A062684 -id:A062684 - cp's OEIS frontend

A062896 Number of addition triangles with apex n (version 2).

1, 2, 2, 4, 4, 7, 7, 12, 12, 18, 19, 27, 28, 39, 41, 54, 58, 74, 78, 99, 106, 129, 139, 168, 179, 214, 229, 268, 289, 335, 357, 414, 443, 504, 540, 612, 653, 737, 786, 878, 938, 1045, 1111, 1234, 1313, 1444, 1539, 1692, 1795, 1965, 2082, 2273, 2414

Offset: 1

Naohiro Nomoto, Feb 11 2002

An addition triangle has any set of positive numbers as base; other rows are formed by adding pairs of adjacent numbers.

Reversing the base does not count as a different triangle.

			For n = 5:
    5
   2,3     5     5
  1,1,2   4,1   2,3   5.
with four different bases, so a(5) = 4.

Seiichi Manyama, Table of n, a(n) for n = 1..200

See A062684 for version 1 (counts reversals).
Equivalent sequences with restrictions on rows: A337765 (weakly increasing), A337766 (strongly increasing).
Equivalent sequence where n is the sum of all numbers in the triangle: A337787.
Cf. A028307, A066411.

Extended and edited by John W. Layman, Feb 14 2002

A066411 Form a triangle with the numbers [0..n] on the base, where each number is the sum of the two below; a(n) is the number of different possible values for the apex.

Original entry on oeis.org

1, 1, 3, 5, 23, 61, 143, 215, 995, 2481, 5785, 12907, 29279, 64963, 144289, 158049, 683311, 1471123, 3166531, 6759177, 14404547, 30548713

Offset: 0

Naohiro Nomoto, Dec 25 2001

a(n) is the number of different possible sums of c_k * (n choose k) where the c_k are a permutation of 0 through n. - Joshua Zucker, May 08 2006

			For n = 2 we have three triangles:
..4.......5.......3
.1,3.....2,3.....2,1
0,1,2...0,2,1...2,0,1
with three different values for the apex, so a(2) = 3.

Cf. A062684, A062896, A099325, A189162, A189390 (maximum apex value), A189391 (minimum apex value).

import Data.List (permutations, nub)
a066411 0 = 1
a066411 n = length $ nub $ map
   apex [perm | perm <- permutations [0..n], head perm < last perm] where
   apex = head . until ((== 1) . length)
                       (\xs -> (zipWith (+) xs $ tail xs))
-- Reinhard Zumkeller, Jan 24 2012

for n=0:9
size(unique(perms(0:n)*diag(fliplr(pascal(n+1)))),1)
end % Nathaniel Johnston, Apr 20 2011
(C++)
#include 
#include 
#include 
#include 
using namespace std;
inline long long pascApx(const vector & s)
{
    const int n = s.size() ;
    vector scp(n) ;
    for(int i=0; i s;
        for(int i=0;i apx;
        do
        {
            apx.insert( pascApx(s)) ;
        } while( next_permutation(s.begin(),s.end()) ) ;
        cout << n << " " << apx.size() << endl ;
    }
    return 0 ;
} /* R. J. Mathar, Jan 24 2012 */

g[s_List] := Plus @@@ Partition[s, 2, 1]; f[n_] := Block[{k = 1, lmt = 1 + (n + 1)!, lst = {}, p = Permutations[Range[0, n]]}, While[k < lmt, AppendTo[ lst, Nest[g, p[[k]], n][[1]]]; k++]; lst]; Table[ Length@ Union@ f@ n, {n, 0, 10}] (* Robert G. Wilson v, Jan 24 2012 *)

