A302907 -id:A302907 - cp's OEIS frontend

A301516 Numbers n with decimal expansion (d_1, ..., d_k) such that the convex hull of the set of points { (i, d_i), i = 1..k } has positive area.

100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 148, 149, 150, 151, 152, 153, 154, 155

Offset: 1

Rémy Sigrist, Dec 16 2018

This sequence is the complement of the straight-line numbers (0..99 alongside A135643).

This sequence first differs from A134999 at n = 857: a(857) = 1001 whereas A134999(857) = 1011.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A134999, A135643, A302907.

is(n, base=10) = my (d=digits(n, base)); for (i=1, #d-2, if (d[i]+d[i+2]-2*d[i+1], return (1))); return (0)

def ok(n):
    d = list(map(int, str(n)))
    return any(d[i]+d[i+2]-2*d[i+1] != 0 for i in range(len(d)-2))
print([k for k in range(1002) if ok(k)]) # Michael S. Branicky, Aug 03 2022 after Rémy Sigrist

A322629 For a nonnegative number m with decimal digits (d_1, ..., d_k), let s(m) be the area of the convex hull of the set of points { (i, d_i), i = 1..k }; a(n) = 2 * s(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 15, 14, 13

Offset: 100

Rémy Sigrist, Dec 21 2018

The data section starts at offset 100, however the sequence is well-defined for smaller values of n: a(n) = 0 for n = 0...99.

			For n = 1212:
- the corresponding convex hull is as follows:
      (2,2) +-----+ (4,2)
           /     /
          /     /
   (1,1) +-----+ (3,1)
- it has area 2, hence a(1212) = 4.

Rémy Sigrist, Table of n, a(n) for n = 100..10000
Rémy Sigrist, PARI program for A322629

Cf. A301516, A302907.

PARI
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A302907(n) = a(prime(n)) where n denotes the n-th prime number.
a(10^n) = n-1 for any n > 0.
a(n) > 0 iff n belongs to A301516.

cp's OEIS Frontend

A301516 Numbers n with decimal expansion (d_1, ..., d_k) such that the convex hull of the set of points { (i, d_i), i = 1..k } has positive area.

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