A386250 Total number of ones in runs of 1's of length >= 4 over all binary strings of length n.

0, 0, 0, 0, 4, 13, 36, 92, 224, 528, 1216, 2752, 6144, 13568, 29696, 64512, 139264, 299008, 638976, 1359872, 2883584, 6094848, 12845056, 27000832, 56623104, 118489088, 247463936, 515899392, 1073741824, 2231369728, 4630511616, 9596567552, 19864223744, 41070624768, 84825604096, 175019917312

Offset: 0

Author

Félix Balado, Aug 14 2025

Keywords

• nonn
• easy

Examples

			For n=6 there are eight binary strings that contain runs of 1s of length >= 4: 001111, 011110, 011111, 101111, 111100, 111101, 111110 and 111111; the runs of length >= 4 in these strings contain a(6) = 36 ones.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-4).

Crossrefs

Cf. A001787, A066373, A128135, A053220.

Programs

Mathematica

LinearRecurrence [{4,-4}, {4,13}, 30] (* Hugo Pfoertner, Aug 14 2025 *)

Formula

For n>=4, a(n) = (5*n-12)*2^(n-5).

G.f.: -x^4*(3*x-4)/(2*x-1)^2. - Alois P. Heinz, Aug 14 2025