A066411(n)={my(u=0,o=A189391(n),v,b=vector(n++,i,binomial(n-1,i-1))~);sum(k=1,n!\2,!bittest(u,numtoperm(n,k)*b-o) & u+=1<<(numtoperm(n,k)*b-o))}  \\ M. F. Hasler, Jan 24 2012

from sympy import binomial
def partitionpairs(xlist): # generator of all partitions into pairs and at most 1 singleton, returning the sums of the pairs
    if len(xlist) <= 2:
        yield [sum(xlist)]
    else:
        m = len(xlist)
        for i in range(m-1):
            for j in range(i+1,m):
                rem = xlist[:i]+xlist[i+1:j]+xlist[j+1:]
                y = [xlist[i]+xlist[j]]
                for d in partitionpairs(rem):
                    yield y+d
def A066411(n):
    b = [binomial(n,k) for k in range(n//2+1)]
    return len(set((sum(d[i]*b[i] for i in range(n//2+1)) for d in partitionpairs(list(range(n+1)))))) # Chai Wah Wu, Oct 19 2021

More terms from John W. Layman, Jan 07 2003
a(10) from Nathaniel Johnston, Apr 20 2011
a(11) from Alois P. Heinz, Apr 21 2011
a(12) and a(13) from Joerg Arndt, Apr 21 2011
a(14)-a(15) from Alois P. Heinz, Apr 27 2011
a(0)-a(15) verified by R. H. Hardin Jan 27 2012
a(16) from Alois P. Heinz, Jan 28 2012
a(17)-a(21) from Graeme McRae, Jan 28, Feb 01 2012

A337766 Number of addition triangles with apex n where all rows are strongly increasing.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 9, 10, 11, 13, 14, 16, 17, 19, 22, 24, 25, 28, 31, 33, 35, 39, 43, 46, 48, 52, 57, 60, 63, 69, 75, 78, 82, 88, 94, 99, 104, 111, 119, 124, 129, 137, 147, 153, 160, 169, 179, 187, 194, 204, 216, 224, 233, 246, 259, 267, 277, 292, 308, 318, 329, 343, 361

Offset: 1

Seiichi Manyama, Sep 19 2020

nonn

An addition triangle has any finite sequence of positive numbers as base; other rows are formed by adding pairs of adjacent numbers.
If the bottom row is strongly increasing, then every row is strongly increasing.
    8
   3<5
  1<2<3

			For n = 5:
   5     5
  1,4   2,3   5
For n = 6:
   6     6
  1,5   2,4   6
For n = 7:
   7     7     7
  1,6   2,5   3,4   7
For n = 8:
    8
   3,5     8     8     8
  1,2,3   1,7   2,6   3,5   8
For n = 9:
    9
   3,6     9     9     9     9
  1,2,4   1,8   2,7   3,6   4,5   9

Seiichi Manyama, Table of n, a(n) for n = 1..500

Equivalent sequences with different restrictions on rows: A062684 (none, except terms are positive), A062896 (not a reversal of a counted row), A337765 (weakly increasing).

Cf. A346523.

def A(n)
  f_ary = [[n]]
  cnt = 1
  while f_ary.size > 0
    b_ary = []
    f_ary.each{|i|
      s = i.size
      (1..i[0] - 1).each{|j|
        a = [j]
        (0..s - 1).each{|k|
          num = i[k] - a[k]
          if num > 0
            a << num
          else
            break
          end
        }
        b_ary << a if a.size == s + 1 && a == a.uniq.sort
      }
    }
    f_ary = b_ary
    cnt += f_ary.size
  end
  cnt
end
def A337766(n)
  (1..n).map{|i| A(i)}
end
p A337766(50)

A337765 Number of addition triangles with apex n where all rows are weakly increasing.

