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 power-of-ten 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
Scott R. Shannon, Line graph of the terms for n=0 to 1000000.
FORMULA
Let d_m,d_(m-1),..,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 power-of-ten 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
Scott R. Shannon, Apr 28 2020
STATUS
approved