A377091 -id:A377091 - cp's OEIS frontend

A379882 a(n) is the length of the n-th maximal run of consecutive terms of A377091 with the same sign.

1, 2, 2, 3, 2, 3, 8, 5, 6, 5, 6, 10, 8, 13, 8, 9, 10, 11, 10, 11, 15, 13, 23, 13, 14, 15, 16, 15, 16, 17, 18, 17, 18, 38, 20, 21, 20, 21, 22, 23, 22, 23, 24, 25, 50, 26, 26, 26, 28, 28, 28, 29, 29, 29, 31, 29, 5, 32, 26, 4, 32, 28, 2, 3, 33, 31, 34, 34, 4, 36

Offset: 1

Rémy Sigrist, Jan 05 2025

nonn

			A377091 begins:
    0, 1, 2, -2, -1, 3, 4, 5, -4, -3, 6, 7, 8, -8.
So the present sequence begins:
    1, 2,    2,      3,       2,      3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C++ program

Cf. A377091.

(* A377091list is defined at A377091 *)
Map[Length, Most[SplitBy[A377091list[2000], Sign]]] (* Paolo Xausa, Jan 24 2025 *)

A379789 Sequence Sb of the eight sequences defining the blocks of terms in A377091.

Original entry on oeis.org

1, 3, 6, 13, 18, 25, 32, 42, 51, 62, 73, 98, 113, 128, 145, 162, 181, 201, 222, 243, 266, 290, 337, 366, 394, 419, 452, 479, 478, 516, 544, 546, 578, 613, 612, 651, 684, 721, 720, 761, 763, 804, 801, 842, 883, 926, 970, 1057, 1058, 1105, 1156, 1153, 1205, 1202, 1251, 1302, 1353, 1457, 1459, 1513, 1568, 1625, 1683, 1742, 1801

Offset: 1

N. J. A. Sloane, Jan 17 2025

nonn

See the comments in A379788 (Sequence Sa) for further information.

A380837 Sequence Sg of the eight sequences defining the blocks of terms in A377091.

Original entry on oeis.org

4, 9, 20, 31, 42, 60, 81, 100, 121, 147, 183, 210, 241, 272, 307, 342, 400, 441, 484, 529, 576, 651, 703, 757, 813, 871, 931, 965, 1023, 1059, 1089, 1125, 1190, 1228, 1296, 1369, 1408, 1448, 1520, 1525, 1598, 1603, 1681, 1764, 1849, 1936, 2070, 2117, 2209, 2302

Offset: 1

Paolo Xausa, Feb 05 2025

nonn

See the comments in A379788 (Sequence Sa) for further information.

Apart from A379066(1) = 0, the beginning of this sequence coincides with A379066 up to a(27) = A379066(28). a(28) = 965 is the first term not present in A379066 (cf. A380838).

Paolo Xausa, Table of n, a(n) for n = 1..1000

Cf. A377091, A379066, A379788, A379789, A379790, A379791, A379792, A379793, A379794, A380838.

SequencePosition[A377091list[5000], {?Negative, ?Positive}][[All, 1]] - 1

a(n) = A379788(n+1) - 1.

A379788 Sequence Sa of the eight sequences defining the blocks of terms in A377091.

Original entry on oeis.org

1, 5, 10, 21, 32, 43, 61, 82, 101, 122, 148, 184, 211, 242, 273, 308, 343, 401, 442, 485, 530, 577, 652, 704, 758, 814, 872, 932, 966, 1024, 1060, 1090, 1126, 1191, 1229, 1297, 1370, 1409, 1449, 1521, 1526, 1599, 1604, 1682, 1765, 1850, 1937, 2071, 2118, 2210, 2303, 2308, 2400, 2405, 2501, 2602, 2708, 2863, 2917, 2973

Offset: 1

N. J. A. Sloane, Jan 17 2025

nonn

For this discussion let R(n) (n >= 0) denote A377091(n). Sequence A377091 starts with R(0) = 0. From then on the sequence consists of blocks of consecutive terms all with the same sign. The blocks are defined by eight sequences denoted by Sa, Sb, ..., Sh. The (2k-1)-st block (k >= 1) consists of positive terms running from R(Sa(k)) = Sb(k) to R(Sc(k)) = Sd(k). This is followed by the (2*k)-th block (k >= 1) which consists of negative terms running from R(e(k)) = -Sf(k) to R(g(k)) = -Sh(k).
The sequences Sa, ..., Sh are Sa = A379788, Sb = A379789, Sc = A379790, Sd = A379791, Se = A379792, Sf = A379793, Sg = A380837, and Sh = A379794.
The sequences are related by Sa(k) = Sg(k-1)+1 and Se(k) = Sc(k)+1, and one of each pair could be omitted from the OEIS. However, since the structure of A377091 is already sufficiently confusing, at present all eight sequences have their own entries. This also makes it simpler to define the block lengths, etc., and to use the Plot2 OEIS command.