Original entry on oeis.org

1, 2, 2, 4, 4, 5, 6, 9, 9, 11, 12, 15, 16, 18, 20, 26, 27, 29, 32, 37, 39, 43, 47, 53, 55, 60, 65, 72, 75, 80, 88, 99, 102, 108, 114, 125, 132, 141, 148, 159, 166, 176, 187, 200, 206, 218, 232, 249, 257, 268, 282, 301, 313, 327, 340, 360, 374, 393, 410, 429, 444, 465, 487, 516, 530, 550

Offset: 1

Seiichi Manyama, Sep 19 2020

nonn

An addition triangle has any set of positive numbers as base; other rows are formed by adding pairs of adjacent numbers.
If the bottom row are weakly increasing, then every rows are weakly increasing.
     5
   2<=3
  1<=1<=2

			For n = 5:
    5
   2,3     5     5
  1,1,2   1,4   2,3   5
For n = 6:
    6
   2,4     6     6     6
  1,1,3   1,5   2,4   3,3   6
For n = 7:
    7       7
   2,5     3,4     7     7     7
  1,1,4   1,2,2   1,6   2,5   3,4   7
For n = 8:
     8
    4,4       8       8       8
   2,2,2,    2,6     3,5     4,4     8     8     8     8
  1,1,1,1   1,1,5   1,2,3   2,2,2   1,7   2,6   3,5   4,4   8
For n = 9:
     9
    4,5       9       9       9
   2,2,3,    2,7     3,6     4,5     9     9     9     9
  1,1,1,2   1,1,6   1,2,4   2,2,3   1,8   2,7   3,6   4,5   9

Seiichi Manyama, Table of n, a(n) for n = 1..500

Cf. A062684, A062896, A337766.

def A(n)
  f_ary = [[n]]
  cnt = 1
  while f_ary.size > 0
    b_ary = []
    f_ary.each{|i|
      s = i.size
      (1..i[0] - 1).each{|j|
        a = [j]
        (0..s - 1).each{|k|
          num = i[k] - a[k]
          if num > 0
            a << num
          else
            break
          end
        }
        b_ary << a if a.size == s + 1 && a == a.sort
      }
    }
    f_ary = b_ary
    cnt += f_ary.size
  end
  cnt
end
def A337765(n)
  (1..n).map{|i| A(i)}
end
p A337765(50)

A337787 Number of addition triangles whose sum is n (version 2).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 6, 3, 5, 2, 8, 2, 8, 4, 8, 3, 12, 3, 11, 6, 11, 5, 15, 4, 16, 9, 14, 7, 20, 8, 18, 11, 20, 12, 25, 8, 25, 18, 24, 12, 31, 16, 32, 19, 29, 21, 39, 19, 36, 28, 38, 25, 47, 25, 46, 33, 46, 34, 55, 31, 56, 44, 55, 39, 67, 42, 66, 52, 66, 53, 76, 50, 81, 65, 77, 57

Offset: 1

Seiichi Manyama, Sep 21 2020

An addition triangle has any set of positive numbers as base; other rows are formed by adding pairs of adjacent numbers.

Reversing the base does not count as a different triangle.

			   n |
-----+-------------------------------
   1 |  1
-----+-------------------------------
   2 |  2
-----+-------------------------------
   3 |  3
-----+-------------------------------
   4 |      2
     |  4  1,1
-----+-------------------------------
   5 |  5
-----+-------------------------------
   6 |      3
     |  6  1,2
-----+-------------------------------
   7 |  7
-----+-------------------------------
   8 |      4    4
     |  8  1,3  2,2
-----+-------------------------------
   9 |  9
-----+-------------------------------
  10 |      5    5
     | 10  1,4  2,3
-----+-------------------------------
  11 |       4
     |      2,2
     | 11  1,1,1
-----+-------------------------------
  12 |      6    6    6
     | 12  1,5  2,4  3,3
-----+-------------------------------
  13 | 13
-----+-------------------------------
  14 |                      5
     |      7    7    7    2,3
     | 14  1,6  2,5  3,4  1,1,2
-----+-------------------------------
  15 | 15
-----+-------------------------------
  16 |                           6
     |      8    8    8    8    3,3
     | 16  1,7  2,6  3,5  4,4  1,2,1
-----+-------------------------------
  17 |       6      6
     |      2,4    3,3
     | 17  1,1,3  2,1,2
-----+-------------------------------
  18 |      9    9    9    9
     | 18  1,8  2,7  3,6  4,5
-----+-------------------------------
  19 |       7
     |      3,4
     | 19  1,2,2

