A005044 -id:A005044 - cp's OEIS frontend

A070113 Numbers k such that [A070080(k), A070081(k), A070082(k)] is a scalene integer triangle with relatively prime side lengths.

8, 13, 17, 20, 21, 25, 29, 30, 33, 36, 37, 41, 42, 44, 45, 49, 53, 56, 57, 59, 60, 62, 66, 67, 69, 70, 74, 75, 77, 78, 79, 80, 83, 86, 89, 90, 92, 96, 97, 99, 100, 101, 102, 105, 106, 110, 111, 113, 114, 115, 119, 122, 123, 125, 126

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			36 is a term [A070080(36), A070081(36), A070082(36)]=[3<6<7], A070084(36)=gcd(3,6,7)=1.

Jean-François Alcover, Table of n, a(n) for n = 1..596
Reinhard Zumkeller, Integer-sided triangles

Cf. A005044, A070122, A070131, A070112, A070110.

m = 50 (* max perimeter *);
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1] & // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];
Position[triangles, {a_, b_, c_} /; a < b < c && GCD[a, b, c] == 1] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

A070203 Number of scalene integer triangles with perimeter n having integral inradius.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 8, 0, 0, 0, 1, 0, 3

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A070201(n) - A070204(n).

Seiichi Manyama, Table of n, a(n) for n = 1..5000
Mohammad K. Azarian, Solution to Problem S125: Circumradius and Inradius, Math Horizons, Vol. 16, Issue 2, November 2008, p. 32.
Eric Weisstein's World of Mathematics, Incircle.
Eric Weisstein's World of Mathematics, Scalene Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A024153, A005044.

def A(n)
  cnt = 0
  (1..n / 3).each{|a|
    (a + 1..(n - a) / 2).each{|b|
      c = n - a - b
      if a + b > c && b < c
        s = n / 2r
        t = (s - a) * (s - b) * (s - c) / s
        if t.denominator == 1
          t = t.to_i
          cnt += 1 if Math.sqrt(t).to_i ** 2 == t
        end
      end
    }
  }
  cnt
end
def A070203(n)
  (1..n).map{|i| A(i)}
end
p A070203(100) # Seiichi Manyama, Oct 07 2017

A107573 a(n)=least sidelength of n-th triangle listed at A107572.

Original entry on oeis.org

2, 2, 3, 2, 3, 3, 2, 3, 4, 3, 4, 2, 3, 4, 4, 3, 4, 5, 2, 3, 4, 4, 5, 3, 4, 5, 5, 2, 3, 4, 4, 5, 5, 6, 3, 4, 5, 5, 6, 2, 3, 4, 4, 5, 5, 6, 6, 3, 4, 5, 5, 6, 6, 7, 2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 3, 4, 5, 5, 6, 6, 7, 7, 2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 3, 4, 5, 5, 6, 6, 7, 7, 7, 8, 2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7

Offset: 1

Clark Kimberling, May 16 2005

nonn

			The first 5 integer-sided scalene triangles (a,b,c) with a<b<c are (2,3,4), (2,4,5), (3,4,5), (2,5,6), (3,4,6), of which the least sidelengths are 2,2,3,2,3

Cf. A107572, A107574, A107575, A107576, A107577, A005044, A070080.

A107574 a(n)=middle sidelength of n-th triangle listed at A107572.

Original entry on oeis.org

3, 4, 4, 5, 4, 5, 6, 5, 5, 6, 5, 7, 6, 5, 6, 7, 6, 6, 8, 7, 6, 7, 6, 8, 7, 6, 7, 9, 8, 7, 8, 6, 7, 7, 9, 8, 7, 8, 7, 10, 9, 8, 9, 7, 8, 7, 8, 10, 9, 8, 9, 7, 8, 8, 11, 10, 9, 10, 8, 9, 7, 8, 9, 8, 11, 10, 9, 10, 8, 9, 8, 9, 12, 11, 10, 11, 9, 10, 8, 9, 10, 8, 9, 9, 12, 11, 10, 11, 9, 10, 8, 9, 10, 9, 13

Offset: 1

Clark Kimberling, May 16 2005

nonn

			The first 5 integer-sided scalene triangles (a,b,c) with a<b<c are (2,3,4), (2,4,5), (3,4,5), (2,5,6), (3,4,6), of which the middle sidelengths are 3,4,4,5,4.

Cf. A107572, A107573, A107575, A107576, A107577, A005044, A070080.

