OFFSET
1,2
COMMENTS
Suppose that n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See the guide at A297330.
Differs from A029955 first at 739=1011_9 which is not a palindrome in base 9 but has DV(739,9)=UV(793,9) =1. - R. J. Mathar, Jan 23 2018
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..10000
EXAMPLE
446 in base-9: 5,4,5, having DV = 1, UV = 1, so that 446 is in the sequence.
MATHEMATICA
g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
b = 9; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
Take[Flatten[Position[w, -1]], 120] (* A297267 *)
Take[Flatten[Position[w, 0]], 120] (* A297268 *)
Take[Flatten[Position[w, 1]], 120] (* A297269 *)
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Clark Kimberling, Jan 15 2018
STATUS
approved