A181521 Representation of n = sum_k b_k*(k!!) in the double-factorial base by some b_k-fold concatenation of the indices k.

1, 2, 3, 13, 23, 33, 133, 4, 14, 24, 34, 134, 234, 334, 5, 15, 25, 35, 135, 235, 335, 1335, 45, 145, 245, 345, 1345, 2345, 3345, 55, 155, 255, 355, 1355, 2355, 3355, 13355, 455, 1455, 2455, 3455, 13455, 23455, 33455, 555, 1555, 2555, 6, 16, 26, 36, 136, 236, 336

Offset: 1

Author

Paolo P. Lava and Giorgio Balzarotti, Oct 26 2010

Keywords

• easy
• nonn
• base

Comments

The encoding of n is similar to A111095 but uses a double-factorial base A006882 to define the expansion coefficients.

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).

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).

Examples

			a(39) = 1455 because 1!!+4!!+5!!+5!! = 1+8+15+15 = 39
		

Crossrefs

Cf. A111095

Programs

Maple

dblfactfloor := proc(n) local j ; for j from 1 do if doublefactorial(j) > n then return j-1 ; end if; end do: end proc:
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:
read("transforms") ; A181521 := proc(n) digcatL(dblfbase(n)) ; end proc:
seq(A181521(n),n=1..70) ; # R. J. Mathar, Dec 06 2010