Seiichi Manyama, Table of n, a(n) for n = 1..500

Cf. A014430, A062684, A062896, see A337785 for version 1.

def f(n)
  ary = [1]
  (n - 1).times{|i|
    ary = [0] + ary + [0]
    ary = (0..i + 1).map{|j| ary[j] + ary[j + 1] + 1}
  }
  ary
end
def A(n)
  f_ary = (1..n / 2).map{|i| [i]}
  cnt = 2
  s = 1
  while f_ary.size > 0
    s_ary = f(s + 1)
    b_ary = []
    f_ary.each{|i|
      (1..i[0] - 1).each{|j|
        a = [j]
        (0..s - 1).each{|k|
          num = i[k] - a[k]
          if num > 0
            a << num
          else
            break
          end
        }
        if a.size == s + 1
          sum = (0..s).inject(0){|t, m| t + s_ary[m] * a[m]}
          if sum < n
            b_ary << a
          elsif sum == n
            cnt += 1
            cnt += 1 if a == a.reverse
          end
        end
      }
    }
    f_ary = b_ary
    s += 1
  end
  cnt / 2
end
def A337787(n)
  (1..n).map{|i| A(i)}
end
p A337787(50)

A337785 Number of addition triangles whose sum is n (version 1).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 2, 6, 1, 9, 1, 9, 4, 9, 3, 14, 2, 14, 6, 14, 5, 21, 4, 19, 10, 21, 8, 27, 6, 29, 16, 25, 12, 38, 14, 33, 19, 37, 22, 46, 14, 47, 33, 45, 22, 59, 29, 59, 35, 56, 40, 74, 34, 68, 53, 72, 47, 90, 47, 88, 63, 88, 64, 105, 59, 108, 84, 106, 75, 130, 81, 125, 99, 128, 103, 147

Offset: 1

Seiichi Manyama, Sep 21 2020

An addition triangle has any set of positive numbers as base; other rows are formed by adding pairs of adjacent numbers.

Reversing the base counts as a different triangle.

			   n |
-----+------------------------------------------------
   1 |  1
-----+------------------------------------------------
   2 |  2
-----+------------------------------------------------
   3 |  3
-----+------------------------------------------------
   4 |      2
     |  4  1,1
-----+------------------------------------------------
   5 |  5
-----+------------------------------------------------
   6 |      3    3
     |  6  1,2  2,1
-----+------------------------------------------------
   7 |  7
-----+------------------------------------------------
   8 |      4    4    4
     |  8  1,3  2,2  3,1
-----+------------------------------------------------
   9 |  9
-----+------------------------------------------------
  10 |      5    5    5    5
     | 10  1,4  2,3  3,2  4,1
-----+------------------------------------------------
  11 |       4
     |      2,2
     | 11  1,1,1
-----+------------------------------------------------
  12 |      6    6    6    6    6
     | 12  1,5  2,4  3,3  4,2  5,1
-----+------------------------------------------------
  13 | 13
-----+------------------------------------------------
  14 |                                     5      5
     |      7    7    7    7    7    7    2,3    3,2
     | 14  1,6  2,5  3,4  4,3  5,2  6,1  1,1,2  2,1,1

Seiichi Manyama, Table of n, a(n) for n = 1..500

Cf. A014430, A062684, A062896, A337765, A337766, see A337787 for version 2.

