A238986 Ground Pyramidalized Numbers: Write the decimal digits of 'n' (a nonnegative integer) and take successive absolute differences ("pyramidalization"), then sum all digits of each level of the pyramid. If total is greater than 9, repeat the process until result is between 0 and 9, which is 'a(n)' (0 <= a(n) <= 9).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 2, 4, 6, 8, 2, 4, 8, 4, 4, 4, 4, 4, 6, 8, 2, 4, 8, 4, 4, 6, 6, 6, 6, 8, 2, 4, 8, 4, 4, 8, 8, 8, 8, 8, 2, 4, 8, 4, 4, 2, 2, 2, 2, 2, 2, 4, 8, 4, 4

Offset

0,3

Comments

A given nonnegative integer 'n' is decomposed to its digits and the absolute differences between the digits are taken, then the differences between differences between digits (and so on, until the top of the "gap-pyramid" is reached - we could call this process "pyramidalization"). If the sum 's(n)' of the resulting digits is 0 <= s(n) <= 9, it's 'a(n)'; if greater than 9, the same process is applied to the result, and to the subsequent result if necessary, and so on, until the result is smaller than 10.

Links

Giovanni Resta, Table of n, a(n) for n = 0..10000

Formula

a(n)=n, if 0<=n<=9;

b'(n)=n-9*floor(n/10)+|-n+11*floor(n/10)|, if 10<=n<=99;

b'(n)=a(n), if 0<=b'(n)<=9;

else, b''(n)=b'(n)-9*floor(b'(n)/10)+|-b'(n)+11*floor(b'(n)/10)|;

b''(n)=a(n), if 0<=b''(n)<=9;

else, b'''(n)=...

c'(n)=n-9*floor(n/10)-9*floor(n/100)+|-floor(n/10)+11*floor(n/100)|+|-n+11*floor(n/10)-10*floor(n/100)|+||-floor(n/10)+11*floor(n/100)|-|-n+11*floor(n/10)-10*floor(n/100)||, if 100<=n<=999.

c'(n)=a(n), if 0<=c'(n)<=9;

else, if 10<=c'(n)<=99, c''(n)=c'(n)-9*floor(c'(n)/10)+|-c'(n)+11*floor(c'(n)/10)|;

c''(n)=a(n), if 0<=c''(n)<=9

else, ...

Example

If n=364, a(364)=4, for...
.
____1
__3_:_2__ -->b'(364)=3+6+4+|3-6|+|6-4|+||3-6|-|6-4||=3+6+4+3+2+1=19>9
3_:_6_:_4
.
__8
1_:_9  --> b''(364)=1+9|1-9|=1+9+8=18>9
.
__7
1_:_8 --> b'''(364)=1+8+|1-8|=1+8+7=16>9
.
__5
1_:_6 --> b''''(364)=1+6+|1-6|=1+6+5=12>9
.
__1
1_:_2 --> b'''''(364)=1+2+|1-2|=1+2+1=4=a(364)

Mathematica

a[n_] := If[n < 10, n, Block[{d = IntegerDigits@ n, s}, s = Total@ d; While[Length@ d > 1, d = Abs@ Differences@ d; s += Total@d]; If[s < 10, s, a@s]]]; a /@ Range[0, 99] (* Giovanni Resta, Mar 16 2014 *)

Crossrefs

Cf. A227876. The pyramidalization process is applied and reapplied to each term until the result reaches its "ground limit".

Cf. A007318. The pyramidalization process relates to Pascal's Triangle because it is done in the opposite direction (towards the top instead of the base), using the contrary operation (absolute difference instead of sum of the prior terms).

Sequence in context: A297233 A177895 A340184 * A227876 A276716 A189506

Adjacent sequences: A238983 A238984 A238985 * A238987 A238988 A238989

Keyword

nonn,base,uned

Author

Filipi R. de Oliveira, Mar 07 2014

Status

approved