A107575 a(n)=greatest sidelength of n-th triangle listed at A107572.

Original entry on oeis.org

4, 5, 5, 6, 6, 6, 7, 7, 6, 7, 7, 8, 8, 8, 7, 8, 8, 7, 9, 9, 9, 8, 8, 9, 9, 9, 8, 10, 10, 10, 9, 10, 9, 8, 10, 10, 10, 9, 9, 11, 11, 11, 10, 11, 10, 10, 9, 11, 11, 11, 10, 11, 10, 9, 12, 12, 12, 11, 12, 11, 12, 11, 10, 10, 12, 12, 12, 11, 12, 11, 11, 10, 13, 13, 13, 12, 13, 12, 13, 12, 11

Offset: 1

Clark Kimberling, May 16 2005

nonn

			The first 5 integer-sided scalene triangles (a,b,c) with a<b<c are (2,3,4), (2,4,5), (3,4,5), (2,5,6), (3,4,6), of which the greatest sidelengths are 4,5,5,6,6.

Cf. A107572, A107573, A107574, A107576, A107577, A005044, A070080.

A254594 Expansion of 1 / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

Original entry on oeis.org

1, 0, 2, 1, 4, 2, 7, 4, 11, 7, 16, 11, 23, 16, 31, 23, 41, 31, 53, 41, 67, 53, 83, 67, 102, 83, 123, 102, 147, 123, 174, 147, 204, 174, 237, 204, 274, 237, 314, 274, 358, 314, 406, 358, 458, 406, 514, 458, 575, 514, 640, 575, 710, 640, 785, 710, 865, 785, 950

Offset: 0

Michael Somos, Feb 02 2015

Partitions of n into parts of size 3 and size 4 and two kinds of parts of size 2.
The number of quadruples of integers [x, u, v, w] which satisfy x > u > v > w >=0, n+5 = x+u, u+v >= x+w, and x+u+v+w is even.
Euler transform of length 4 sequence [ 0, 2, 1, 1].

			G.f. = 1 + 2*x^2 + x^3 + 4*x^4 + 2*x^5 + 7*x^6 + 4*x^7 + 11*x^8 + 7*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,1,0,-2,-2,0,1,2,0,-1).

Cf. A001400, A002620, A005044, A115264, A165188.

Cf. A241526, A000601, A181120.

I:=[1,0,2,1,4,2,7,4,11,7,16]; [n le 11 select I[n] else 2*Self(n-2)+Self(n-3)-2*Self(n-5)-2*Self(n-6)+Self(n-8)+2*Self(n-9)-Self(n-11): n in [1..60]]; // Vincenzo Librandi, Feb 03 2015

a[ n_] := Quotient[ n^3 + If[ OddQ[n], 12 n^2 + 33 n + 54, 21 n^2 + 132 n + 288], 288];
a[ n_] := Module[{s = 1, m = n}, If[ n < 0, s = -1; m = -11 - n]; s SeriesCoefficient[ 1 / ((1 - x^2)^2 (1 - x^3) (1 - x^4)), {x, 0, m}]];
a[ n_] := Length @ FindInstance[ {x > u, u > v, v > w, w >= 0, x + u == n + 5, u + v >= x + w, x + u + v + w == 2 k}, {x, u, v, w, k}, Integers, 10^9];
CoefficientList[Series[1 / (1 - 2 x^2 - x^3 + 2 x^5 + 2 x^6 - x^8 - 2 x^9 + x^11), {x, 0, 60}], x] (* Vincenzo Librandi, Feb 03 2015 *)

{a(n) = (n^3 + if(n%2, 12*n^2 + 33*n + 54, 21*n^2 + 132*n + 288)) \ 288};

{a(n) = my(s=1); if( n<0, s=-1; n=-11-n); s * polcoeff( 1 / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};

