XProfan

Window Title "**************    CATALAN-ZAHLEN   ******************"
Font 2:randomize:cls rnd(8^8)
set("Decimals",0)
Declare nmax&,n&,p!
zugross:
locate 1,1
print " Catalan-numbers to charge (n_max=511): n =      ";
locate 1,45
input nmax&
case nmax&>511:goto "zugross"
n&=0

While n&<=nmax&

    p!=1

    WhileLoop 1,2*n&-1,2

        p!=p!*&Loop/(&Loop+1)

    EndWhile

    p!=p!*2^(2*n&)/(n&+1)
    'print " C("+st$(n&)+") = ";
    print n&;":";st$(p!)+"  ";
    case %pos>40: print

    if %csrlin>22 : WaitInput 2000: cls rnd(8^8): endif

        inc n&

    EndWhile

    print "\n\nEs follow some Info..."
    WaitInput 6000
    Cls rnd(8^8)
    print " 1.                                            "
    print " Catalan-numbers are benannt to              "
    print " charles Catalan, belgischer Mathematiker      "
    print " (1814-1894). it worked on Kettenbrüchen,   "
    print " Geometrie, Zahlentheorie and Kombinatorik.    "
    print " (Anm.: numbers this follow get already     "
    print " 1751 of Leonhard Euler in a letter on     "
    print " Christian Goldbach described. Euler sought  "
    print " The Number of Opportunities, one konvexes n- "
    print " Eck through Diagonalen in Dreiecke To decompose.)"
    print " --------------------------------------------  "
    print " Catalan-numbers having similar weight       "
    print " How z.B. the Pascal'sche Dreieck or         "
    print " The Fibonacci-follow.                          "
    print " --------------------------------------------  "
    print "                                               "
    WaitInput 20000
    cls rnd(8^8)
    print " 2.                                            "
    print " further Zuschreibungen:                       "
    print " --------------------------------------------  "
    print " The Catalanische supposition (1844) watts       "
    print " first 2002 of Mihailescu proved:            "
    print " 'except 2^3 and 3^2 there no real      "
    print " Potenzen, which circa very 1 discern'  "
    print " --------------------------------------------  "
    print " The Catalan'sche Constant G is the          "
    print " Grenzsumme of -1^n/(2*n+1)^2 for n=0..+Inf.  "
    print " G = 0,915965594177219015054603514932384110::  "
    print " ::77414937428167... (follow A006752 in OEIS)   "
    print " on the 16. april 2009 were  31026000000 comma-   "
    print " to put famous.                              "
    print " --------------------------------------------  "
    print "                                               "
    WaitInput 20000
    cls rnd(8^8)
    print " 3.                                            "
    print " Berechnung of Catalan-numbers:                "
    print " --------------------------------------------  "
    print " The n. Catalan-number C_n errechnet itself To     "
    print " 1/(n+1) * (2n OVER n) = (2*n)!/((n+1)!*n!)    "
    print " where 2n over n = Mittlerer Binomialkoeff.    "
    print " above-mentioned Formel is equivalent To                "
    print " C(n)=(2n OVER n) - (2n OVER n+1)              "
    print " --------------------------------------------  "
    print " only C2=2 and C3=5 are Primzahlen.         "
    print " --------------------------------------------  "
    print "                                               "
    WaitInput 20000
    cls rnd(8^8)
    print " 4.                                            "
    print " Applications with Catalan-numbers:               "
    print " --------------------------------------------  "
    print " Catalan-numbers zurück with Abzählungsaufgaben  "
    print " on, graphentheoretisch with undertow. Bäumen.      "
    print " --------------------------------------------  "
    print " C_n is too The Number of Klammerungen eines"
    print " Produktes,in the n Multiplikationen vorkommen,"
    print " or with n+1 factors so, that always only The  "
    print " Multipl. two factors durchzuführen is. "
    print " --------------------------------------------  "
    print "                                               "
    WaitInput 20000
    cls rnd(8^8)
    print " 5.                                            "
    print " Paths and Irrfahrten                          "
    print " too eindimensionale Irrfahrten of 0 to 2n "
    print " with initially-& Endpunkt in 0 so, that itself the "
    print " way never below the x-axis befindet: 2n=6:"
    print "  ///\\\\\  //\/\\\\ //\\\\/\ /\//\\\\ /\/\/\: C(3)=5  "
    print " --------------------------------------------  "
    print " C_n gives The number the Gitterwege of A to B."
    print " --------------------------------------------  "
    print " amount unterschiedlicher Binärbäumen with 2n+1 "
    print " nodes (or. n+1 flaking):   C(n)            "
    print " --------------------------------------------  "
    print "                                               "
    print "                   end                        "
    WaitInput
    End