A069905 -id:A069905 - cp's OEIS frontend

A099830 Smallest perimeter S such that exactly n distinct Pythagorean triangles with this perimeter can be constructed.

12, 60, 120, 240, 420, 720, 1320, 840, 2640, 1680, 3360, 2520, 4620, 7920, 7560, 5040, 10080, 17160, 10920, 9240, 40320, 25200, 28560, 21840, 18480, 60480, 41580, 46200, 36960, 32760, 27720, 78540, 60060, 129360, 134640, 115920, 85680, 65520, 83160

Offset: 1

Hugo Pfoertner, Oct 27 2004

nonn

Least perimeter common to exactly n distinct Pythagorean triangles. - Lekraj Beedassy, Jun 07 2006

			a(7)=1320 because 1320 is the smallest possible perimeter for which exactly 7 different Pythgorean triangles exist: 1320 = 110+600+610 = 120+594+606 = 220+528+572 = 231+520+569 = 264+495+561 = 330+440+550 = 352+420+548.

Ray Chandler, Table of n, a(n) for n = 1..163
Ron Knott, Pythagorean Triples and Online Calculators
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Index entries related to Pythagorean Triples.

Cf. A099829 first perimeter producing at least n Pythagorean triangles, A009096 ordered perimeters of Pythagorean triangles, A001399, A069905 partitions into 3 parts.

More terms from Ray Chandler, Oct 29 2004

A215521 Number T(n,k) of distinct values of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n with largest part = k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 4, 3, 2, 1, 1, 1, 4, 5, 5, 3, 2, 1, 1, 1, 4, 7, 6, 5, 3, 2, 1, 1, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1, 1, 5, 10, 10, 10, 7, 5, 3, 2, 1, 1, 1, 6, 12, 14, 12, 11, 7, 5, 3, 2, 1, 1, 1, 6, 14, 16, 17, 13, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Alois P. Heinz, Aug 14 2012

Differs from A008284 first at T(11,4).

			T(4,2) = 2 = |{4!/(2!*2!), 4!/(2!*1!*1!)}| = |{6, 12}|.
T(7,4) = 3 = |{35, 105, 210}|.
T(8,3) = 5 = |{560, 1120, 1680, 3360, 6720}|.
T(11,4) = 10 = |{11550, 34650, 46200, 69300, 138600, 207900, 277200, 415800, 831600, 1663200}|.
Triangle T(n,k) begins:
  1;
  1,  1;
  1,  1,  1;
  1,  2,  1,  1;
  1,  2,  2,  1,  1;
  1,  3,  3,  2,  1,  1;
  1,  3,  4,  3,  2,  1,  1;
  1,  4,  5,  5,  3,  2,  1,  1;
  1,  4,  7,  6,  5,  3,  2,  1,  1;
  1,  5,  8,  9,  7,  5,  3,  2,  1,  1;
  1,  5, 10, 10, 10,  7,  5,  3,  2,  1,  1;
  ...

Alois P. Heinz, Rows n = 1..100, flattened
Wikipedia, Multinomial coefficients

Columns k=1-3 give: A000012 (for n>0), A004526, A069905(n) = A001399(n-3) for n>=3.
T(2*n,n) gives: A070289.
Cf. A008284, A215520.

b:= proc(n, i) option remember; `if`(n=0, {1}, `if`(i<1, {},
      {b(n, i-1)[], seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=1..n/i)}))
    end:
T:= (n, k)-> nops(b(n-k, min(k, n-k))):
seq(seq(T(n, k), k=1..n), n=1..15);

b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i < 1, {}, Join[b[n, i - 1], Table[ b[n - i*j, i - 1] *i!^j, {j, 1, n/i}] // Flatten]] // Union]; T[n_, k_] := Length[b[n, k]]; Table[Table[T[n - k, Min[k, n - k]], {k, 1, n}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 21 2015, after Alois P. Heinz *)

A309513 Number of even parts in the partitions of n into 3 parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 4, 7, 8, 12, 12, 18, 18, 24, 24, 31, 32, 41, 40, 49, 50, 60, 60, 72, 72, 84, 84, 97, 98, 113, 112, 127, 128, 144, 144, 162, 162, 180, 180, 199, 200, 221, 220, 241, 242, 264, 264, 288, 288, 312, 312, 337, 338, 365, 364, 391, 392, 420, 420

