

A330859


The additive version of the 'Decade transform' : to obtain a(n) write n as a sum of its poweroften parts and then continue to calculate the sum of the adjacent parts until a single number remains.


2



100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 220, 221, 222
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OFFSET

100,1


COMMENTS

Due to its construction a(n) = n for n=0..109, thus the data section shows a(n) for n >= 100.
To obtain the additive version of the 'Decade transform' of n first write n as a sum of its poweroften parts and then continue to calculate the sum of the adjacent parts until a single number remains. See the Examples for details.
See A334387 for the difference version of the same transform.


LINKS



FORMULA

Let d_m,d_(m1),..,d_1,d_0 be the m decimal digits of n, then a(n) = Sum_{k=0..m} d_k*C(m,k)*10^k.  Giovanni Resta, May 09 2020


EXAMPLE

Let n = 32871. Write n as a sum of its poweroften parts:
32871 = 30000+2000+800+70+1
Now take the sum of adjacent numbers in this sum:
30000+2000+800+70+1 > (30000+2000):(2000+800):(800+70):(70+1) = 32000:2800:870:71
Now repeat this until a single number remains:
32000:2800:870:71 > 34800:3670:941
34800:3670:941 > 38470:4611
38470:4611 > 43081
Thus a(32871) = 43081.
Other examples:
a(100) = 100 as 100 = 100+0+0 thus 100:0:0 > 100:0 > 100. The equality a(n) = n holds for n=0 to 109.
a(110) = 120 as 110 = 100+10+0 thus 100:10:0 > 110:10 > 120.
a(1234) = 1694 as 1234 = 1000+200+30+4 thus 1000:200:30:4 > 1200:230:34 > 1430:264 > 1694.
a(15010) = 30040 as 15010 = 10000+5000+0+10+0 thus 10000:5000:0:10:0 > 15000:5000:10:10 > 20000:5010:20 > 25010:5030 > 30040.


MATHEMATICA

a[n_] := Block[{d = IntegerDigits[n], m}, m = Length[d]  1; Total[d Binomial[ m, Range[0, m]] 10^Range[m, 0, 1]]]; a /@ Range[100, 162] (* Giovanni Resta, May 09 2020 *)


CROSSREFS



KEYWORD

nonn,base


AUTHOR



STATUS

approved



