cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A070104 Number of integer triangles with perimeter n and relatively prime side lengths which are obtuse and scalene.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 2, 3, 1, 4, 3, 6, 2, 7, 4, 8, 4, 8, 6, 10, 6, 12, 8, 14, 8, 16, 11, 18, 11, 17, 14, 21, 12, 25, 18, 25, 15, 30, 19, 32, 20, 32, 25, 38, 23, 40, 28, 41, 28, 47, 31, 51, 34, 46, 40, 55, 35, 61, 44, 58, 41, 68
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Links

Crossrefs

Cf. A070080, A070081, A070082, A070101, A051493, A005044, A070102, A024156, A070084, A070131.

Programs

  • Maple
    f:= proc(n) local a,b,q,bmin,bmax,t;
  t:= 0;
  for a from 1 to n/3 do
    if n::even then bmin:= max(a+1,n/2-a+1) else bmin:= max(a+1,(n+1)/2-a) fi;
    q:= (n^2-2*n*a)/(2*(n-a));
    if q::integer then bmax:= min((n-a)/2, q-1) else bmax:= min((n-a)/2, floor(q)) fi;
    t:= t + nops(select(b -> igcd(a,b,n-a-b) = 1, [$bmin .. bmax]))
  od;
  t
end proc:
map(f, [$1..100]); # Robert Israel, Jul 26 2024

A288165 Expansion of x^4/((1-x^4)*(1-x^3)*(1-x^6)*(1-x^9)).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 2, 1, 1, 3, 2, 1, 5, 3, 2, 6, 5, 3, 9, 6, 5, 11, 9, 6, 15, 11, 9, 18, 15, 11, 23, 18, 15, 27, 23, 18, 34, 27, 23, 39, 34, 27, 47, 39, 34, 54, 47, 39, 64, 54, 47, 72, 64, 54, 84, 72, 64, 94, 84, 72, 108, 94, 84, 120, 108, 94, 136, 120
Offset: 0

Views

Author

Seiichi Manyama, Jun 06 2017

Keywords

Examples

			a(57) = p_4(57/3)     = p_4(19) = A001400(15) = 54,
a(58) = p_4((58+8)/3) = p_4(22) = A001400(18) = 84,
a(59) = p_4((59+4)/3) = p_4(21) = A001400(17) = 72,
a(60) = p_4(60/3)     = p_4(20) = A001400(16) = 64,
a(61) = p_4((61+8)/3) = p_4(23) = A001400(19) = 94,
a(62) = p_4((62+4)/3) = p_4(22) = A001400(18) = 84.

Links

Crossrefs

Cf. A001400, A029253.
Cf. A005044 (k=3), this sequence (k=4), A288166 (k=5).

Formula

a(n) = p_4(n/3) if n == 0 mod 3,
a(n) = p_4((n+8)/3) if n == 1 mod 3,
a(n) = p_4((n+4)/3) if n == 2 mod 3,
where p_4(n) is the number of partitions of n into exactly 4 parts.

A288166 Expansion of x^5/((1-x^5)*(1-x^4)*(1-x^8)*(1-x^12)*(1-x^16)).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 3, 2, 1, 1, 5, 3, 2, 1, 7, 5, 3, 2, 10, 7, 5, 3, 13, 10, 7, 5, 18, 13, 10, 7, 23, 18, 13, 10, 30, 23, 18, 13, 37, 30, 23, 18, 47, 37, 30, 23, 57, 47, 37, 30, 70, 57, 47, 37, 84, 70, 57, 47, 101, 84, 70, 57, 119
Offset: 0

Views

Author

Seiichi Manyama, Jun 06 2017

Keywords

Examples

			a(56) = p_5(56/4)      = p_5(14) = A001401(9)  = 23,
a(57) = p_5((57+15)/4) = p_5(18) = A001401(13) = 57,
a(58) = p_5((58+10)/4) = p_5(17) = A001401(12) = 47,
a(59) = p_5((59+5)/4)  = p_5(16) = A001401(11) = 37,
a(60) = p_5(60/4)      = p_5(15) = A001401(10) = 30,
a(61) = p_5((61+15)/4) = p_5(19) = A001401(14) = 70,
a(62) = p_5((62+10)/4) = p_5(18) = A001401(13) = 57,
a(63) = p_5((63+5)/4)  = p_5(17) = A001401(12) = 47.