Offset: 0

Wesley Ivan Hurt, Aug 05 2019

			Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
                                                          1+1+8
                                                   1+1+7  1+2+7
                                                   1+2+6  1+3+6
                                            1+1+6  1+3+5  1+4+5
                                     1+1+5  1+2+5  1+4+4  2+2+6
                              1+1+4  1+2+4  1+3+4  2+2+5  2+3+5
                       1+1+3  1+2+3  1+3+3  2+2+4  2+3+4  2+4+4
         1+1+1  1+1+2  1+2+2  2+2+2  2+2+3  2+3+3  3+3+3  3+3+4    ...
-----------------------------------------------------------------------
  n  |     3      4      5      6      7      8      9     10      ...
-----------------------------------------------------------------------
a(n) |     0      1      2      5      4      7      8     12      ...
-----------------------------------------------------------------------

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1,1,-1,0,0,-1,1).

Cf. A069905, A128012, A309511.

Table[Sum[Sum[Mod[i - 1, 2] + Mod[j - 1, 2] + Mod[n - i - j - 1, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (((i-1) mod 2) + ((j-1) mod 2) + ((n-i-j-1) mod 2)). [Corrected by Georg Fischer, Mar 11 2025]
From Colin Barker, Aug 06 2019: (Start)
G.f.: x^4*(1 + x + 3*x^2 - x^3 + 2*x^4) / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) + a(n-6) - a(n-7) - a(n-10) + a(n-11) for n>10. (End)
a(n) = 3*A069905(n) - A309511(n). - Ray Chandler, Mar 13 2025

A374702 Number of integer compositions of n whose leaders of maximal weakly decreasing runs sum to 3. Column k = 3 of A374748.

Original entry on oeis.org

0, 0, 0, 2, 3, 6, 9, 13, 17, 23, 28, 35, 42, 50, 58, 68, 77, 88, 99, 111, 123, 137, 150, 165, 180, 196, 212, 230, 247, 266, 285, 305, 325, 347, 368, 391, 414, 438, 462, 488, 513, 540, 567, 595, 623, 653, 682, 713, 744, 776, 808, 842, 875, 910, 945, 981

Offset: 0

Gus Wiseman, Aug 12 2024

The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.

			The a(0) = 0 through a(8) = 17 compositions:
  .  .  .  (3)   (31)   (32)    (33)     (322)     (332)
           (12)  (112)  (122)   (321)    (331)     (3221)
                 (121)  (311)   (1122)   (1222)    (3311)
                        (1112)  (1221)   (3211)    (11222)
                        (1121)  (3111)   (11122)   (12221)
                        (1211)  (11112)  (11221)   (32111)
                                (11121)  (12211)   (111122)
                                (11211)  (31111)   (111221)
                                (12111)  (111112)  (112211)
                                         (111121)  (122111)
                                         (111211)  (311111)
                                         (112111)  (1111112)
                                         (121111)  (1111121)
                                                   (1111211)
                                                   (1112111)
                                                   (1121111)
                                                   (1211111)

Andrew Howroyd, Table of n, a(n) for n = 0..10000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

The version for k = 2 is A004526.
The version for partitions is A069905 or A001399 (shifted).
For reversed partitions we appear to have A137719.
For length instead of sum we have A241627.
For leaders of constant runs we have A373952.
The opposite rank statistic is A374630, row-sums of A374629.
The corresponding rank statistic is A374741 row-sums of A374740.
Column k = 3 of A374748.
A003242 counts anti-run compositions.
A011782 counts integer compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
Cf. A000009, A000041, A101271, A106356, A188900, A238343, A261982, A333213.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#,GreaterEqual]]==3&]],{n,0,15}]

seq(n)={Vec((2 + x + x^2)/((1 + x + x^2)*(1 + x)*(1 - x)^3) + O(x^(n-2)), -n-1)} \\ Andrew Howroyd, Aug 14 2024

G.f.: x^3*(2 + x + x^2)/((1 + x + x^2)*(1 + x)*(1 - x)^3). - Andrew Howroyd, Aug 14 2024

a(27) onwards from Andrew Howroyd, Aug 14 2024

A242089 Number of triples (a,b,c) with 0 < a < b < c < p and a + b + c == 0 mod p, where p = prime(n).

