A003057 -id:A003057 - cp's OEIS frontend

A014410 Elements in Pascal's triangle (by row) that are not 1.

2, 3, 3, 4, 6, 4, 5, 10, 10, 5, 6, 15, 20, 15, 6, 7, 21, 35, 35, 21, 7, 8, 28, 56, 70, 56, 28, 8, 9, 36, 84, 126, 126, 84, 36, 9, 10, 45, 120, 210, 252, 210, 120, 45, 10, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 13, 78

Offset: 2

Mohammad K. Azarian

Also, rows of triangle formed using Pascal's rule except begin and end n-th row with n+2. - Asher Auel.
Row sums are A000918. - Roger L. Bagula and Gary W. Adamson, Jan 15 2009
Given the triangle signed by rows (+ - + ...) = M, with V = a variant of the Bernoulli numbers starting [1/2, 1/6, 0, -1/30, 0, 1/42, ...]; M*V = [1, 1, 1, ...]. - Gary W. Adamson, Mar 05 2012
Also A014410 * [1/2, 1/6, 0, -1/30, 0, 1/42, 0, ...] = [1, 2, 3, 4, ...]. For an alternative way to derive the Bernoulli numbers from a modified version of Pascal's triangle see A135225. - Peter Bala, Dec 18 2014
T(n,k) mod n = A053201(n,k), k=1..n-1. - Reinhard Zumkeller, Aug 17 2013
From Wolfdieter Lang, May 22 2015: (Start)
This is Johannes Scheubel's (1494-1570) (also Scheybl, Schöblin) version of the arithmetical triangle from his 1545 book "De numeris et diversis rationibus". See the Kac reference, p. 396 and the Table 12.1 on p. 395.
The row sums give 2*A000225(n-1) = A000918(n) = 2*(2^n - 1), n >= 2. (See the second comment above).
The alternating row sums give repeat(2,0) = 2*A059841(n), n >= 2. (End)
T(n+1,k) is the number of k-facets of the n-simplex. - Jianing Song, Oct 22 2023

			The triangle T(n,k) begins:
n\k  1  2   3   4    5    6    7    8   9  10 11
2:   2
3:   3  3
4:   4  6   4
5:   5 10  10   5
6:   6 15  20  15    6
7:   7 21  35  35   21    7
8:   8 28  56  70   56   28    8
9:   9 36  84 126  126   84   36    9
10: 10 45 120 210  252  210  120   45  10
11: 11 55 165 330  462  462  330  165  55  11
12: 12 66 220 495  792  924  792  495 220  66 12
... reformatted. - _Wolfdieter Lang_, May 22 2015

Victor J. Kac, A History of Mathematics, third edition, Addison-Wesley, 2009, pp. 395, 396.

Reinhard Zumkeller, Rows n=2..150 of triangle, flattened
Carl McTague, On the Greatest Common Divisor of binomial(qn, q), binomial(qn,2q), ..., binomial(qn, qn-q), arXiv:1510.06696 [math.CO], 2015.
Wikipedia, Johannes Scheubel (in German).
Wikipedia, Simplex

Cf. A007318, A000918, A027641.
A180986 is the same sequence but regarded as a square array.
Cf. A000225,A059841, A257241 (Stifel's version).

a014410 n k = a014410_tabl !! (n-2) !! (k-1)
a014410_row n = a014410_tabl !! (n-2)
a014410_tabl = map (init . tail) $ drop 2 a007318_tabl
-- Reinhard Zumkeller, Mar 12 2012

for i from 0 to 12 do seq(binomial(i, j)*1^(i-j), j = 1 .. i-1) od; # Zerinvary Lajos, Dec 02 2007

Select[ Flatten[ Table[ Binomial[ n, i ], {n, 0, 13}, {i, 0, n} ] ], #>1& ]

T(n,k) = binomial(n,k) = A007318(n,k), n >= 2, k = 1, 2, ..., n-1.
a(n) = C(A003057(n),A002260(n)) = C(A003057(n),A004736(n)). - Lekraj Beedassy, Jul 29 2006
T(n,k) = A028263(n,k) - A007318(n,k). - Reinhard Zumkeller, Mar 12 2012
gcd_{k=1..n-1} T(n, k) = A014963(n), see Theorem 1 of McTague link. - Michel Marcus, Oct 23 2015

More terms from Erich Friedman

A119741 A008279, with the first and last of each row removed.