Links

Crossrefs

Cf. A001401.
Cf. A005044 (k=3), A288165 (k=4), this sequence (k=5).

Programs

  • Mathematica
    CoefficientList[Series[x^5/((1-x^4)(1-x^5)(1-x^8)(1-x^12)(1-x^16)),{x,0,120}],x] (* or *) LinearRecurrence[ {0,0,0,1,1,0,0,1,-1,0,0,0,-1,0,0,0,0,0,0,-2,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,1,-1,0,0,-1,-1,0,0,0,1},{0,0,0,0,0,1,0,0,0,1,1,0,0,2,1,1,0,3,2,1,1,5,3,2,1,7,5,3,2,10,7,5,3,13,10,7,5,18,13,10,7,23,18,13,10},120] (* Harvey P. Dale, Apr 22 2019 *)

Formula

a(n) = p_5(n/4) if n == 0 mod 4,
a(n) = p_5((n+15)/4) if n == 1 mod 4,
a(n) = p_5((n+10)/4) if n == 2 mod 4,
a(n) = p_5((n+5)/4) if n == 3 mod 4,
where p_5(n) is the number of partitions of n into exactly 5 parts.

A307828 Number of integer-sided triangles with perimeter n whose smallest side-length divides n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 4, 3, 1, 3, 1, 5, 4, 1, 1, 8, 4, 1, 4, 6, 1, 8, 1, 7, 4, 1, 8, 12, 1, 1, 4, 14, 1, 9, 1, 8, 12, 1, 1, 18, 5, 8, 4, 9, 1, 10, 10, 17, 4, 1, 1, 28, 1, 1, 13, 15, 11, 11, 1, 11, 4, 18, 1, 31, 1, 1, 15, 12, 11, 12, 1, 32
Offset: 1

Views

Author

Wesley Ivan Hurt, May 15 2019

Keywords

Links

Crossrefs

Cf. A005044.

Programs

  • Mathematica
    Table[Sum[Sum[(1 - Ceiling[n/k] + Floor[n/k]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
  • PARI
    A307828(n) = sum(k=1, n\3, sum(i=k, (n-k)\2, sign(floor((i+k)/(n-i-k+1)) * (1 - ceil(n/k) + floor(n/k))))); \\ Antti Karttunen, Dec 05 2021

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1)) * (1 - ceiling(n/k) + floor(n/k))).

A308454 Number of integer-sided triangles with perimeter n whose largest side length is squarefree.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 1, 1, 0, 3, 2, 5, 4, 7, 5, 4, 3, 2, 1, 6, 5, 10, 9, 8, 7, 13, 11, 17, 15, 21, 18, 17, 14, 22, 19, 18, 15, 24, 20, 19, 16, 26, 22, 33, 29, 40, 36, 35, 31, 30, 26, 38, 35, 33, 30, 28, 25, 38, 35, 48, 45, 58, 54, 51, 48, 62, 58, 73, 69, 84
Offset: 1

Views

Author

Wesley Ivan Hurt, May 27 2019

Keywords

Examples

			There exist A005044(11) = 4 integer-sided triangles with perimeter = 11; these four triangles have respectively sides: (1, 5, 5); (2, 4, 5); (3, 3, 5); (3, 4, 4). Only the last one: (3, 4, 4) has a largest side length = 4 that is not squarefree, so a(11) = 3. - _Bernard Schott_, Jan 24 2023

Links

Crossrefs

Cf. A005044, A008683, A308453.

