A135643 Straight-line numbers > 99.

111, 123, 135, 147, 159, 210, 222, 234, 246, 258, 321, 333, 345, 357, 369, 420, 432, 444, 456, 468, 531, 543, 555, 567, 579, 630, 642, 654, 666, 678, 741, 753, 765, 777, 789, 840, 852, 864, 876, 888, 951, 963, 975, 987, 999, 1111, 1234

Offset: 1

Author

Omar E. Pol, Nov 30 2007, Dec 09 2008, Nov 14 2009

Keywords

• nonn
• base

Comments

Numbers with more than two digits whose digits are in arithmetic progression. The structure of digits represents a straight line. In the graphic representation the points are connected by imaginary line segments. For a(1) to a(45) this sequence is equal to A034840. Each term of this sequence that is greater than 9876543210 is a repdigit number (A010785).

Note that the sequence of straight-line numbers starts: 10, 11, 12, ..., 98, 99, 111, 123, ... All 2-digit numbers are straight-line numbers, but here the numbers < 100 are omitted. - Omar E. Pol, Nov 14 2009

Examples

			The number 3579 is a straight-line number:
  . . . 9
  . . . .
  . . 7 .
  . . . .
  . 5 . .
  . . . .
  3 . . .
  . . . .
  . . . .
  . . . .
		

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 1..168

Crossrefs

Cf. A010785, A034840.

Cf. A135600, A135601, A135602, A135603, A135641, A135642, A163278, A167847. - Omar E. Pol, Nov 14 2009

Cf. A247616 (subsequence).

Programs

Haskell

a135643 n = a135643_list !! (n-1)
a135643_list = filter f [100..] where
   f x = all (== 0) ws where
         ws = zipWith (-) (tail vs) vs
         vs = zipWith (-) (tail us) us
         us = map (read . return) $ show x
-- Reinhard Zumkeller, Sep 21 2014

Mathematica

Select[Range[100,1300],Length[Union[Differences[IntegerDigits[#]]]]==1&] (* Harvey P. Dale, May 09 2012 *)

PARI

is(n) = my (d=digits(n), cvx=0, ccv=0, str=0); for (i=1, #d-2, my (x=d[i]+d[i+2]-2*d[i+1]); if (x>0, cvx++, x<0, ccv++, str++)); return (cvx==0 && ccv==0 && str>0) \\ Rémy Sigrist, Aug 09 2017

Python

from itertools import count, islice
def agen():
    progressions = ["".join(map(str, range(i, j+1, d))) for i in range(10) for d in range(1, 10-i) for j in range(i+2*d, 10)]
    s =  [p for p in progressions if p[0] != "0"]          # up
    s += [p[::-1] for p in progressions]                   # down
    s += [d*i for d in "123456789" for i in range(3, 11)]  # flat
    yield from sorted(set(int(w) for w in s))
    yield from (int(f*d) for d in count(11) for f in "123456789")
print(list(islice(agen(), 178))) # Michael S. Branicky, Aug 03 2022