A220664 First differences of A030299.

11, 9, 102, 9, 81, 18, 81, 9, 913, 9, 81, 18, 81, 9, 702, 9, 171, 27, 72, 18, 693, 18, 72, 27, 171, 9, 702, 9, 81, 18, 81, 9, 8024, 9, 81, 18, 81, 9, 702, 9, 171, 27, 72, 18, 693, 18, 72, 27, 171, 9, 702, 9, 81, 18, 81, 9, 5913, 9, 81, 18, 81, 9, 1602, 9, 261

Offset

1,1

Comments

From M. F. Hasler, Jan 12 2013: (Start)

Note [updated Mar 03 2013]: The definition of sequence A030299 has been slightly modified in Jan. 2013, and as a consequence the following properties remain valid beyond the first A007489(9)-1 = 409112 terms, which had not been the case before, when A030299 had been defined through concatenation of the lexicographically ordered permutations, which in case of elements >= 10 broke up the nice mathematical properties (esp. of the sequence A219664 = 9*A217626 cited below).

This sequence taken modulo 9 is zero except (possibly) at indices where a run of permutations ends in A030299. (These indices are given by A007489(n), n>0.) There it equals (mod 9) the "n" of the following run. E.g., a(1)=2 (mod 9), and A030299(1+1)=12 is the start of the run for n=2; a(3)=3 (mod 9) and A030299(3+1)=123 is the start of the run for n=3, a(9)=4 (mod 9) and A030299(9+1)=1234 is the start of the run for n=4, etc.

The subsequence between these indices (A007489(n)+1,...,A007489(n+1)-1), always starts with the same terms, listed in A219664 = 9*A217626 (= A209280 = A107346 where the latter are defined). (End)

Links

Antti Karttunen, Table of n, a(n) for n = 1..5912

Formula

a(n) = A030299(n+1) - A030299(n).

a(n) = A219664(n-A007489(k)), for A007489(k) < n < A007489(k+1). - M. F. Hasler, Jan 13 2013

Example

A030299 starts (1, 12, 21, 123, 132, 213, 231, 312, ...), the first differences thereof yield (11, 9, 102, 9, 81, 18, 81, ...).

Maple

(l-> seq(l[j]-l[j-1], j=2..nops(l)))([seq(map(x-> parse(cat(x[])),
     combinat[permute](n))[], n=0..5)])[];  # Alois P. Heinz, Nov 09 2021

Prog

(Scheme): (define (A220664 n) (- (A030299 (+ 1 n)) (A030299 n)))
(PARI) {A030299=concat( vector( 5, k, vecsort( vector( (#k=vector(k, j, 10^j)~\10)!, i, numtoperm(#k, i-1)*k )))); A220664=vecextract(A030299, "^1")-vecextract(A030299, "^-1")} \\ M. F. Hasler, Jan 12 2013
(Python)
from itertools import permutations
def pmap(s, m): return sum(s[i-1]*10**(m-i) for i in range(1, len(s)+1))
def agen():
    m = 1
    while True:
        for s in permutations(range(1, m+1)): yield pmap(s, m)
        m += 1
def aupton(terms):
    alst, g = [], agen()
    t = next(g)
    while len(alst) < terms:
        t, prevt = next(g), t
        alst += [t - prevt]
    return alst
print(aupton(65)) # Michael S. Branicky, Nov 09 2021

Crossrefs

The repeating part is given by A219664, equal to A107346 for indices < 5!.

Sequence in context: A210284 A219746 A335854 * A038323 A121154 A326587

Adjacent sequences: A220661 A220662 A220663 * A220665 A220666 A220667

Keyword

nonn,base

Author

Antti Karttunen, Dec 17 2012

Status

approved