Programs

  • Maple
    f:= proc(n) local p, v;
  v:= add(1/2*(3*p-n+1)+`if`((n-p)::even, 1/2, 0),
     p = select(numtheory:-issqrfree, [$ceil(n/3)..floor((n-1)/2)]));
end proc:
map(f, [$1..100]); # Robert Israel, Jan 16 2023
  • Mathematica
    Table[Sum[Sum[ MoebiusMu[n - i - k]^2* Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * mu(n-i-k)^2, where mu is the Möbius function (A008683).

A333917 Perimeters of integer-sided triangles whose altitude from their longest side is an integer.

Original entry on oeis.org

16, 18, 32, 36, 48, 50, 54, 60, 64, 70, 72, 80, 90, 96, 98, 100, 108, 112, 120, 126, 128, 132, 140, 144, 150, 154, 160, 162, 168, 176, 180, 192, 196, 198, 200, 208, 210, 216, 220, 224, 234, 240, 242, 250, 252, 256, 260, 264, 270, 272, 280, 286, 288, 290, 294, 300
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 09 2020

Keywords

Examples

			16 is in the sequence since it is the perimeter of the triangle [5,5,6], whose altitude from 6 (the longest side) is 4 (an integer).
18 is in the sequence since it is the perimeter of the triangle [5,5,8], whose altitude from 8 (the longest side) is 3 (an integer).
48 is in the sequence since it is the perimeter of the triangles [15,15,18] and [10,17,21], whose altitudes from their longest sides are 12 and 8 respectively (both integers).

Links

Crossrefs

Cf. A005044, A333918, A333919.

Programs

  • Mathematica
    Flatten[Table[If[Sum[Sum[(1 - Ceiling[2*Sqrt[(n/2) (n/2 - i) (n/2 - k) (n/2 - (n - i - k))]/(n - i - k)] + Floor[2*Sqrt[(n/2) (n/2 - i) (n/2 - k) (n/2 - (n - i - k))]/(n - i - k)]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}] > 0, n, {}], {n, 100}]]

A333918 Perimeters of integer-sided triangles whose altitude from their shortest side is an integer.

Original entry on oeis.org

12, 24, 30, 32, 36, 40, 42, 44, 48, 50, 54, 56, 60, 64, 66, 68, 70, 72, 76, 80, 84, 88, 90, 96, 98, 100, 104, 108, 112, 120, 126, 128, 130, 132, 136, 140, 144, 150, 152, 154, 156, 160, 162, 164, 168, 170, 172, 174, 176, 180, 182, 186, 190, 192, 196, 198, 200
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 09 2020

Keywords

Links

Crossrefs

Cf. A005044, A333917, A333919.

Programs

  • Mathematica
    Flatten[Table[If[Sum[Sum[(1 - Ceiling[2*Sqrt[(n/2) (n/2 - i) (n/2 - k) (n/2 - (n - i - k))]/k] + Floor[2*Sqrt[(n/2) (n/2 - i) (n/2 - k) (n/2 - (n - i - k))]/k]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}] > 0, n, {}], {n, 100}]]

Formula

12 is in the sequence since it is the perimeter of the triangle [3,4,5], whose altitude from 3 (the shortest side) is 4 (an integer).
24 is in the sequence since it is the perimeter of the triangle [6,8,10], whose altitude from 6 (the shortest side) is 8 (an integer).
54 is in the sequence since it is the perimeter of the triangles [3,25,26] and [12,17,25] whose altitudes from their shortest sides are 24 and 15 respectively (both integers).

A333919 Perimeters of integer-sided triangles with side lengths a <= b <= c whose altitude from side b is an integer.

Original entry on oeis.org

12, 24, 30, 36, 40, 42, 48, 56, 60, 70, 72, 78, 80, 84, 90, 96, 104, 108, 110, 112, 114, 120, 126, 132, 136, 140, 144, 150, 154, 156, 160, 162, 168, 176, 180, 182, 186, 192, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 230, 232, 234, 238, 240, 250, 252
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 09 2020

Keywords

Examples

			12 is in the sequence since it is the perimeter of the triangle [3,4,5], whose altitude from 4 (its "middle" side) is 3 (an integer).
24 is in the sequence since it is the perimeter of the triangle [6,8,10], whose altitude from 8 (its "middle" side) is 6 (an integer).
60 is in the sequence since it is the perimeter of the triangles [10,24,26] and [15,20,25], whose altitudes (from their "middle" sides) are 10 and 15 respectively (both integers).

