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.

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, 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, 10, 10, 9, 9, 8, 10, 10, 9, 9, 9, 11, 10, 10, 9, 9, 8, 10, 10, 9, 10, 9, 9

Offset

3,1

Comments

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.

The number of patterns on each n-th step is A053260(n).

Links

Table of n, a(n) for n=3..90.

Kival Ngaokrajang, Illustration for n = 42, k = 1..8

Example

For n = 3..6.
   [1]    [1]    [1|2]     [1]     [1]       [1]
  [2|3] [2|3|4] [3|4|5] [2|3|4|5]  [2]      [2|3]
                                 [3|4|5]   [4|5|6]
  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, ...
The triangle begins:
n/k 1 2 3 4 5
3   2
4   3
5   0
6   3
7   4
8   4
9   5
10  4
11  5
12  5
13  6 5
14  6
15  6 5
16  7 6
17  6
18  7 6
19  7 6
20  7 7
21  8 7 6
22  8 7 7
23  8 7
24  8 8 7
25  9 8 7
26  8 8 7
27  9 8 8
28  9 8 8 7
29  9 9 8 8
30  9 9 8 8
31  10 9 9 9 8
...
For n = 42, T(n,k) = 11, 11, 11, 10, 11, 10, 10, 9; see illustration in links.

Prog

(Small Basic)
x[0]=1
y[0]=1
for i = 1 To 12
a=math.Power(2, i-1)-2
b=math.Power(2, i)-2
  For j = 1 To math.Power(2, i)
   m=Math.Remainder(j, 2)
   k=math.Round(j/2+(1/2)*m)
   y[j+b]=y[k+a]-m+2
   x[j+b]=x[k+a]+y[j+b]
  EndFor
EndFor
For n = 3 To 50
  a[n]=0
  c=1
  For nn=1 To j+b
    If n=x[nn] Then
      a[n]=a[n]+1
      aa[c]=y[nn]
      c=c+1
    Else
      aa[c]=" "
      EndIf
  EndFor
  For cc=1 To c
    TextWindow.Write(aa[cc]+" ")
  endfor
  TextWindow.WriteLine(" ")
EndFor

Crossrefs

Cf. A053260.

Sequence in context: A082116 A079777 A224909 * A047773 A279416 A331781

Adjacent sequences: A227533 A227534 A227535 * A227537 A227538 A227539

Keyword

nonn,tabf

Author

Kival Ngaokrajang, Jul 15 2013

Status

approved