A135603 Obtuse-angled numbers with an internal digit as the vertex.

100, 110, 112, 113, 114, 115, 116, 117, 118, 119, 122, 124, 125, 126, 127, 128, 129, 133, 134, 136, 137, 138, 139, 144, 145, 146, 148, 149, 155, 156, 157, 158, 166, 167, 168, 169, 177, 178, 179, 188, 189, 199, 200, 211, 220, 221, 223, 224, 225, 226, 227, 228, 229, 233

Offset: 1

Author

Omar E. Pol, Dec 02 2007

Keywords

• base
• nonn

Comments

The structure of digits represents an obtuse angle. The vertex is an internal digit. In the graphic representation the points are connected by imaginary line segments from left to right.

For each k >= 11, there are 354 k-digit terms. - Michael S. Branicky, Aug 03 2022

Examples

			Illustration of the number 9753111:
  9 . . . . . .
  . . . . . . .
  . 7 . . . . .
  . . . . . . .
  . . 5 . . . .
  . . . . . . .
  . . . 3 . . .
  . . . . . . .
  . . . . 1 1 1
  . . . . . . .
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10232

Crossrefs

Cf. A134810, A134853, A134941, A134970, A135641, A135642, A135643, A135600, A135601, A135602.

Programs

Python

from itertools import count, islice
def ok3(n):
    if n < 100: return False
    d = list(map(int, str(n)))
    m1, m2 = (d[1]-d[0], d[-1]-d[-2])
    return len({m1, m2}) == 2 and m1*m2 >= 0
def agen():
    seeds = [k for k in range(100, 1000) if ok3(k)]
    for digits in count(4):
        yield from sorted(seeds)
        new, pow10 = set(), 10**(digits-1)
        for q in seeds:
            d = list(map(int, str(q)))
            e1, e2 = d[0] - (d[1]-d[0]), d[-1] + (d[-1]-d[-2])
            if 9 >= e1 > 0: new.add(e1*pow10 + q)
            if 9 >= e2 >= 0: new.add(10*q + e2)
        seeds = new
print(list(islice(agen(), 54))) # Michael S. Branicky, Aug 03 2022

Extensions

a(49) and beyond from Michael S. Branicky, Aug 03 2022