Links

Crossrefs

Cf. A005044, A333917, A333918.

Programs

  • Mathematica
    Flatten[Table[If[Sum[Sum[(1 - Ceiling[2*Sqrt[(n/2) (n/2 - i) (n/2 - k) (n/2 - (n - i - k))]/i] + Floor[2*Sqrt[(n/2) (n/2 - i) (n/2 - k) (n/2 - (n - i - k))]/i]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}] > 0, n, {}], {n, 100}]]

A334761 Perimeters of Pythagorean triangles whose hypotenuse divides the difference of squares of its long and short legs.

Original entry on oeis.org

60, 120, 180, 240, 300, 360, 390, 420, 480, 540, 600, 660, 680, 720, 780, 840, 900, 960, 1020, 1080, 1140, 1170, 1200, 1260, 1320, 1360, 1380, 1400, 1440, 1500, 1560, 1620, 1680, 1740, 1800, 1860, 1920, 1950, 1980, 2030, 2040, 2100, 2160, 2220, 2280, 2340, 2400
Offset: 1

Views

Author

Wesley Ivan Hurt, May 10 2020

Keywords

Comments

The smallest terms corresponding to 2,...,5 triangles are a(15) = 780, a(191) = 9360, a(3324) = 159120, and a(19433) = 928200, respectively. - Giovanni Resta, May 11 2020

Examples

			a(1) = 60; the triangle [15,20,25] has perimeter 60. The difference of squares of its long and short leg lengths is (20^2 - 15^2) = 400 - 225 = 175 and 25|175.

Links

Crossrefs

Cf. A005044, A010814.

Programs

  • Mathematica
    Reap[Do[s = Solve[x^2 + y^2 == (p-x-y)^2 && z^2 == x^2 + y^2 && 0 0, {x, y, z}, Integers]; If[s != {} && AnyTrue[{x, y , z} /. s, Mod[#[[2]]^2 - #[[1]]^2, #[[3]]] == 0 &], Print@Sow@p], {p, 12, 1000, 2}]][[2, 1]] (* Giovanni Resta, May 11 2020 *)

Extensions

Terms a(31) and beyond from Giovanni Resta, May 11 2020

A371973 a(n) is the number of distinct areas > 0 of triangles with integer sides and perimeter n.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 13, 19, 14, 21, 19, 23, 20, 27, 23, 30, 27, 32, 29, 35, 32, 39, 34, 44, 39, 48, 43, 52, 47, 55, 51, 60, 53, 63, 59, 69, 58, 74, 67, 78, 73, 84, 75, 90, 81, 92, 88, 101, 91, 108, 93, 112, 106
Offset: 3

Views

Author

Hugo Pfoertner, Apr 16 2024

Keywords

Links

Crossrefs

Cf. A005044, A007237, A024153, A317182, A371070.
See the formula section for the relationships with A026810, A070083, A135622 (which has many crossrefs related to areas of triangles).

Programs

  • PARI
    A2(a,b,c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
a371973(n) = {my (A=List()); forpart (v=n, listput(A, A2(v[1],v[2],v[3])), [1,(n-1)\2], [3,3]); #Set(A)};
  • Python
    def A371973(n): return len(set((2*(b+c)-n)*(n-2*b)*(n-2*c) for c in range((n+2)//3, (n+1)//2) for b in range((n-c+1)//2, c+1))) # David Radcliffe, Aug 01 2025

Formula

a(n) = |{A135622(k) : A070083(k) = n}| = |{A135622(k) : A026810(n) < k <= A026810(n+1)}|. - Peter Munn, Jul 29 2025

Extensions

b-file corrected by David Radcliffe, Aug 01 2025
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