A297253 Numbers whose base-4 digits having equal up-variation and down-variation; see Comments.

1, 2, 3, 5, 10, 15, 17, 21, 25, 29, 34, 38, 42, 46, 51, 55, 59, 63, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 195, 199, 203, 207, 211, 215, 219, 223

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.

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

Example

223 in base-4:  3,2,3,3, having DV = 1, UV = 1, so that 223 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 = 4; 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]   (* A297252 *)
Take[Flatten[Position[w, 0]], 120]    (* A297253 *)
Take[Flatten[Position[w, 1]], 120]    (* A297254 *)

Crossrefs

Cf. A297330, A297252, A297254.

Sequence in context: A048301 A043707 A296694 * A014192 A250746 A048315

Adjacent sequences: A297250 A297251 A297252 * A297254 A297255 A297256

Keyword

nonn,base,easy

Author

Clark Kimberling, Jan 15 2018

Status

approved