A117571 -id:A117571 - cp's OEIS frontend

A220196 T(n,k) = Sum of neighbor maps: log base 2 of the number of n X k binary arrays indicating the locations of corresponding elements equal to the sum mod 2 of their horizontal and vertical neighbors in a random 0..1 n X k array.

1, 1, 1, 3, 4, 3, 4, 4, 4, 4, 4, 8, 9, 8, 4, 6, 9, 12, 12, 9, 6, 7, 12, 12, 12, 12, 12, 7, 7, 12, 18, 20, 20, 18, 12, 7, 9, 16, 21, 24, 23, 24, 21, 16, 9, 10, 17, 22, 28, 30, 30, 28, 22, 17, 10, 10, 20, 27, 32, 31, 36, 31, 32, 27, 20, 10, 12, 20, 30, 32, 39, 42, 42, 39, 32, 30, 20, 12, 13, 24

Offset: 1

R. H. Hardin Dec 07 2012

Table starts
..1..1..3..4..4..6..7..7..9..10.10..12.13.13.15.16.16.18.19.19
..1..4..4..8..9.12.12.16.17..20.20..24.25.28.28.32.33.36.36
..3..4..9.12.12.18.21.22.27..30.30..36.39.40.45.48.48.54
..4..8.12.12.20.24.28.32.32..40.44..48.52.52.60.64.68
..4..9.12.20.23.30.31.39.44..50.51..60.64.69.71.80
..6.12.18.24.30.36.42.42.54..60.66..72.78.84.90
..7.12.21.28.31.42.49.54.63..70.70..84.91.96
..7.16.22.32.39.42.54.64.71..80.86..96.97
..9.17.27.32.44.54.63.71.73..90.98.108
.10.20.30.40.50.60.70.80.90.100
.10.20.30.44.51.66.70.86.98
.12.24.36.48.60.72.84.96

			Some solutions for n=3 k=4
..0..0..0..0....0..0..0..0....1..1..1..0....0..1..0..0....0..1..0..0
..0..1..0..0....0..0..0..0....1..0..0..1....0..0..0..1....0..1..1..1
..0..1..0..0....0..0..0..1....1..0..1..1....0..0..0..0....0..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..199

Column 1 is A117571.

Empirical for column k:
k=1: a(n) = a(n-1) +a(n-3) -a(n-4) increment period 3: 0 2 1
k=2: a(n) = a(n-1) +a(n-4) -a(n-5) increment period 4: 3 0 4 1
k=3: a(n) = a(n-1) +a(n-6) -a(n-7) increment period 6: 1 5 3 0 6 3
k=4: a(n) = a(n-1) +a(n-5) -a(n-6) increment period 5: 4 4 0 8 4
k=5: a(n) = a(n-3) +a(n-8) -a(n-11) increment period 24: 5 3 8 3 7 1 8 5 6 1 9 4 5 2 9 3 7 2 7 5 6 0 10 4
k=6: a(n) = a(n-1) +a(n-9) -a(n-10) increment period 9: 6 6 6 6 6 6 0 12 6
k=7: a(n) = a(n-1) +a(n-12) -a(n-13) increment period 12: 5 9 7 3 11 7 5 9 7 0 14 7

A325253 Number of integer partitions of n with adjusted frequency depth ceiling(sqrt(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 6, 8, 17, 26, 25, 44, 53, 63, 83, 128, 168, 212, 273, 344, 429, 525, 662, 796, 684, 910, 1211, 1595, 2060, 2663, 3406, 4315, 5426, 6784, 8417, 0, 0, 0, 0, 0, 1, 5, 14, 36, 76, 143, 269, 446, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Apr 22 2019

nonn

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is one plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

			The a(2) = 1 through a(11) = 26 partitions:
    11  111  22    32  42    43   53    54      433        443
             1111  41  51    52   62    63      442        533
                       321   61   71    72      622        551
                       2211  421  431   81      811        722
                                  521   432     3331       911
                                  3311  531     4222       3332
                                        621     7111       5222
                                        222111  61111      8111
                                                222211     32222
                                                322111     33311
                                                331111     44111
                                                511111     71111
                                                2221111    222221
                                                4111111    322211
                                                22111111   332111
                                                31111111   422111
                                                211111111  611111
                                                           2222111
                                                           3221111
                                                           3311111
                                                           5111111
                                                           22211111
                                                           41111111
                                                           221111111
                                                           311111111
                                                           2111111111

Cf. A117571, A181819, A225485, A323014, A323023, A325245, A325246, A325252, A325258, A325271, A325278, A325280, A325282.

