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 A330859 The additive version of the 'Decade transform' : to obtain a(n) 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. 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 (list; graph; refs; listen; history; text; internal format)
 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 Table of n, a(n) for n=100..162. 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 Cf. A334387, A011557, A040115, A053392. Sequence in context: A301516 A204581 A135641 * A252480 A323142 A220401 Adjacent sequences: A330856 A330857 A330858 * A330860 A330861 A330862 KEYWORD nonn,base AUTHOR Scott R. Shannon, Apr 28 2020 STATUS approved

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Last modified November 29 00:31 EST 2023. Contains 367422 sequences. (Running on oeis4.)