Paolo Xausa, Table of n, a(n) for n = 1..1000 (terms 1..513 from N. J. A. Sloane).

Cf. A377091, A379789, A379790, A379791, A379792, A379793, A379794, A380837.

(* A377091list is defined at A377091 *)
SequencePosition[A377091list[5000], {?NonPositive, ?Positive}][[All, 1]] (* Paolo Xausa, Feb 05 2025 *)

A379794 Sequence Sh of the eight sequences defining the blocks of terms in A377091.

Original entry on oeis.org

1, 3, 12, 18, 24, 32, 39, 49, 59, 71, 98, 112, 128, 144, 162, 180, 199, 219, 241, 263, 286, 339, 363, 390, 422, 448, 482, 483, 508, 545, 543, 578, 612, 613, 645, 685, 723, 724, 760, 758, 796, 799, 839, 881, 923, 966, 1059, 1058, 1104, 1148, 1151, 1196, 1199, 1249, 1299, 1351, 1459, 1457, 1512, 1568, 1624, 1681, 1739, 1799, 1859

Offset: 1

N. J. A. Sloane, Jan 18 2025

nonn

See the comments in A379788 (Sequence Sa) for further information.

Paolo Xausa, Table of n, a(n) for n = 1..1000 (terms 1..512 from N. J. A. Sloane).

Cf. A377091, A379066, A379788, A379789, A379790, A379791, A379792, A379793, A380837.

(* A377091list is defined at A377091 *)
-SequenceCases[A377091list[5000], {?Negative, ?Positive}][[All, 1]] (* Paolo Xausa, Feb 05 2025 *)

A379066 Indices in A377091 where there is a record upward jump from a negative term to a positive term.

Original entry on oeis.org

0, 4, 9, 20, 31, 42, 60, 81, 100, 121, 147, 183, 210, 241, 272, 307, 342, 400, 441, 484, 529, 576, 651, 703, 757, 813, 871, 931, 1023, 1059, 1125, 1190, 1296, 1369, 1408, 1520, 1598, 1681, 1764, 1849, 1936, 2070, 2209, 2302, 2399, 2500, 2601, 2707, 2862, 2972, 3081, 3192, 3364, 3481, 3600, 3721, 3906, 4033, 4160, 4291, 4422, 4560, 4761, 4900, 5041, 5256, 5403, 5550

Offset: 1

N. J. A. Sloane, Dec 28 2024

nonn

These are the indices of record (positive) high points in A379061.

The k-th high point corresponds to a jump in A379061 of height approximately k^2. These heights (whose square roots are 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, ...) do not warrant an OEIS entry of their own.

			The third term, 9, corresponds to the jump at the 9th term in A377091 from A377091(9) = -3 to A377091(10) = 6. (The fact that this is a jump of 9 = 3^2 is just a coincidence.)
The fourth term, 20, corresponds to the jump of height 25 = 5^2 at the 20th term in A377091 from A377091(20) = -12 to A377091(21) = 13.

Paolo Xausa, Table of n, a(n) for n = 1..888 (terms 1..130 from N. J. A. Sloane).

Cf. A377091, A379061, A379062.

A379059 Index of n in A377091, or -1 if n does not appear there.

Original entry on oeis.org

0, 1, 2, 5, 6, 7, 10, 11, 12, 22, 23, 24, 25, 21, 33, 34, 35, 36, 32, 46, 45, 44, 48, 47, 52, 43, 49, 50, 51, 64, 63, 62, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 82, 83, 84, 85, 86, 87, 88, 89, 90, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 188

Offset: 0

N. J. A. Sloane, Dec 26 2024

nonn

It is conjectured that every integer appears in A377091.
Conjecture 1: For n >= 10, |a(n) - 2*n| < 2*sqrt(n); for n >= 1000, |a(n) - 2*n| < 1.59*sqrt(n) (compare A379786). - N. J. A. Sloane, Jan 19 2025
Conjecture 2: lim sup |a(n) - 2*n|/sqrt(n) = sqrt(2) as n -> oo. - N. J. A. Sloane and Paolo Xausa, Feb 02 2025

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Paolo Xausa, Scatterplot of |a(n) - 2*n| for n <= 100000. The orange line is sqrt(2*n).