Original entry on oeis.org

2, 3, 6, 4, 12, 24, 5, 20, 60, 120, 6, 30, 120, 360, 720, 7, 42, 210, 840, 2520, 5040, 8, 56, 336, 1680, 6720, 20160, 40320, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800, 11, 110, 990, 7920, 55440, 332640, 1663200, 6652800, 19958400, 39916800

Offset: 2

Lekraj Beedassy, Jul 29 2006

Triangle read by rows: T(n,k) (n>=2, k=1..n-1) is the number of topologies t on n points having exactly k+2 open sets such that t contains exactly one open set of size m for each m in {0,1,2,...,s,n} where s is the size of the largest proper open set in t. - N. J. A. Sloane, Jan 29 2016 [clarified by Geoffrey Critzer, Feb 19 2017]

			Triangle begins:
   2;
   3,  6;
   4, 12,  24;
   5, 20,  60,  120;
   6, 30, 120,  360,   720;
   7, 42, 210,  840,  2520,   5040;
   8, 56, 336, 1680,  6720,  20160,  40320;
   9, 72, 504, 3024, 15120,  60480, 181440,  362880;
  10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 rows)
G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.

Row sums give A038156.
Triangles in this series: A268216, A268217, A268221, A268222, A268223.
Cf. A008279, A014631, A282507.

T:= (n, k)-> n!/(n-k)!:
seq(seq(T(n,k), k=1..n-1), n=2..11);  # Alois P. Heinz, Aug 22 2025

Table[FactorialPower[n, k], {n, 2, 11}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 21 2020 *)

a(n) = (A003057(n))!/(A004736(n))! = (A002260(n))!*(A014410(n)).

T(n,k) = A173333(n+1,n-k+1), 1<=k<=n. - Reinhard Zumkeller, Feb 19 2010

Edited by Don Reble, Aug 01 2006

A349082 The number of two-term Egyptian fractions of rational numbers, x/y, 0 < x/y < 1, ordered as below. The sequence is the number of (p,q) pairs such that x/y = 1/p + 1/q where p and q are integers with p < q.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 4, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 2, 2, 1, 1, 1, 0, 2, 2, 1, 1, 1, 1, 0, 0, 4, 1, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 7, 4, 2, 1, 2, 1, 2, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 1, 3, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 4, 4, 1, 3, 1, 1, 0, 2, 1, 1, 0, 0, 0, 0, 4, 3, 2, 2, 1, 2, 0, 1, 1, 1, 0, 1, 0, 0, 0

Offset: 1

Jud McCranie, Nov 07 2021

The sequence are the terms in a triangle, where the rows correspond to the denominator of the rational number (starting with row 2, column 1) and the columns correspond to the numerators:
                    x=1  2  3  4  5       rationals x/y:
  Row 1 (y=2):  1                   1/2
  Row 2 (y=3):  1, 1                1/3, 2/3
  Row 3 (y=4):  2, 1, 1             1/4, 2/4, 3/4
  Row 4 (y=5):  1, 1, 1, 0          1/5, 2/5, 3/5, 4/5
  Row 5 (y=6):  4, 1, 1, 1, 1       1/6, 2/6, 3/6, 4/6, 5/6
Alternatively, order the rational numbers, x/y, 0 < x/y < 1, in this order: 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, 2/5, ... For example, in this ordering, the sixth rational number is 3/4. The numerators of the n-th rational number are A002260(n) and the denominators are A003057(n).
A018892 is a subsequence (for x/y = 1/n).

