for row r+, r, + r, + r, r,. But r, is eve for ll etries i row r- by the iductio hypothesis, therefore r, (mod ). Hece, ll etries i row r+ re eve. For row r+ we ow tht r, will be eve except for the two cosecutive etries i the middle of the row. This mes tht ll etries i row r+ will be eve except for 8

9 9

10 0

11 the two cosecutive etries i the middle, which will be cogruet to (mod ), which implies tht they re odd. Thus, we coclude tht the iductio hypothesis is true. The remiig results llow us to uderstd the structure of the trigle mod, d how it behves., (mod ) if is eve d, (mod ) if is odd. Proof: (By iductio) We ow, - for >. o, for,, (mod ) d for 4, 4, (mod ). Now ssume, (mod ) if is eve d, (mod ) if is odd. The cosider +,,. If, (mod ), the +, (mod ); d if, (mod ), the +, (mod ). Thus, by iductio, we coclude tht if is eve,, (mod ) d if is odd,, (mod ). From the followig two lemms, we re ble to determie the structure of the etire trigle mod. Lemm: If,, + (mod ), the +, +, (mod ). Proof: Let, 0 d, + 0 (mod ). We ow +, +, +, + -,. o, +, + (0) + (0) 0 (mod ) which mes +, + 0 (mod ). Let, (mod ) d, + (mod ). The we c write +, + () + () 0 (mod ), which mes +, + (mod ). Filly, let, d, + (mod ). The this mes, +, + () +() 0 (mod ). Hece, +, + (mod ).

12 Lemm: If,, + (mod ), the, +, +, + (mod ). Proof: Cosider, (mod ) d, + (mod ). We ow +, +, +, + -,. o, +, + () + ()- 0 (mod ). Thus, +, + 0 (mod ). Now cosider, (mod ) d, + 0 (mod ). The +, + () + (0)- 0 (mod ). Hece, +, + (mod ). Filly, cosider, 0 (mod ) d, + (mod ). This mes +, + (0) + ()- 0 (mod ). Thus, +, + (mod ). I dditio, we hve two cojectures: Cojecture: I the trigle mod, trigulr formtios composed of zeroes will begi to te form if d oly if the row umber is multiple of 9. Cojecture: I the trigle mod where row 4 is ow cosidered row, there will be oly oe trigulr formtio composed of zeroes if d oly if the row umber c be writte s for.

13 V. ummry d Future Reserch To coclude, the followig re the properties tht hve bee foud: The trigle is symmetricl (i.e.:, +, ). Lemm:,,, for. for. Lemm: +,, + for., (+) 4 for 4. Lemm:,, +, +, for 4., ( ) 7 for 4. Lemm: Lemm: for. for. (/9),, for where d. +,, + - for 5. + for 4. The oly odd umbers i the trigle re, d etries of the form, d, +. These re cosecutive umbers i the middle of every eve umbered row.

14 , Lemm: If Lemm: If (mod ) if is eve d, (mod ) if is odd.,, + (mod ), the +, +, (mod ).,, + (mod ), the, +, +, + (mod ). Cojecture: I the trigle mod, trigulr formtios composed of zeroes will begi to te form if d oly if the row umber is multiple of 9. Cojecture: I the trigle mod where row 4 is ow cosidered row, there will be oly oe trigulr formtio composed of zeroes if d oly if the row umber c be writte s for. A good veue to te i cotiuig this reserch would be to ) prove the cojectures, ) serch for wys i which the trigle my be useful i vrious fields of mthemtics, s Pscl s trigle is, d ) cotiue drwig prllels betwee Pscl s trigle d the vrit. I do ot ecessrily ow wht these my etil, but they seem lie logicl ext step. 4

15 VI. Appedix ome results were prove iitilly by iductio d the more cocise proof ws discovered t lter time. The followig ltertive proofs re the oes tht were discovered first durig the reserch., (+) 4 for 4. Proof: (By Iductio) Whe 4, 4, () + (+) 4 is true for some. ice +,, + (+) 0 5 (4+), d we coclude by iductio tht, (+) 0. Now ssume tht, we hve +, ((+) 4 for 4. 4 ) +, ( ) 7 for 4. Proof: (By Iductio) First, for 5, 5, 4, +, +, (5) (-4 + 7(5) + 5 ) -. Now ssume tht, ( ) 7 is true for some. Proceedig iductively, +,, +, +, +, ( ) , [( ) + + ] 6 [(/) + (9/8) + ( /8)] ( ) 6, d we coclude by iductio tht, ( ) 7 for 4. (/9) for where d. Proof: (By Iductio) First, for, () 6 (/9). Now ssume tht (/9) for some. o, + (/9) + (/9) +. Thus, by iductio, + (/9) + for. 5