A135601 Acute-angled numbers with an internal digit as the vertex.

102, 103, 104, 105, 106, 107, 108, 109, 120, 130, 131, 132, 140, 141, 142, 143, 150, 151, 152, 153, 154, 160, 161, 162, 163, 164, 165, 170, 171, 172, 173, 174, 175, 176, 180, 181, 182, 183, 184, 185, 186, 187, 190, 191, 192, 193, 194, 195

Offset: 1

Author

Omar E. Pol, Dec 02 2007

Keywords

• base
• fini
• nonn
• full

Comments

The structure of digits represents an acute angle. The vertex is an internal digit. In the graphic representation the points are connected by imaginary line segments from left to right. This sequence is finite. The final term has 14 digits: 98765432102468.

Examples

			Illustration using the final term of this sequence:
  9 . . . . . . . . . . . . .
  . 8 . . . . . . . . . . . 8
  . . 7 . . . . . . . . . . .
  . . . 6 . . . . . . . . 6 .
  . . . . 5 . . . . . . . . .
  . . . . . 4 . . . . . 4 . .
  . . . . . . 3 . . . . . . .
  . . . . . . . 2 . . 2 . . .
  . . . . . . . . 1 . . . . .
  . . . . . . . . . 0 . . . .
		

Links

• Micah Manary, Table of n, a(n) for n = 1..2337
• Micah Manary, Python program

Crossrefs

Cf. A134810, A134853, A134941, A134970, A135641, A135642, A135643, A135600, A135602, A135603.

Programs

Python

progressions = set(tuple(range(i, j+1, d)) for i in range(10) for d in range(1, 10-i) for j in range(i+d, 10))
s = set()
for left in progressions:
    for right in progressions:
        dl, dr = left[1] - left[0], right[1] - right[0]
        if dl + dr > 2:
            if left[-1] == right[-1]: s.add(left[:-1] + right[::-1])
            if left[0] == right[0]: s.add(left[::-1] + right[1:])
afull = sorted(int("".join(map(str, t))) for t in s if t[0] != 0)
print(afull[:53]) # Michael S. Branicky, Aug 02 2022

Formula

If a(n) does not end in 0, then A004086(a(n)) is a term; if a(n) does not start with 9, then A061601(a(n)) is a term. - Michael S. Branicky, Aug 02 2022