A033075 Positive numbers all of whose pairs of consecutive decimal digits differ by 1.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 101, 121, 123, 210, 212, 232, 234, 321, 323, 343, 345, 432, 434, 454, 456, 543, 545, 565, 567, 654, 656, 676, 678, 765, 767, 787, 789, 876

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• base
• easy

Comments

Number of n-digit terms: 9, 17, 32, 61, 116, 222, 424 (= A090994).

Also called 10-esthetic numbers (where in general a q-esthetic number has the property that the consecutive digits of its base-q representation differ by 1, see "Esthetic numbers" by J. M. De Koninck and N. Doyon). - Narad Rampersad, Aug 03 2018

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• J. M. De Koninck and N. Doyon, Esthetic numbers, Annales des Sciences mathématiques du Québec, 33 (2009), 155-164.

Crossrefs

Cf. A090994, A048398 (primes), A048411 (squares), A207954 (palindromes).

Programs

Haskell

-- import Data.Set (fromList, deleteFindMin, insert)
a033075 n = a033075_list !! (n-1)
a033075_list = f (fromList [1..9]) where
   f s | d == 0    = m : f (insert (10*m+1) s')
       | d == 9    = m : f (insert (10*m+8) s')
       | otherwise = m : f (insert (10*m+d-1) (insert (10*m+d+1) s'))
       where (m,s') = deleteFindMin s
             d = mod m 10
-- Reinhard Zumkeller, Feb 21 2012

Mathematica

Join[Range[9],Select[Range[2000],Union[Abs[Differences[IntegerDigits[#]]]]=={1}&]] (* Harvey P. Dale, Dec 28 2011 *)

PARI

diff(v)=vector(#v-1,i,v[i+1]-v[i])
is(n)=if(n>9, Set(abs(diff(digits(n))))==[1], n>0) \\ Charles R Greathouse IV, Mar 11 2014

Python

def ok(n):
    s = str(n)
    return all(abs(int(s[i]) - int(s[i+1])) == 1 for i in range(len(s)-1))
print(list(filter(ok, range(1, 877)))) # Michael S. Branicky, Aug 22 2021

Python

# faster version for initial segment of sequence
def gen(d, s=None): # generate remaining d digits, from start digit s
    if d == 0:
        yield tuple()
        return
    if s == None:
        yield from [(i, ) + g for i in range(1, 10) for g in gen(d-1, s=i)]
    else:
        if s > 0:
            yield from [(s-1, ) + g for g in gen(d-1, s=s-1)]
        if s < 9:
            yield from [(s+1, ) + g for g in gen(d-1, s=s+1)]
def agentod(digits):
    for d in range(1, digits+1):
        yield from [int("".join(map(str, g))) for g in gen(d, s=None)]
print(list(agentod(11))) # Michael S. Branicky, Aug 22 2021

Formula

a(n) >> n^3.53267..., where the exponent is log 10/log k and k is the largest root of x^5 - x^4 - 4x^3 + 3x^2 + 3x - 1. - Charles R Greathouse IV, Mar 11 2014