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Patrick2788's avatar
Patrick2788
Silver Contributor
May 12, 2023

Create a 'Triforce' array (Sierpiński Triangle) with a formula

I've been studying 'pyramid' arrays lately and with a new Zelda title out, I was inspired to generate a left-justified 'Triforce' array with a formula. Sierpiński triangle - Wikipedia Triforce - Wi...
excel
Formulas and Functions
office 365
mtarler's avatar
mtarler
Silver Contributor
May 12, 2023
ok this ISODD() (pun intended). there are a couple of errors in your output and I don't think due to the formula. In particular on cell(26,11) the calculation is COMBIN(25,10) which = 3268760 and the result of ISODD is TRUE. This is the classic roundoff / representation inside of excel causing issues so you need to add INT() around that COMBIN (or around the N)
mtarler's avatar
mtarler
Silver Contributor
May 12, 2023

On another note I complicated the formula a bit for aesthetic improvements:

 

=MAKEARRAY(K2,K2,LAMBDA(r,c, IF(AND(r = 1, c = 1), "\", IF(c = r,"\",IF(c<r, IF(ISODD(INT(COMBIN(r - 1, c - 1)) ), IF(ISODD(INT(COMBIN(r - 2, c-1 )*COMBIN(r-1, c ))),"X","\"), ""), "")))))

 

or using characters I found in the Cambria Math font I could get:

 

 

  • Patrick2788's avatar
    Patrick2788
    Silver Contributor
    May 12, 2023

    mtarler 

    The left justified one is straightforward and maybe too easy!  This one below (Image) would take some work to add spaces where needed for the sake of appearance in Excel:
                 1

             1       1

         1       1      1

     

     

     

    • mtarler's avatar
      mtarler
      Silver Contributor
      May 12, 2023

      Patrick2788  here is a start:

       

      =LET(s,K2,MAKEARRAY(s,2*s,LAMBDA(r,i,LET(c,ROUND((r+i-s)/2,0),IF(c<1,"",IF(AND(r=1,c=1),K3,IF(c=r,K3,IF(c<r,IF(ISODD(INT(COMBIN(r-1,c-1))),IF(ISODD(INT(COMBIN(r-2,c-1)*COMBIN(r-1,c))),K4,K3),""),""))))))))

       

       

      getting closer:

      =LET(s,K2,MAKEARRAY(s,2*s,LAMBDA(r,i,LET(cc,(r+i-s)/2,c,ROUND(cc,0),cp,ROUND(cc+0.5,0),cm,ROUND(cc-0.5,0),
IF(c<1,"", IF(AND(r=1,c=1,cp=1),K5,IF(AND(c=r,cm=r),K6,
IF(c<=r,IF(ISODD(INT(COMBIN(r-1,c-1))),
           IF(ISODD(INT(IFERROR(COMBIN(r-1,cm-1),0))),
                IF(ISODD(INT(COMBIN(r-1,cp-1))),K7,
                IF(ISODD(INT(COMBIN(r-2,c-1))),K8,K6)),
                IF(ISODD(INT(IFERROR(COMBIN(r-2,c-1),0))),K5,K9)),""),""))))))))

       

       

      • Patrick2788's avatar
        Patrick2788
        Silver Contributor
        May 13, 2023

        mtarler 

        This is my solution:

         

        'Triforce
=DROP(
    REDUCE(
        "",
        SEQUENCE(32),
        LAMBDA(a, v,
            LET(
                binom, INT(COMBIN(v - 1, TAKE(Seq, , v * 2))),
                padding, EXPAND("", 1, 64 - v, ""),
                resize, EXPAND(1, 1, 2, 1),
                VSTACK(
                    a,
                    EXPAND(HSTACK(padding, IF(v = 1, resize, IF(ISODD(binom), 1, ""))), 1, 96, "")
                )
            )
        )
    ),
    1,
    32
)

         

        I achieved the 'doubling' with some help from Seq which is:

        Seq
=INT(SEQUENCE(, 64, 0, 0.5))

         

        Rolled it into a 'Triangle' Lambda where you can pick left justified or Triforce.

         

         

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