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
Table[Length[Select[IntegerPartitions[n],fdadj[#]==Ceiling[Sqrt[n]]&]],{n,0,30}]

A325271 Number of integer partitions of n with frequency depth round(sqrt(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 4, 6, 8, 11, 11, 19, 44, 53, 63, 83, 113, 124, 171, 190, 344, 429, 525, 662, 796, 981, 1182, 1442, 1709, 2096, 2663, 3406, 4315, 5426, 6784, 8417, 10466, 12824, 15721, 19104, 23267, 27981, 5, 14, 36, 76, 143, 269, 446, 738, 1143, 1754

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

			The a(2) = 1 through a(10) = 11 partitions:
  (2)  (111)  (22)    (11111)  (33)      (43)   (53)    (54)      (64)
              (1111)           (222)     (52)   (62)    (63)      (73)
                               (111111)  (61)   (71)    (72)      (82)
                                         (421)  (431)   (81)      (91)
                                                (521)   (432)     (532)
                                                (3311)  (531)     (541)
                                                        (621)     (631)
                                                        (222111)  (721)
                                                                  (3322)
                                                                  (4321)
                                                                  (4411)

Cf. A117571, A181819, A225485, A323014, A323023, A325245, A325246, A325272, A325253, A325258, A325278, A325280, A325282.

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
Table[Length[Select[IntegerPartitions[n],fdadj[#]==Round[Sqrt[n]]&]],{n,0,30}]

A274516 Place n equally-spaced points around a circle, labeled 0,1,2,...,n-1. For each i = 0..n-1 such that 5i != i mod n, draw an (undirected) chord from i to (5i mod n). Then a(n) is the total number of distinct chords.

Original entry on oeis.org

0, 0, 0, 1, 0, 4, 2, 6, 2, 7, 8, 10, 4, 12, 12, 13, 10, 16, 14, 18, 16, 19, 20, 22, 10, 24, 24, 25, 24, 28, 26, 30, 26, 31, 32, 34, 28, 36, 36, 37, 34, 40, 38, 42, 40, 43, 44, 46, 34, 48, 48, 49, 48, 52, 50, 54, 50, 55, 56, 58, 52, 60, 60

Offset: 0

Brooke Logan, Jun 25 2016

nonn

Brooke Logan, Table of n, a(n) for n = 0..10000

If 5i in the definition is replaced by 2i we get A117571, if 5i is replaced by 3i we get A273724, and if 5i is replaced by 4i we get A274462.

We argue as in A273724. There are n-1 choices for i.
For nontrivial chords we need i != 5i mod n, which means 4i != 0 mod n, and so when n == 0 mod 4 we must subtract 3 from n-1 and when n == 2 mod 4 we must subtract 1 from n-1.
A chord occurs twice (but must be counted only once) when j==5i mod n and i==5j mod n, thus when 24i == 0 mod n. If n == +/- 3, +/- 9 mod 24 then subtract another 1, if n == +/- 6, +/- 8 mod 24 then subtract another 2, if n==12 mod 24 subtract 4, and if n == 0 mod 24 then subtract another 10.
Putting the pieces together, we obtain the g.f.
x^2/(1-x)^2-(3+x^2)/(1-x^4)-(x^3+x^9+x^15+x^21)/(1-x^24)-2(x^6+x^8+x^16+x^18)/(1-x^24)-(4*x^12+10)/(1-x^24)+13.
The g.f. can also be written as
(14*x^25 - 12*x^24 + 2*x^23 + x^22 + 3*x^21 - 2*x^20 + 2*x^19 + 4*x^17 - x^16 + x^15 + 8*x^13 - 6*x^12 + 2*x^11 + x^10 + 3*x^9 - 2*x^8 + 2*x^7 + 4*x^5 - x^4 + x^3 + 2*x - 2) / ((1-x)*(1-x^24)).

A325252 Number of integer partitions of n with frequency depth floor(sqrt(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 3, 8, 11, 11, 19, 17, 25, 29, 83, 113, 124, 171, 190, 242, 289, 368, 399, 796, 981, 1182, 1442, 1709, 2096, 2469, 2990, 3545, 4276, 5037, 8417, 10466, 12824, 15721, 19104, 23267, 27981, 33856, 40515, 48508, 57826, 68982, 81493, 446, 738

Offset: 0

Gus Wiseman, Apr 22 2019

nonn

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

			The a(2) = 1 through a(12) = 19 partitions (A = 10, B = 11):
  2  3  22    11111  33      1111111  44        54      64    65    75
        1111         222              2222      63      73    74    84
                     111111           11111111  72      82    83    93
                                                81      91    92    A2
                                                432     532   A1    B1
                                                531     541   542   543
                                                621     631   632   642
                                                222111  721   641   651
                                                        3322  731   732
                                                        4321  821   741
                                                        4411  5321  831
                                                                    921
                                                                    4422
                                                                    5421
                                                                    5511
                                                                    6321
                                                                    332211
                                                                    333111
                                                                    22221111

Cf. A117571, A181819, A225485, A323014, A323023, A325245, A325246, A325253, A325258, A325271, A325278, A325280, A325282.