Original entry on oeis.org

0, 0, 0, 2, 10, 16, 32, 42, 66, 112, 130, 192, 240, 266, 322, 416, 522, 560, 682, 770, 816, 962, 1066, 1232, 1472, 1600, 1666, 1802, 1872, 2016, 2562, 2730, 2992, 3082, 3552, 3650, 3952, 4266, 4482, 4816, 5162, 5280, 5890, 6016, 6272, 6402, 7210, 8066, 8362, 8512

Offset: 1

Jonathan Sondow, Jun 16 2014

a(n) is even. (Proof. Each triple (a,b,c) with b < p/2 pairs uniquely with a triple (a',b',c') = (p-c,p-b,p-a) with b' > p/2.)

			For prime(4) = 7 there are 2 triples (a,b,c) with 0 < a < b < c < 7 and a + b + c == 0 mod 7, namely, 1+2+4 = 7 and 3+5+6 = 2*7, so a(4) = 2.

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..1000
Steven J. Miller, Combinatorial and Additive Number Theory Problem Sessions, arXiv:1406.3558 [math.NT], 2014-2023. See Nathan Kaplan's Problem 2014.1.4 on p. 30.

Cf. A069905, A242090.

Table[ Length[ Reduce[ Mod[a + b + c, Prime[n]] == 0 && 0 < a < b < c < Prime[n], {a, b, c}, Integers]], {n, 40}]

a(n) = 2 * round((prime(n) - 3)^2/12) \\ David A. Corneth, May 27 2025

a(n) = 2*A242090(n).

a(n) = 2*A069905(prime(n)-3) = 2 * round((prime(n) - 3)^2/12). - David A. Corneth, May 27 2025

a(41)-a(50) from Fausto A. C. Cariboni, Sep 30 2018

A306403 The number of distinct products that can be formed by multiplying the parts of a partition of n into 3 positive parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 13, 14, 19, 20, 23, 27, 29, 32, 34, 39, 43, 47, 51, 53, 59, 58, 67, 73, 75, 81, 88, 91, 93, 106, 109, 114, 117, 128, 131, 133, 145, 154, 163, 166, 174, 181, 180, 201, 206, 209, 219, 231, 240, 238, 252, 267, 272, 289, 290, 300, 299, 323, 328, 345, 349, 366, 376

Offset: 0

R. J. Mathar, Feb 13 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000
R. J. Mathar, Java program that computes a b-file

Row sums of A317578.

Cf. A069905.

a:= proc(n) option remember; local m, c, i, j, h, w;
      m, c:= proc() true end, 0; forget(m);
      for i to iquo(n, 3) do for j from i to iquo(n-i, 2) do
        h:= i*j*(n-j-i); w:= m(h);
        if w then m(h):= false; c:= c+1 fi
      od od; c
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Feb 13 2019

a[n_] := a[n] = Module[{m, c = 0, i, j, h, w}, m[_] = True; For[i = 1, i <= Quotient[n, 3], i++, For[j = i, j <= Quotient[n - i, 2], j++, h = i*j*(n - j - i); w = m[h]; If[w, m[h] = False; c++]]]; c];
a /@ Range[0, 80] (* Jean-François Alcover, Feb 24 2020, after Alois P. Heinz *)

a(n) <= A069905(n).

A307018 Total number of parts of size 3 in the partitions of n into parts of size 2 and 3.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 2, 4, 2, 4, 6, 4, 6, 9, 6, 9, 12, 9, 12, 16, 12, 16, 20, 16, 20, 25, 20, 25, 30, 25, 30, 36, 30, 36, 42, 36, 42, 49, 42, 49, 56, 49, 56, 64, 56, 64, 72, 64, 72, 81, 72, 81, 90, 81, 90, 100, 90, 100, 110, 100, 110, 121, 110, 121, 132

Offset: 0

Andrew Ivashenko, Mar 19 2019

Index entries for linear recurrences with constant coefficients, signature (0,1,2,0,-2,-1,0,1).

Cf. A000041, A002620, A103221, A114209, A321202.

Cf. A008133, A069905, A008611.

a:=[0,0,0,1,0,1,2,1];; for n in [9..80] do a[n]:=a[n-2]+2*a[n-3] -2*a[n-5]-a[n-6]+a[n-8]; od; a; # G. C. Greubel, Apr 03 2019

R:=PowerSeriesRing(Integers(), 80); [0,0,0] cat Coefficients(R!( x^3/((1-x^2)*(1-x^3)^2) )); // G. C. Greubel, Apr 03 2019

