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%I #6 Aug 24 2012 10:50:03
%S 1,2,3,13,23,33,133,4,14,24,34,134,234,334,5,15,25,35,135,235,335,
%T 1335,45,145,245,345,1345,2345,3345,55,155,255,355,1355,2355,3355,
%U 13355,455,1455,2455,3455,13455,23455,33455,555,1555,2555,6,16,26,36,136,236,336
%N Representation of n = sum_k b_k*(k!!) in the double-factorial base by some b_k-fold concatenation of the indices k.
%C The encoding of n is similar to A111095 but uses a double-factorial base A006882 to define the expansion coefficients.
%C The expansion coefficients b_k in n = sum_{k>=1} b_k * A006882(k) are defined "greedily" by taking the largest A006882(k) which is <=n, choosing b_k as large as possible such that b_k*A006882(k) remains <=n, subtracing b_k*A006882(k) from n to define a remainder, and recursively slicing the remainder to generate b_{k-1}, then b_{k-2} etc until the remainder reduces to zero. This produces the b_k for each n equivalent to A019513(n).
%C This representation A019513 is then scanned from the least to the most-significant b_k, i.e., along increasing k, and for each nonzero b_k, b_k copies of k are appended to a string representation -- starting from an empty string. This final representation is interpreted as a base-10 number a(n).
%e a(39) = 1455 because 1!!+4!!+5!!+5!! = 1+8+15+15 = 39
%p dblfactfloor := proc(n) local j ; for j from 1 do if doublefactorial(j) > n then return j-1 ; end if; end do: end proc:
%p dblfbase := proc(n) local nshf,L,f; nshf := n ; L := [] ; while nshf > 0 do f := dblfactfloor(nshf) ; L := [f,op(L)] ; nshf := nshf-doublefactorial(f) ; end do: L ; end proc:
%p read("transforms") ; A181521 := proc(n) digcatL(dblfbase(n)) ; end proc:
%p seq(A181521(n),n=1..70) ; # _R. J. Mathar_, Dec 06 2010
%Y Cf. A111095
%K easy,nonn,base
%O 1,2
%A _Paolo P. Lava_ and _Giorgio Balzarotti_, Oct 26 2010
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