Cf. A377091, A379060, A379786.

A379791 Sequence Sd of the eight sequences defining the blocks of terms in A377091.

Original entry on oeis.org

2, 5, 8, 12, 17, 24, 41, 50, 61, 72, 85, 97, 112, 127, 144, 161, 200, 221, 242, 265, 288, 311, 338, 362, 390, 420, 448, 481, 512, 513, 545, 543, 577, 609, 647, 686, 685, 722, 762, 760, 800, 802, 841, 882, 925, 968, 1011, 1059, 1103, 1152, 1154, 1201, 1203, 1250, 1301, 1352, 1405, 1458, 1456, 1512, 1567, 1682, 1741, 1800, 1861

Offset: 1

N. J. A. Sloane, Jan 18 2025

nonn

See the comments in A379788 (Sequence Sa) for further information.

Paolo Xausa, Table of n, a(n) for n = 1..1000 (terms 11..513 from N. J. A. Sloane).

Cf. A377091, A379788, A379789, A379790, A379792, A379793, A379794, A380837.

(* A377091list is defined at A377091 *)
SequenceCases[A377091list[5000], {?Positive, ?Negative}][[All, 1]] (* Paolo Xausa, Feb 05 2025 *)

A379793 Sequence Sf of the eight sequences defining the blocks of terms in A377091.

Original entry on oeis.org

2, 4, 8, 13, 19, 25, 40, 50, 60, 72, 84, 99, 113, 129, 145, 163, 200, 220, 242, 264, 288, 314, 338, 367, 394, 421, 452, 480, 512, 511, 544, 546, 579, 616, 649, 683, 684, 722, 759, 761, 800, 798, 840, 882, 924, 968, 1014, 1057, 1106, 1152, 1150, 1200, 1198, 1250, 1300, 1352, 1404, 1458, 1460, 1513, 1569, 1682, 1740, 1800, 1860

Offset: 1

N. J. A. Sloane, Jan 18 2025

nonn

See the comments in A379788 (Sequence Sa) for further information.

Paolo Xausa, Table of n, a(n) for n = 1..1000 (terms 1..513 from N. J. A. Sloane).

Cf. A377091, A379066, A379788, A379789, A379790, A379791, A379792, A379794, A380837.

(* A377091list is defined at A377091 *)
-SequenceCases[A377091list[5000], {?Positive, ?Negative}][[All, 2]] (* Paolo Xausa, Feb 05 2025 *)

A379790 Sequence Sc of the eight sequences defining the blocks of terms in A377091.

Original entry on oeis.org

2, 7, 12, 25, 36, 52, 73, 90, 111, 132, 160, 196, 225, 256, 289, 324, 380, 421, 462, 507, 552, 601, 677, 729, 785, 842, 900, 960, 997, 1027, 1087, 1092, 1156, 1224, 1264, 1332, 1371, 1444, 1486, 1522, 1563, 1601, 1641, 1722, 1807, 1892, 1981, 2115, 2163, 2257, 2305, 2353, 2402, 2450, 2551, 2652, 2760, 2914, 2919, 3025

Offset: 1

N. J. A. Sloane, Jan 17 2025

nonn

See the comments in A379788 (Sequence Sa) for further information.

Paolo Xausa, Table of n, a(n) for n = 1..1000 (terms 1..513 from N. J. A. Sloane).

Cf. A377091, A379788, A379789, A379791, A379792, A379793, A379794, A380837.

(* A377091list is defined at A377091 *)
SequencePosition[A377091list[5000], {?Positive, ?Negative}][[All, 1]] - 1 (* Paolo Xausa, Feb 05 2025 *)

cp's OEIS Frontend

A379882 a(n) is the length of the n-th maximal run of consecutive terms of A377091 with the same sign.

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A379789 Sequence Sb of the eight sequences defining the blocks of terms in A377091.

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A380837 Sequence Sg of the eight sequences defining the blocks of terms in A377091.

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A379788 Sequence Sa of the eight sequences defining the blocks of terms in A377091.

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A379794 Sequence Sh of the eight sequences defining the blocks of terms in A377091.

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A379066 Indices in A377091 where there is a record upward jump from a negative term to a positive term.

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A379059 Index of n in A377091, or -1 if n does not appear there.

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A379791 Sequence Sd of the eight sequences defining the blocks of terms in A377091.

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A379793 Sequence Sf of the eight sequences defining the blocks of terms in A377091.

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