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A178476 Permutations of 123456: Numbers having each of the decimal digits 1,...,6 exactly once, and no other digit.

%I A178476 #23 Jul 04 2019 03:33:41
%S A178476 123456,123465,123546,123564,123645,123654,124356,124365,124536,
%T A178476 124563,124635,124653,125346,125364,125436,125463,125634,125643,
%U A178476 126345,126354,126435,126453,126534,126543,132456,132465,132546,132564,132645,132654
%N A178476 Permutations of 123456: Numbers having each of the decimal digits 1,...,6 exactly once, and no other digit.
%C A178476 This finite sequence contains 6!=720 terms.
%C A178476 This is a subsequence of A030299, consisting of elements A030299(154)..A030299(873).
%C A178476 If individual digits are be split up into separate terms, we get a subsequence of A030298.
%C A178476 It would be interesting to conceive simple and/or efficient functions which yield (a) the n-th term of this sequence: f(n)=a(n), (b) for a given term, the subsequent one: f(a(n)) = a(1 + (n mod 6!)).
%C A178476 The expression a(n+6) - a(n) takes only 18 different values for n = 1..6!-6.
%C A178476 An efficient procedure for generating the n-th term of this sequence can be found at A178475. - _Nathaniel Johnston_, May 19 2011
%C A178476 From _Hieronymus Fischer_, Feb 13 2013: (Start)
%C A178476 The sum of all terms as decimal numbers is 279999720.
%C A178476 General formula for the sum of all terms (interpreted as decimal permutational numbers with exactly d different digits from the range 1..d < 10): sum = (d+1)!*(10^d-1)/18.
%C A178476 If the terms are interpreted as base-7 numbers the sum is 49412160.
%C A178476 General formula for the sum of all terms of the corresponding sequence of base-p permutational numbers (numbers with exactly p-1 different digits excluding the zero digit): sum = (p-2)!*(p^p-p)/2. (End)
%H A178476 Nathaniel Johnston, <a href="/A178476/b178476.txt">Table of n, a(n) for n = 1..720</a> (full sequence)
%F A178476 a(n) + a(6! + 1 - n) = 777777.
%F A178476 floor( a(n) / 10^5 ) = ceiling( n / 5! ).
%F A178476 a(n) = A030299(n+153).
%F A178476 a(n) == 3 (mod 9).
%F A178476 a(n) = 3 + 9*A178486(n).
%t A178476 Take[FromDigits/@Permutations[Range[6]],40] (* _Harvey P. Dale_, Jun 05 2012 *)
%o A178476 (PARI) v=vector(6,i,10^(i-1))~; A178476=vecsort(vector(6!,i,numtoperm(6,i)*v));
%o A178476 is_A178476(x)= { vecsort(Vec(Str(x)))==Vec("123456") }
%o A178476 forstep( m=123456,654321,9, is_A178476(m) & print1(m","))
%Y A178476 Cf. A030298, A030299, A055089, A060117.
%K A178476 fini,full,easy,nonn,base
%O A178476 1,1
%A A178476 _M. F. Hasler_, May 28 2010