|
%I #16 Apr 28 2016 12:42:00
%S 2,3,0,3,4,4,5,4,5,5,6,5,6,6,5,7,6,6,7,6,7,6,7,7,8,7,6,8,7,7,8,7,8,8,
%T 7,9,8,7,8,8,7,9,8,8,9,8,8,7,9,9,8,8,10,9,9,9,8,10,9,9,8,10,9,9,9,8,
%U 10,10,9,9,8,10,10,9,9,9,11,10,10,9,9,8,10,10,9,10,9,9
%N Irregular triangle read by rows, T(n,k) is the sum of base boxes of Pyramid arranged by n-boxes in k-th patterns.
%C The rules for Pyramid arrangement are: (1) boxes shall be arranged in symmetrical forms; (2) each step width shall be 0.5 or 1, where boxes width = 1.
%C The number of patterns on each n-th step is A053260(n).
%H Kival Ngaokrajang, <a href="/A227536/a227536.jpg">Illustration for n = 42, k = 1..8</a>
%e For n = 3..6.
%e [1] [1] [1|2] [1] [1] [1]
%e [2|3] [2|3|4] [3|4|5] [2|3|4|5] [2] [2|3]
%e [3|4|5] [4|5|6]
%e T(3,1) = 2, T(4,1) = 3, T(5,k) = 0 {no pattern exist due to step width vilolations i.e. [0.5,2], [1.5,1] & [1,0,1]}, T(6,1) = 3, ...
%e The triangle begins:
%e n/k 1 2 3 4 5
%e 3 2
%e 4 3
%e 5 0
%e 6 3
%e 7 4
%e 8 4
%e 9 5
%e 10 4
%e 11 5
%e 12 5
%e 13 6 5
%e 14 6
%e 15 6 5
%e 16 7 6
%e 17 6
%e 18 7 6
%e 19 7 6
%e 20 7 7
%e 21 8 7 6
%e 22 8 7 7
%e 23 8 7
%e 24 8 8 7
%e 25 9 8 7
%e 26 8 8 7
%e 27 9 8 8
%e 28 9 8 8 7
%e 29 9 9 8 8
%e 30 9 9 8 8
%e 31 10 9 9 9 8
%e ...
%e For n = 42, T(n,k) = 11, 11, 11, 10, 11, 10, 10, 9; see illustration in links.
%o (Small Basic)
%o x[0]=1
%o y[0]=1
%o for i = 1 To 12
%o a=math.Power(2,i-1)-2
%o b=math.Power(2,i)-2
%o For j = 1 To math.Power(2,i)
%o m=Math.Remainder(j,2)
%o k=math.Round(j/2+(1/2)*m)
%o y[j+b]=y[k+a]-m+2
%o x[j+b]=x[k+a]+y[j+b]
%o EndFor
%o EndFor
%o For n = 3 To 50
%o a[n]=0
%o c=1
%o For nn=1 To j+b
%o If n=x[nn] Then
%o a[n]=a[n]+1
%o aa[c]=y[nn]
%o c=c+1
%o Else
%o aa[c]=" "
%o EndIf
%o EndFor
%o For cc=1 To c
%o TextWindow.Write(aa[cc]+" ")
%o endfor
%o TextWindow.WriteLine(" ")
%o EndFor
%Y Cf. A053260.
%K nonn,tabf
%O 3,1
%A _Kival Ngaokrajang_, Jul 15 2013
|