G.f.: 1 / (1 - 2*x^2 - x^3 + 2*x^5 + 2*x^6 - x^8 - 2*x^9 + x^11).
a(n) = -a(-11-n) for all n in Z.
a(n+3) - a(n) = 0 if n even else floor((n+7)^2 / 16).
0 = a(n) - 2*a(n+2) - a(n+3) + 2*a(n+5) + 2*a(n+6) - a(n+8) - 2*a(n+9) + a(n+11) for all n in Z.
a(n) - a(n-2) = A005044(n+3) for all n in Z.
a(n) + a(n-1) = A001400(n) for all n in Z.
a(n) + a(n-2) = A165188(n+1) for all n in Z.
a(n) = A115264(n) - A115264(n-1) for all n in Z.
a(2*n) - a(2*n-6) = a(2*n+3) - a(2*n-3) = A002620(n+2) for all n in Z. - Michael Somos, Feb 11 2015
a(n) = (2*n^3+33*n^2+181*n+234+3*(3*n^2+33*n+86)*(-1)^n+84*(-1)^((2*n+1-(-1)^n)/4)-96*((1+(-1)^n)*floor(((2*n+9+(-1)^n-6*(-1)^((2*n+3+(-1)^n)/4))/24))+(1-(-1)^n)*floor(((2*n+5+(-1)^n-6*(-1)^((2*n-1+(-1)^n)/4))/24))))/576. - Luce ETIENNE, May 22 2015

A325695 Number of length-3 strict integer partitions of n such that the largest part is not the sum of the other two.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 2, 5, 5, 8, 7, 12, 11, 16, 15, 21, 20, 27, 25, 33, 32, 40, 38, 48, 46, 56, 54, 65, 63, 75, 72, 85, 83, 96, 93, 108, 105, 120, 117, 133, 130, 147, 143, 161, 158, 176, 172, 192, 188, 208, 204, 225, 221, 243, 238, 261, 257, 280, 275

Offset: 0

Gus Wiseman, May 15 2019

nonn

			The a(7) = 1 through a(15) = 12 partitions (A = 10, B = 11, C = 12):
  (421)  (521)  (432)  (631)  (542)  (543)  (643)   (653)   (654)
                (531)  (721)  (632)  (732)  (652)   (842)   (753)
                (621)         (641)  (741)  (742)   (851)   (762)
                              (731)  (831)  (751)   (932)   (843)
                              (821)  (921)  (832)   (941)   (852)
                                            (841)   (A31)   (861)
                                            (931)   (B21)   (942)
                                            (A21)           (951)
                                                            (A32)
                                                            (A41)
                                                            (B31)
                                                            (C21)

Cf. A000041, A001399, A005044, A008642, A069905, A124278.

Cf. A325686, A325690, A325691, A325694, A325696.

Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&&#[[1]]!=#[[2]]+#[[3]]&]],{n,0,30}]

Conjectures from Colin Barker, May 15 2019: (Start)
G.f.: x^7*(1 + x + 2*x^2) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>9.
(End)
a(n) = A325696(n)/6. - Alois P. Heinz, Jun 18 2020

A385736 a(n) is the number of distinct nondegenerate triangles with perimeter n whose side lengths are triangular numbers.

Original entry on oeis.org

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

Offset: 0

Felix Huber, Jul 16 2025

0, 1, 6, 10, 28, 55 are the only triangular numbers <= 10^6 that are not perimeters of triangles whose side lengths are triangular numbers. Conjecture: There are no other triangular numbers that have this property.

			The a(31) = 2 distinct nondegenerate triangles with perimeter 31 and whose side lengths are triangular numbers are [1, 15, 15] and [6, 10, 15].

Felix Huber, Table of n, a(n) for n = 0..10000
Felix Huber, Maple program to compute the counted triangles
Eric Weisstein's World of Mathematics, Triangular Number

Cf. A000217, A005044, A051516, A070088, A366398, A380875, A385737, A385860, A385872.

A385736:=proc(N) # To get the first N + 1 terms.
    local p,x,y,z,i;
    p:=[];
    for z to floor((sqrt(24*N+9)-3)/6) do
        for x from z to floor((sqrt(4*N-3)-1)/2) do
            for y from max(z,floor((sqrt(1+4*(x^2+x-z^2-z))-1)/2)+1) to min(x,floor((sqrt(1+4*(2*N-x^2-x-z^2-z))-1)/2)) do
                p:=[op(p),z*(z+1)/2+y*(y+1)/2+x*(x+1)/2]
            od
        od
    od;
    return seq(numboccur(p,i),i=0..N)
end proc;
A385736(87);

Trivial upper bound: a(n) <= A005044(n).

a(A385737(n)) >= 1.