			The fourth rational number is 1/4, 1/4 = 1/5 + 1/20 = 1/6 + 1/12, so a(4)=2.

Jud McCranie, Table of n, a(n) for n = 1..990

Cf. A002260, A003057.

Columns: A018892 (x=1), A046079 (x=2).

A350361 2-tone chromatic number of a tree with maximum degree n.

Original entry on oeis.org

4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15

Offset: 1

Allan Bickle, Dec 26 2021

The 2-tone chromatic number of a graph G is the smallest number of colors for which G has a coloring where every vertex has two distinct colors, no adjacent vertices have a common color, and no pair of vertices at distance 2 have two common colors.

a(n) is also the 2-tone chromatic number of a star with n leaves.

			For a star with three leaves, label the leaves 12, 13, and 23.  Label the other vertex 45.  A total of 5 colors are used, so a(3)=5.

A. Bickle and B. Phillips, t-Tone Colorings of Graphs, Utilitas Math, 106 (2018) 85-102.
Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer. 217 (2013), 171-190.
N. Fonger, J. Goss, B. Phillips, and C. Segroves, Math 6450: Final Report, (2009).

Cf. A003057, A350362.

Table[Ceiling[(5 + Sqrt[1 + 8*n])/2],{n,71}] (* Stefano Spezia, Dec 27 2021 *)

a(n) = A003057(n-1) + 2.

a(n) = ceiling((5 + sqrt(1 + 8*n))/2).

A351120 Pair chromatic number of a cycle graph with n vertices.

Original entry on oeis.org

6, 6, 5, 5, 6, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14

Offset: 3

Allan Bickle, Feb 01 2022

nonn

The pair chromatic number of a graph G is the smallest number of colors for which G has a coloring where every vertex has two distinct colors, no adjacent vertices have a common color, and no pair of colors is repeated.

There is no pair 5-coloring for cycles of length 3, 4, 7, or 10 since the Petersen graph does not contain cycles of these lengths.

			The colorings for (broken) cycles with orders 3 through 9 are shown below.
  -12-34-56-
  -12-34-15-36-
  -12-34-51-23-45-
  -12-34-15-32-14-35-
  -12-34-56-13-24-35-46-
  -12-34-15-23-14-25-13-45-
  -12-34-15-32-14-25-13-24-35-

Paolo Xausa, Table of n, a(n) for n = 3..10000
Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer., Vol. 217 (2013), pp. 171-190.
Allan Bickle and Ben Phillips, t-Tone Colorings of Graphs, Utilitas Math, Vol. 106 (2018), pp. 85-102.

Cf. A003057, A350361, A350362.

A351120[n_]:=If[n<11,{6,6,5,5,6,5,5,6}[[n-2]],Ceiling[(1+Sqrt[1+8n])/2]];Array[A351120,100,3] (* Paolo Xausa, Nov 30 2023 *)

a(n) = ceiling((1 + sqrt(1 + 8*n))/2) for n > 10.

A349090 Where zeros occur in A349082. These correspond to rationals, 0 < p/q < 1, that have no solution p/q = 1/x + 1/y, 0 < x < y.

Original entry on oeis.org

10, 18, 20, 21, 28, 35, 36, 44, 45, 50, 52, 53, 54, 55, 66, 69, 70, 71, 72, 74, 75, 76, 77, 78, 84, 88, 89, 90, 91, 98, 102, 103, 104, 105, 112, 116, 118, 119, 120, 124, 125, 127, 128, 130, 131, 132, 133, 134, 135, 136, 149, 150, 152, 153, 156, 159, 160, 161

Offset: 1

Jud McCranie, Dec 12 2021

nonn

For index k, p/q = A002260(k)/A003057(k).

			10 is a term because A349082(10)=0, indicating that 4/5 = 1/x + 1/y has no solution.