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
Table[Length[Select[IntegerPartitions[n],fdadj[#]==Floor[Sqrt[n]]&]],{n,0,30}]

A325279 Number of integer partitions of n whose maximum multiplicity is one greater than their minimum multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 5, 6, 9, 10, 18, 18, 31, 34, 48, 57, 80, 86, 122, 138, 183, 211, 275, 311, 402, 461, 576, 663, 825, 942, 1163, 1334, 1621, 1865, 2248, 2566, 3084, 3532, 4193, 4794, 5674, 6472, 7617, 8685, 10153, 11576, 13483, 15320, 17790, 20200, 23342

Offset: 0

Gus Wiseman, Apr 18 2019

nonn

The Heinz numbers of these partitions are given by A325241.

For example, the partition (44111) has two multiplicities (2 and 3) which differ by 1, so is counted under a(11).

			The a(4) = 1 through a(11) = 18 partitions:
  (211)  (221)  (411)  (322)    (332)    (441)    (433)      (443)
         (311)         (331)    (422)    (522)    (442)      (533)
                       (511)    (611)    (711)    (622)      (551)
                       (3211)   (3221)   (3321)   (811)      (722)
                       (22111)  (4211)   (4221)   (5221)     (911)
                                (22211)  (4311)   (5311)     (4322)
                                         (5211)   (6211)     (4331)
                                         (32211)  (33211)    (4421)
                                         (33111)  (42211)    (5411)
                                                  (2221111)  (6221)
                                                             (6311)
                                                             (7211)
                                                             (33221)
                                                             (33311)
                                                             (43211)
                                                             (44111)
                                                             (52211)
                                                             (2222111)

Cf. A047966, A062770, A071625, A098859, A117571, A127002, A325242, A325244, A325245, A325280.

Table[Length[Select[IntegerPartitions[n],Max@@Length/@Split[#]-Min@@Length/@Split[#]==1&]],{n,0,30}]

A354698 T(n,k) is the number of points with integer coordinates strictly inside the triangle with vertices (0,0), (n,0), (n,k), where T(n,k) is a triangle read by rows, 2 <= k <= n.

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 2, 4, 6, 6, 2, 4, 7, 10, 10, 3, 6, 9, 12, 15, 15, 3, 7, 9, 14, 17, 21, 21, 4, 7, 12, 16, 19, 24, 28, 28, 4, 9, 13, 16, 22, 27, 31, 36, 36, 5, 10, 15, 20, 25, 30, 35, 40, 45, 45, 5, 10, 15, 22, 25, 33, 37, 43, 49, 55, 55, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 66

Offset: 2

Hugo Pfoertner, Jun 06 2022

T(n,n) = T(n,n-1) because all grid points with m = n lie on a side of the triangle and thus not strictly inside.

			The triangle begins:
  0;
  1,  1;
  1,  3,  3;
  2,  4,  6,  6;
  2,  4,  7, 10, 10;
  3,  6,  9, 12, 15, 15;
  3,  7,  9, 14, 17, 21, 21;
  4,  7, 12, 16, 19, 24, 28, 28;
  4,  9, 13, 16, 22, 27, 31, 36, 36;
  5, 10, 15, 20, 25, 30, 35, 40, 45, 45;
  5, 10, 15, 22, 25, 33, 37, 43, 49, 55, 55

Cf. A000217 (right diagonal), A074148 (3rd diagonal).

Cf. A004526 (column 2), A117571 (column 3).

T(n, m) = sum(i=1, n-1, sum(j=1, m-1, (i/j > n/m))); \\ Michel Marcus, Jun 07 2022

cp's OEIS Frontend

A220196 T(n,k) = Sum of neighbor maps: log base 2 of the number of n X k binary arrays indicating the locations of corresponding elements equal to the sum mod 2 of their horizontal and vertical neighbors in a random 0..1 n X k array.

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A325253 Number of integer partitions of n with adjusted frequency depth ceiling(sqrt(n)).

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A325271 Number of integer partitions of n with frequency depth round(sqrt(n)).

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A274516 Place n equally-spaced points around a circle, labeled 0,1,2,...,n-1. For each i = 0..n-1 such that 5i != i mod n, draw an (undirected) chord from i to (5i mod n). Then a(n) is the total number of distinct chords.

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A325252 Number of integer partitions of n with frequency depth floor(sqrt(n)).

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Mathematica

A325279 Number of integer partitions of n whose maximum multiplicity is one greater than their minimum multiplicity.

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Mathematica

A354698 T(n,k) is the number of points with integer coordinates strictly inside the triangle with vertices (0,0), (n,0), (n,k), where T(n,k) is a triangle read by rows, 2 <= k <= n.

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