A005041 A self-generating sequence.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

See A008620 for run lengths: each k occurs A008620(k+2) times. - Reinhard Zumkeller, Mar 16 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
James Propp, Problem 1047, Math. Mag., 52 (1979), 265.
Jeffrey Shallit, Letter to N. J. A. Sloane, Nov 10 1979. Attached: James Propp, Problem 1047, Math. Mag., 52 (1979), 265. [Annotated scanned copy]
Aaron Snook, Augmented Integer Linear Recurrences, Thesis, 2012. - From _N. J. A. Sloane_, Dec 19 2012

Cf. A005038, A005039, A005040, A005043, A005044, A055086, A001462, A082462, A024417, A084500.

a005041 n = a005041_list !! n
a005041_list = 1 : f 1 1 (tail ts) where
   f y i gs'@((j,a):gs) | i < j  = y : f y (i+1) gs'
                        | i == j = a : f a (i+1) gs
   ts = [(6*k + 3*k*(k-1) `div` 2 + r*(k+2), 3*k+r+1) |
         k <- [0..], r <- [0,1,2]]
-- Reinhard Zumkeller, Mar 16 2012

Table[n+1, {n, 0, 20}, {Ceiling[(n+1)/3]+1}] // Flatten (* Jean-François Alcover, Dec 10 2014 *)

For any k in {0, 1, 2, ...} and r in {0, 1, 2}, we have: if n = 6*k + (3/2)*k*(k-1) + r*(k+2), then a(n) = 3*k + r + 1. E.g., for k=3 and r=1, we have n = 6*3 + (3/2)*3*(3-1) + 1*(3+2) = 32 and so a(32) = 3*3 + 1 + 1 = 11. - Francois Jooste (phukraut(AT)hotmail.com), Mar 12 2002

More terms from Samuel Hilliard (sam_spade1977(AT)hotmail.com), Apr 11 2004

A069981 Hermite's problem: number of positive integral solutions to x + y + z = n subject to x <= y + z, y <= z + x and z <= x + y.

Original entry on oeis.org

0, 0, 0, 1, 3, 3, 7, 6, 12, 10, 18, 15, 25, 21, 33, 28, 42, 36, 52, 45, 63, 55, 75, 66, 88, 78, 102, 91, 117, 105, 133, 120, 150, 136, 168, 153, 187, 171, 207, 190, 228, 210, 250, 231, 273, 253, 297, 276, 322, 300, 348, 325, 375, 351, 403, 378, 432

Offset: 0

N. J. A. Sloane, May 06 2002

G. Pólya and G. Szegő, Problems and Theorems in Analysis I, Springer-Verlag, Part I, Chap. 1, Problem 31.

William A. Tedeschi, Table of n, a(n) for n = 0..10000
G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package, p. 16.
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis: The Omega Package, Europ. J. Combin., 22 (2001), 887-904.

Cf. A005044.

[0] cat [(2*n^2 + 6*n - 17 + 3*(2*n - 5)*(-1)^n)/16: n in [1..60]]; // G. C. Greubel, Jun 10 2018

f[n_]:=If[EvenQ[n],((n+8)(n-2))/8,(n^2-1)/8];Join[{0},Array[f,60]] (* Harvey P. Dale, Jul 26 2011 *)

for(n=0, 60, print1(if(n==0, 0, (2*n^2 + 6*n - 17 + 3*(2*n - 5)*(-1)^n)/16), ", ")) \\ G. C. Greubel, Jun 10 2018

G.f.: x^3*(1 + 2*x - 2*x^2)/(1 - x)/(1 - x^2)^2.
a(n) = (n+8)*(n-2)/8 for n even, (n^2-1)/8 for n odd.
a(n) = (2*n^2 + 6*n - 17 + 3*(2*n - 5)*(-1)^n)/16 for n>0. - Luce ETIENNE, Jun 29 2015
E.g.f.: (16 + (x^2 + x - 16)*cosh(x) + (x^2 + 7*x - 1)*sinh(x))/8. - Stefano Spezia, May 09 2022

cp's OEIS Frontend

A070113 Numbers k such that [A070080(k), A070081(k), A070082(k)] is a scalene integer triangle with relatively prime side lengths.

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A070203 Number of scalene integer triangles with perimeter n having integral inradius.

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A107573 a(n)=least sidelength of n-th triangle listed at A107572.

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A107574 a(n)=middle sidelength of n-th triangle listed at A107572.

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A107575 a(n)=greatest sidelength of n-th triangle listed at A107572.

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A254594 Expansion of 1 / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

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A325695 Number of length-3 strict integer partitions of n such that the largest part is not the sum of the other two.

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Mathematica

Formula

A385736 a(n) is the number of distinct nondegenerate triangles with perimeter n whose side lengths are triangular numbers.

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Maple

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A005041 A self-generating sequence.

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