A245596 -id:A245596 - cp's OEIS frontend

A256701 Positive part of the minimal alternating binary representation of n (defined at A245596).

1, 2, 4, 4, 9, 8, 8, 8, 17, 18, 20, 16, 17, 16, 16, 16, 33, 34, 36, 36, 41, 40, 40, 32, 33, 34, 36, 32, 33, 32, 32, 32, 65, 66, 68, 68, 73, 72, 72, 72, 81, 82, 84, 80, 81, 80, 80, 64, 65, 66, 68, 68, 73, 72, 72, 64, 65, 66, 68, 64, 65, 64, 64, 64, 129, 130

Offset: 1

Clark Kimberling, Apr 09 2015

			R(1) = 1; positive part 1, nonpositive part 0
R(2) = 2; positive part 2, nonpositive part 0
R(3) = 4 - 1; positive part 4, nonpositive part 1
R(11) = 16 - 8 + 4 - 1; positive part 16+4 = 20; nonpositive part 8 + 1 = 9

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A245596, A073122, A256702.

b[n_] := 2^n; bb = Table[b[n], {n, 0, 40}];
s[n_] := Table[b[n + 1], {k, 1, b[n]}];
h[0] = {1}; h[n_] := Join[h[n - 1], s[n - 1]];
g = h[10]; Take[g, 100]; r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]]
Table[Total[Abs[r[n]]], {n, 1, 100}] (* A073122 *)
u = Table[Total[(Abs[r[n]] + r[n])/2], {n, 1, 100}]  (* A256701 *)
v = Table[Total[(Abs[r[n]] - r[n])/2], {n, 1, 100}]  (* A256702 *)

A256701(n) - A256702(n) = n.

A248880 Number of tilings of an n X 1 rectangle by tiles of dimension 1 X 1 and 2 X 1 such that every tile shares an equal-length edge with a tile of the same size.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 3, 7, 7, 13, 14, 24, 28, 45, 56, 86, 111, 165, 218, 317, 426, 611, 831, 1181, 1619, 2286, 3150, 4428, 6123, 8582, 11896, 16641, 23105, 32278, 44865, 62620, 87103, 121499, 169087, 235761, 328214, 457508, 637064, 887857, 1236500, 1723054

Offset: 0

Paul Tek, Mar 05 2015

			A 3 X 1 rectangle can be tiled in three ways:
  +-+-+-+  +-+---+     +---+-+
  | | | |, | |   | and |   | |.
  +-+-+-+  +-+---+     +---+-+
The first tiling is acceptable, as every 1 X 1 tile is next to another 1 X 1 tile (and there are no 2 X 1 tiles).
The second and third tilings are not acceptable, as the 1 X 1 tiles are not next to other 1 X 1 tiles.
Hence, a(3)=1.

Paul Tek, Table of n, a(n) for n = 0..1000
Paul Tek, Illustration of the formula
Paul Tek, Illustration of the first terms
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,0,1).

Cf. A245596.

Vec(-(x^2-x+1)*(x^4-x^2+1)/(x^6-x^3+x^2+x-1) + O(x^100)) \\ Colin Barker, Mar 05 2015

                         [ 0 1 0 1 0 0 0 ]       [1]
                         [ 0 0 1 0 0 0 0 ]       [0]
                         [ 0 0 1 1 0 0 0 ]       [1]
a(n) = [1 0 0 0 0 0 0] * [ 0 0 0 0 1 0 0 ] ^ n * [0], for any n>=0.
                         [ 0 0 0 0 0 1 0 ]       [0]
                         [ 0 0 0 0 0 0 1 ]       [0]
                         [ 0 1 0 0 0 1 0 ]       [1]
G.f.: -(x^2-x+1)*(x^4-x^2+1) / (x^6-x^3+x^2+x-1). - Colin Barker, Mar 05 2015

cp's OEIS Frontend

A256701 Positive part of the minimal alternating binary representation of n (defined at A245596).

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