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

1,2

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.

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

Mathematica

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

Prog

(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
(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

Crossrefs

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

Sequence in context: A342262 A255734 A357142 * A215014 A292439 A132577

Adjacent sequences: A033072 A033073 A033074 * A033076 A033077 A033078

Keyword

nonn,base,easy

Author

Clark Kimberling

Status

approved