Jud McCranie, Table of n, a(n) for n = 1..691

Cf. A349082, A002260, A003057, A349091, A349092.

A350715 2-tone chromatic number of a wheel graph with n vertices.

Original entry on oeis.org

8, 8, 7, 7, 8, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15

Offset: 4

Allan Bickle, Feb 02 2022

nonn

The 2-tone chromatic number of a graph G is the smallest number of colors for which G has a coloring where every vertex has two distinct colors, no adjacent vertices have a common color, and no pair of vertices at distance 2 have two common colors.

			The central vertex always requires two distinct colors.  All vertices on the cycle require distinct pairs.
The colorings for small (broken) cycles are shown below.
  -12-34-56-
  -12-34-15-36-
  -12-34-51-23-45-
  -12-34-15-32-14-35-
  -12-34-56-13-24-35-46-
  -12-34-15-23-14-25-13-45-
  -12-34-15-32-14-25-13-24-35-

Paolo Xausa, Table of n, a(n) for n = 4..10000
Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer. 217 (2013), 171-190.
Allan Bickle, 2-Tone Coloring of Planar Graphs, Bull. Inst. Combin. Appl. 103 (2025) 114-129.
Allan Bickle and B. Phillips, t-Tone Colorings of Graphs, Utilitas Math, 106 (2018) 85-102.
N. Fonger, J. Goss, B. Phillips, and C. Segroves, Math 6450: Final Report (2009).

Cf. A003057, A351120 (pair coloring).

Cf. A350361 (trees), A350362 (cycles), A350715 (wheels), A366727 (outerplanar), A366728 (square of cycles), A381562 (maximal planar), A381563 (double wheels).

A350715[n_]:=If[n<12,{8,8,7,7,8,7,7,8}[[n-3]],Ceiling[(5+Sqrt[8n-7])/2]];Array[A350715,100,4] (* Paolo Xausa, Nov 30 2023 *)

a(n) = A351120(n-1) + 2

a(n) = ceiling((5 + sqrt(8*n - 7))/2) for n > 11.

A385882 Values of v in the (1,3)-quartals (m,u,v,w) having m=1; i.e., values of v for solutions to m^1 + u^3 = v^1 + w^3, in positive integers, with m

Original entry on oeis.org

8, 20, 27, 38, 57, 64, 62, 99, 118, 125, 92, 153, 190, 209, 216, 128, 219, 280, 317, 336, 343, 170, 297, 388, 449, 486, 505, 512, 218, 387, 514, 605, 666, 703, 722, 729, 272, 489, 658, 785, 876, 937, 974, 993, 1000, 332, 603, 820, 989, 1116, 1207, 1268, 1305

Offset: 1

Clark Kimberling, Jul 21 2025

nonn

A 4-tuple (m,u,v,w) is a (p,q)-quartal if m,u,v,w are positive integers such that m

			First thirty (1,3)-quartals (1,u,v,w):
  m   u    v   w
  1   2    8   1
  1   3   20   2
  1   3   27   1
  1   4   38   3
  1   4   57   2
  1   4   64   1
  1   5   62   4
  1   5   99   3
  1   5  118   2
  1   5  125   1
  1   6   92   5
  1   6  153   4
  1   6  190   3
  1   6  209   2
  1   6  216   1
  1   7  128   6
  1   7  219   5
  1   7  280   4
  1   7  317   3
  1   7  336   2
  1   7  343   1
  1   8  170   7
  1   8  297   6
  1   8  388   5
  1   8  449   4
  1   8  486   3
  1   8  505   2
  1   8  512   1
  1   9  218   8
  1   9  387   7
1^1 + 4^3 = 57^1 + 2^3, so (1,4,57,2) is in the list.