LinearRecurrence[{0,1,2,0,-2,-1,0,1}, {0,0,0,1,0,1,2,1}, 80] (* G. C. Greubel, Apr 03 2019 *)
Table[(6n(2+n)-5-27(-1)^n+8(4+3n)Cos[2n Pi/3]-8Sqrt[3]n Sin[2n Pi/3])/216,{n,0,66}] (* Stefano Spezia, Apr 21 2022 *)

my(x='x+O('x^80)); concat([0,0,0], Vec(x^3/((1-x^2)*(1-x^3)^2))) \\ G. C. Greubel, Apr 03 2019

(x^3/((1-x^2)*(1-x^3)^2)).series(x, 80).coefficients(x, sparse=False) # G. C. Greubel, Apr 03 2019

a(n+2) = A321202(n) - A114209(n+1).
a(3n+1) = A002620(n+2).
a(3n+2) = A002620(n+1).
a(3n+3) = A002620(n+2).
G.f.: x^3/((1+x)*(1+x+x^2)^2*(1-x)^3). - Alois P. Heinz, Mar 19 2019
a(n) = a(n-2) + 2*a(n-3) - 2*a(n-5) - a(n-6) + a(n-8). - G. C. Greubel, Apr 03 2019
a(n) = (6*n*(2 + n) + 8*(4 + 3*n)*cos(2*n*Pi/3) - 8*sqrt(3)*n*sin(2*n*Pi/3) - 5 - 27*(-1)^n)/216. - Stefano Spezia, Apr 21 2022
From Ridouane Oudra, Nov 24 2024: (Start)
a(n) = (7*n/2 - 7*n^2/2 - 9*floor(n/2) + (6*n+4)*floor(2*n/3) + 4*floor(n/3))/18.
a(n) = A008133(n) - A069905(n-1).
a(n) = A002620(A008611(n)). (End)

More terms from Alois P. Heinz, Mar 19 2019

A307872 Sum of the smallest parts in the partitions of n into 3 parts.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 5, 7, 11, 13, 17, 23, 27, 33, 42, 48, 57, 69, 78, 90, 106, 118, 134, 154, 170, 190, 215, 235, 260, 290, 315, 345, 381, 411, 447, 489, 525, 567, 616, 658, 707, 763, 812, 868, 932, 988, 1052, 1124, 1188, 1260, 1341, 1413, 1494, 1584, 1665

Offset: 1

Wesley Ivan Hurt, May 02 2019

			Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
                                                          1+1+8
                                                   1+1+7  1+2+7
                                                   1+2+6  1+3+6
                                            1+1+6  1+3+5  1+4+5
                                     1+1+5  1+2+5  1+4+4  2+2+6
                              1+1+4  1+2+4  1+3+4  2+2+5  2+3+5
                       1+1+3  1+2+3  1+3+3  2+2+4  2+3+4  2+4+4
         1+1+1  1+1+2  1+2+2  2+2+2  2+2+3  2+3+3  3+3+3  3+3+4    ...
-----------------------------------------------------------------------
  n  |     3      4      5      6      7      8      9     10      ...
-----------------------------------------------------------------------
a(n) |     1      1      2      4      5      7     11     13      ...
-----------------------------------------------------------------------

Michael De Vlieger, Table of n, a(n) for n = 1..3000
Index entries for sequences related to partitions

Cf. A069905.

Table[Sum[Sum[k, {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
Table[Total[IntegerPartitions[n,{3}][[;;,-1]]],{n,100}] (* Harvey P. Dale, Jan 14 2024 *)

a(n) = sum(k=1, n\3, sum(i=k, (n-k)\2, k)); \\ Michel Marcus, May 02 2019

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} k.
Conjectures from Colin Barker, May 02 2019: (Start)
G.f.: x^3 / ((1 - x)^4*(1 + x)*(1 + x + x^2)^2).
a(n) = a(n-1) + a(n-2) + a(n-3) - 2*a(n-4) - 2*a(n-5) + a(n-6) + a(n-7) + a(n-8) - a(n-9) for n > 9.
(End)
a(n) = ((-1)^n*(-1+(-1)^r)+2*r*((-1)^(n+r)+(1+r)*(1+2*n-4*r)))/16, where r = floor(n/3). - Wesley Ivan Hurt, Oct 24 2021

A309405 Number of prime parts in the partitions of n into 3 parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 5, 7, 8, 12, 12, 16, 17, 21, 22, 29, 29, 34, 35, 41, 42, 50, 50, 58, 59, 67, 68, 77, 78, 86, 87, 96, 97, 108, 108, 119, 120, 130, 131, 144, 144, 155, 156, 168, 169, 182, 183, 197, 198, 212, 213, 228, 228, 242, 243, 258, 259, 275, 275, 291

