Félix Balado has authored 7 sequences.
A386250
Total number of ones in runs of 1's of length >= 4 over all binary strings of length n.
Original entry on oeis.org
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
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.
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LinearRecurrence [{4,-4}, {4,13}, 30] (* Hugo Pfoertner, Aug 14 2025 *)
A386270
Total number of ones in runs of 1s of length <=2 over all binary strings of length n.
Original entry on oeis.org
0, 1, 4, 9, 22, 52, 120, 272, 608, 1344, 2944, 6400, 13824, 29696, 63488, 135168, 286720, 606208, 1277952, 2686976, 5636096, 11796480, 24641536, 51380224, 106954752, 222298112, 461373440, 956301312, 1979711488, 4093640704, 8455716864, 17448304640, 35970351104
Offset: 0
For n=4, the binary strings that contain runs of 1s of length <=2 are: 0001, 0010, 0011, 0100, 0101, 0110, 1000, 1001, 1010, 1011, 1100, and 1101. The total number of ones in the runs of 1s of length <=2 is thus a(4)=22.
A386878
Number of runs of 1's of length <= 3 over all binary strings of length n.
Original entry on oeis.org
0, 1, 3, 8, 19, 45, 104, 236, 528, 1168, 2560, 5568, 12032, 25856, 55296, 117760, 249856, 528384, 1114112, 2342912, 4915200, 10289152, 21495808, 44826624, 93323264, 193986560, 402653184, 834666496, 1728053248, 3573547008, 7381975040, 15233712128, 31406948352
Offset: 0
For n=3, the breakdown of the 8 runs of 1s is as follows: 001 (1), 010 (1), 011 (1), 100 (1), 101 (2), 110 (1) and 111 (1).
For n=4, the breakdown of the 19 runs of 1s is as follows: 0001 (1), 0010 (1), 0011 (1), 0100 (1), 0101 (2), 0110 (1), 0111 (1), 1000 (1), 1001 (2), 1010 (2), 1011 (2), 1100 (1), 1101 (2) and 1110 (1).
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LinearRecurrence[{4, -4}, {0, 1, 3, 8, 19, 45}, 40] (* Paolo Xausa, Aug 19 2025 *)
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def A386878(n): return (0,1,3,8,19)[n] if n<5 else 3+7*(n+1)<Chai Wah Wu, Aug 19 2025
A384497
a(n) is the number of binary strings of length n which contain exactly one run of 1s of even length.
Original entry on oeis.org
0, 0, 1, 2, 6, 12, 28, 56, 119, 236, 479, 940, 1859, 3612, 7028, 13538, 26051, 49820, 95098, 180774, 342944, 648648, 1224517, 2306338, 4336449, 8138516, 15250965, 28535528, 53320792, 99504804, 185474501, 345332950, 642310142, 1193510160, 2215702468, 4109801864
Offset: 0
a(3)=2 because of the strings 011 and 110.
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CoefficientList[Series[((1 - x^2) * x^2)/(1 - x - 2 * x^2 + x^3)^2,{x,0,35}],x] (* Stefano Spezia, Jun 02 2025 *)
A384155
a(n) is the number of binary strings of length n whose shortest run of 1s is of length 3.
Original entry on oeis.org
0, 0, 0, 1, 2, 3, 4, 6, 11, 21, 38, 65, 108, 179, 299, 502, 842, 1406, 2337, 3872, 6403, 10575, 17445, 28742, 47293, 77720, 127578, 209210, 342768, 561131, 917910, 1500476, 2451158, 4001723, 6529439, 10648199, 17356589, 28278426
Offset: 0
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LinearRecurrence[{4, -6, 4, 0, -1, -1, 1, 0, -1}, {0, 0, 0, 1, 2, 3, 4, 6, 11}, 50] (* Amiram Eldar, May 31 2025 *)
A384153
a(n) is the number of binary strings of length n whose shortest run of 1s is of length 1.
Original entry on oeis.org
0, 1, 2, 4, 9, 20, 43, 91, 191, 398, 824, 1697, 3480, 7111, 14487, 29439, 59694, 120820, 244153, 492716, 993171, 1999923, 4023679, 8089182, 16251760, 32632321, 65490672, 131377999, 263452079, 528125695, 1058395038, 2120551916, 4247705401, 8506995748, 17034321659
Offset: 0
A384154
a(n) is the number of binary strings of length n whose shortest run of 1s is of length 2.
Original entry on oeis.org
0, 0, 1, 2, 3, 5, 10, 20, 38, 70, 128, 234, 427, 776, 1404, 2531, 4550, 8161, 14608, 26099, 46550, 82901, 147441, 261913, 464759, 823902, 1459287, 2582615, 4567357, 8072082, 14257631, 25169443, 44410452, 78325112, 138082349, 243339192, 428683436, 754961473
Offset: 0
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LinearRecurrence[{4,-6,5,-2,-1,1,-1},{0,0,1,2,3,5,10},40] (* Harvey P. Dale, Jun 24 2025 *)