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A227536 Irregular triangle read by rows, T(n,k) is the sum of base boxes of Pyramid arranged by n-boxes in k-th patterns.

%I A227536 #16 Apr 28 2016 12:42:00
%S A227536 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 A227536 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 A227536 10,10,9,9,8,10,10,9,9,9,11,10,10,9,9,8,10,10,9,10,9,9
%N A227536 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 A227536 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 A227536 The number of patterns on each n-th step is A053260(n).
%H A227536 Kival Ngaokrajang, <a href="/A227536/a227536.jpg">Illustration for n = 42, k = 1..8</a>
%e A227536 For n = 3..6.
%e A227536    [1]    [1]    [1|2]     [1]     [1]       [1]
%e A227536   [2|3] [2|3|4] [3|4|5] [2|3|4|5]  [2]      [2|3]
%e A227536                                  [3|4|5]   [4|5|6]
%e A227536   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 A227536 The triangle begins:
%e A227536 n/k 1 2 3 4 5
%e A227536 3   2
%e A227536 4   3
%e A227536 5   0
%e A227536 6   3
%e A227536 7   4
%e A227536 8   4
%e A227536 9   5
%e A227536 10  4
%e A227536 11  5
%e A227536 12  5
%e A227536 13  6 5
%e A227536 14  6
%e A227536 15  6 5
%e A227536 16  7 6
%e A227536 17  6
%e A227536 18  7 6
%e A227536 19  7 6
%e A227536 20  7 7
%e A227536 21  8 7 6
%e A227536 22  8 7 7
%e A227536 23  8 7
%e A227536 24  8 8 7
%e A227536 25  9 8 7
%e A227536 26  8 8 7
%e A227536 27  9 8 8
%e A227536 28  9 8 8 7
%e A227536 29  9 9 8 8
%e A227536 30  9 9 8 8
%e A227536 31  10 9 9 9 8
%e A227536 ...
%e A227536 For n = 42, T(n,k) = 11, 11, 11, 10, 11, 10, 10, 9; see illustration in links.
%o A227536 (Small Basic)
%o A227536 x[0]=1
%o A227536 y[0]=1
%o A227536 for i = 1 To 12
%o A227536 a=math.Power(2,i-1)-2
%o A227536 b=math.Power(2,i)-2
%o A227536   For j = 1 To math.Power(2,i)
%o A227536    m=Math.Remainder(j,2)
%o A227536    k=math.Round(j/2+(1/2)*m)
%o A227536    y[j+b]=y[k+a]-m+2
%o A227536    x[j+b]=x[k+a]+y[j+b]
%o A227536   EndFor
%o A227536 EndFor
%o A227536 For n = 3 To 50
%o A227536   a[n]=0
%o A227536   c=1
%o A227536   For nn=1 To j+b
%o A227536     If n=x[nn] Then
%o A227536       a[n]=a[n]+1
%o A227536       aa[c]=y[nn]
%o A227536       c=c+1
%o A227536     Else
%o A227536       aa[c]=" "
%o A227536       EndIf
%o A227536   EndFor
%o A227536   For cc=1 To c
%o A227536     TextWindow.Write(aa[cc]+" ")
%o A227536   endfor
%o A227536   TextWindow.WriteLine(" ")
%o A227536 EndFor
%Y A227536 Cf. A053260.
%K A227536 nonn,tabf
%O A227536 3,1
%A A227536 _Kival Ngaokrajang_, Jul 15 2013