Offset: 0

Wesley Ivan Hurt, Jul 30 2019

nonn

			Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
                                                          1+1+8
                                                   1+1+7  1+2+7
                                                   1+2+6  1+3+6
                                            1+1+6  1+3+5  1+4+5
                                     1+1+5  1+2+5  1+4+4  2+2+6
                              1+1+4  1+2+4  1+3+4  2+2+5  2+3+5
                       1+1+3  1+2+3  1+3+3  2+2+4  2+3+4  2+4+4
         1+1+1  1+1+2  1+2+2  2+2+2  2+2+3  2+3+3  3+3+3  3+3+4    ...
-----------------------------------------------------------------------
  n  |     3      4      5      6      7      8      9     10      ...
-----------------------------------------------------------------------
a(n) |     0      1      3      5      7      8     12     12      ...
-----------------------------------------------------------------------

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for sequences related to partitions

Cf. A010051, A069905.

Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) + (PrimePi[j] - PrimePi[j - 1]) + (PrimePi[n - i - j] - PrimePi[n - i - j - 1]), {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
Table[Count[Flatten[IntegerPartitions[n,{3}]],?PrimeQ],{n,0,60}] (* _Harvey P. Dale, Jun 13 2025 *)

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (c(i) + c(j) + c(n-i-j)), where c = A010051.

A348536 Number of partitions of n into 3 parts that divide n.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0

Offset: 1

Wesley Ivan Hurt, Oct 21 2021

From David A. Corneth, Oct 08 2022: (Start)
Proof of formula: suppose we have a partition d_1 + d_2 + d_3 = n where 0 < d_1 <= d_2 <= d_3 and d_1 | n, d_2 | n and d_3 | n.
Then let m_i * d_i = n for i in (1, 2, 3).
Dividing d_1 + d_2 + d_3 = n by n gives d_1/n + d_2/n + d_3/n = n/n, i.e., 1/m_1 + 1/m_2 + 1/m_3 = 1. There are 3 solutions to that equation namely (m_1, m_2, m_3) in {(2, 4, 4), (2, 3, 6), (3, 3, 3)}. The lcm of these numbers is 12 so the sequence is periodic with a period of 12 and then calculating the first 12 terms defines the sequence.
Alternative name: number of divisors of n from {3, 4, 6}. (End)

			a(12) = 3 via 2 + 4 + 6 = 3 + 3 + 6 = 4 + 4 + 4. - _David A. Corneth_, Oct 08 2022

David A. Corneth, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions

Cf. A002966, A069905, A355200.

Block[{c}, c[n_] := 1 - Ceiling[n] + Floor[n]; Array[Sum[Sum[c[#/j]*c[#/i]*c[#/(# - i - j)], {i, j, Floor[(# - j)/2]} ], {j, Floor[#/3]} ] &, 105]] (* Michael De Vlieger, Oct 21 2021 *)
Table[Count[IntegerPartitions[n,{3}],?(Mod[n,#]=={0,0,0}&)],{n,100}] (* _Harvey P. Dale, Apr 07 2025 *)

a(n) = [0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 3][(n-1)%12 + 1] \\ David A. Corneth, Oct 08 2022

a(n) = { my(d = divisors(n), res = 0); d = d[^#d]; forvec(x = vector(2, i, [1, #d]), s = d[x[1]] + d[x[2]]; if(n - s >= d[x[2]], if(n % (n - s) == 0, print([d[x[1]], d[x[2]], n-s]); res++ ) ) , 1 ); res } \\ David A. Corneth, Oct 08 2022

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} c(n/j) * c(n/i) * c(n/(n-i-j)), where c(n) = 1 - ceiling(n) + floor(n).

a(n + 12) = a(n) where n >= 1. - David A. Corneth, Oct 08 2022

cp's OEIS Frontend

A099830 Smallest perimeter S such that exactly n distinct Pythagorean triangles with this perimeter can be constructed.

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A215521 Number T(n,k) of distinct values of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n with largest part = k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A309513 Number of even parts in the partitions of n into 3 parts.

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A374702 Number of integer compositions of n whose leaders of maximal weakly decreasing runs sum to 3. Column k = 3 of A374748.

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A242089 Number of triples (a,b,c) with 0 < a < b < c < p and a + b + c == 0 mod p, where p = prime(n).

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A306403 The number of distinct products that can be formed by multiplying the parts of a partition of n into 3 positive parts.

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A307018 Total number of parts of size 3 in the partitions of n into parts of size 2 and 3.

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