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

1,2

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).

Links

Table of n, a(n) for n=1..54.

Example

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

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

Crossrefs

Cf. A111095

Sequence in context: A136260 A296932 A288054 * A133201 A215302 A103150

Adjacent sequences: A181518 A181519 A181520 * A181522 A181523 A181524

Keyword

easy,nonn,base

Author

Paolo P. Lava and Giorgio Balzarotti, Oct 26 2010

Status

approved