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A181521
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Representation of n = sum_k b_k*(k!!) in the double-factorial base by some b_k-fold concatenation of the indices k.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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EXAMPLE
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a(39) = 1455 because 1!!+4!!+5!!+5!! = 1+8+15+15 = 39
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MAPLE
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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:
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CROSSREFS
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KEYWORD
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easy,nonn,base
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AUTHOR
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STATUS
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approved
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