def f(n)
  ary = [1]
  (n - 1).times{|i|
    ary = [0] + ary + [0]
    ary = (0..i + 1).map{|j| ary[j] + ary[j + 1] + 1}
  }
  ary
end
def A(n)
  f_ary = (1..n / 2).map{|i| [i]}
  cnt = 1
  s = 1
  while f_ary.size > 0
    s_ary = f(s + 1)
    b_ary = []
    f_ary.each{|i|
      (1..i[0] - 1).each{|j|
        a = [j]
        (0..s - 1).each{|k|
          num = i[k] - a[k]
          if num > 0
            a << num
          else
            break
          end
        }
        if a.size == s + 1
          sum = (0..s).inject(0){|t, m| t + s_ary[m] * a[m]}
          if sum < n
            b_ary << a
          elsif sum == n
            cnt += 1
          end
        end
      }
    }
    f_ary = b_ary
    s += 1
  end
  cnt
end
def A337785(n)
  (1..n).map{|i| A(i)}
end
p A337785(50)

A337719 The number of maximally large absolute-difference triangles consisting of positive integers <= n.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 44, 72, 128, 220, 380, 620, 1232, 2400, 3988, 7008, 14260, 25512, 50944, 105560, 197880, 381432, 785984, 1443992, 2981200, 6623144, 13044340, 26020924, 55781760, 108592260, 231819360, 526660160, 1071224176, 2231977656, 4950184948, 10009562624

Offset: 1

Samuel B. Reid, Sep 16 2020

nonn

a(17) is the first term that is more than twice its predecessor.
All terms after a(2) are divisible by four. This is because valid starting layers (of length greater than two) produce distinct valid starting layers when subjected to either or both of two transformations.
  .
      1       1
     2 1     1 2
    1 3 2   2 3 1
      *   |   *
        * | *
    *     |     *
    ------|------
    *     |     *
        * | *
      *   |   *
    3 1 2   2 1 3
     2 1     1 2
      1       1
  .
  There is the obvious reflection about the y-axis (reversal), and there is the somewhat less obvious reflection about the x-axis. Reflection about the x-axis is valid because absolute differences are maintained. Note that it is not possible for a solution to be equivalent to any of its own transformations. If it were, the base layer or the layer that succeeds it would need to be palindromic. This is invalid because any absolute-difference triangle with a palindromic base and a height greater than one is topped with a zero.

			a(5) = 16
.
  1 2 5 1 2   3 2 5 1 2   3 4 1 5 4   5 4 1 5 4
   1 3 4 1     1 3 4 1     1 3 4 1     1 3 4 1
    2 1 3       2 1 3       2 1 3       2 1 3
     1 2         1 2         1 2         1 2
      1           1           1           1
.
  3 1 5 4 2   3 5 1 2 4   1 4 5 1 4   5 2 1 5 2
   2 4 1 2     2 4 1 2     3 1 4 3     3 1 4 3
    2 3 1       2 3 1       2 3 1       2 3 1
     1 2         1 2         1 2         1 2
      1           1           1           1
.
  2 4 5 1 3   4 2 1 5 3   2 5 1 2 5   4 1 5 4 1
   2 1 4 2     2 1 4 2     3 4 1 3     3 4 1 3
    1 3 2       1 3 2       1 3 2       1 3 2
     2 1         2 1         2 1         2 1
      1           1           1           1
.
  2 1 5 2 1   2 1 5 2 3   4 5 1 4 3   4 5 1 4 5
   1 4 3 1     1 4 3 1     1 4 3 1     1 4 3 1
    3 1 2       3 1 2       3 1 2       3 1 2
     2 1         2 1         2 1         2 1
      1           1           1           1
.

Samuel B. Reid, Python program for A337719
Samuel B. Reid, C program for A337719

Cf. A062684, A094225.

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A062896 Number of addition triangles with apex n (version 2).

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A066411 Form a triangle with the numbers [0..n] on the base, where each number is the sum of the two below; a(n) is the number of different possible values for the apex.

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A337766 Number of addition triangles with apex n where all rows are strongly increasing.

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A337765 Number of addition triangles with apex n where all rows are weakly increasing.

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A337787 Number of addition triangles whose sum is n (version 2).

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A337785 Number of addition triangles whose sum is n (version 1).

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A337719 The number of maximally large absolute-difference triangles consisting of positive integers <= n.

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