Guide to related sequences:
   m |    u    |    v    |    w
   --+---------+---------+--------
   1 | A003057 | A385882 | A004736
   2 | A003057 | A386215 | A004736
   3 | A003057 | A386217 | A004736
   4 | A003057 | A386219 | A004736
  --+---------+---------+---------
Cf. A003057, A004736.

quartals[m_, p_, q_, max_] := Module[{ans = {}, lhsD = <||>, lhs, v, u, w, rhs},
   For[u = 1, u <= max, u++, lhs = m^p + u^q;
    AssociateTo[lhsD, lhs -> Append[Lookup[lhsD, lhs, {}], u]];];
   For[v = m + 1, v <= max, v++,
    For[w = 1, w <= max, w++, rhs = v^p + w^q; If[KeyExistsQ[lhsD, rhs],
       Do[AppendTo[ans, {m, u, v, w}], {u, lhsD[rhs]}];];];];
   ans = SortBy[ans, #[[2]] &];
   Do[Print["Solution ", i, ": ", ans[[i]], " (", m, "^", p, " + ",
     ans[[i, 2]], "^", q, " = ", ans[[i, 3]], "^", p, " + ",
     ans[[i, 4]], "^", q, " = ", m^p + ans[[i, 2]]^q, ")"], {i,
     Length[ans]}]; ans];
solns = quartals[1, 1, 3, 2000] (* Solutions restricted to v<2000 *)
Grid[solns]
u1 = Map[#[[2]] &, solns]   (*u, A003057 *)
v1 = Map[#[[3]] &, solns]   (*v, A385882 *)
w1 = Map[#[[4]] &, solns]   (*w, A004736 *)
(* Peter J. C. Moses, Jun 20 2025 *)

A386215 Values of v in the (1,3)-quartals (m,u,v,w) having m=2; i.e., values of v for solutions to m + u^3 = v + w^3, in positive integers, with m

Original entry on oeis.org

9, 21, 28, 39, 58, 65, 63, 100, 119, 126, 93, 154, 191, 210, 217, 129, 220, 281, 318, 337, 344, 171, 298, 389, 450, 487, 506, 513, 219, 388, 515, 606, 667, 704, 723, 730, 273, 490, 659, 786, 877, 938, 975, 994, 1001, 333, 604, 821, 990, 1117, 1208, 1269

Offset: 1

Clark Kimberling, Jul 22 2025

nonn

A 4-tuple (m,u,v,w) is a (p,q)-quartal if m,u,v,w are positive integers such that m

			First thirty (1,3)-quartals (2,u,v,w):
  m     u      v     w
  2     2      9     1
  2     3     21     2
  2     3     28     1
  2     4     39     3
  2     4     58     2
  2     4     65     1
  2     5     63     4
  2     5    100     3
  2     5    119     2
  2     5    126     1
  2     6     93     5
  2     6    154     4
  2     6    191     3
  2     6    210     2
  2     6    217     1
  2     7    129     6
  2     7    220     5
  2     7    281     4
  2     7    318     3
  2     7    337     2
  2     7    344     1
  2     8    171     7
  2     8    298     6
  2     8    389     5
  2     8    450     4
  2     8    487     3
  2     8    506     2
  2     8    513     1
  2     9    219     8
  2     9    388     7
2^1 + 3^3 = 21^1 + 2^3, so (2,3,21,2) is in the list.

Cf. A003057, A004736, A385882.

seq(seq(2 + u^3 - w^3, w = u-1 .. 1,-1),u=2..20); # Robert Israel, Jul 27 2025

quartals[m_, p_, q_, max_] :=
  Module[{ans = {}, lhsD = <||>, lhs, v, u, w, rhs},
   For[u = 1, u <= max, u++, lhs = m^p + u^q;
    AssociateTo[lhsD, lhs -> Append[Lookup[lhsD, lhs, {}], u]];];
   For[v = m + 1, v <= max, v++,
    For[w = 1, w <= max, w++, rhs = v^p + w^q;
      If[KeyExistsQ[lhsD, rhs],
       Do[AppendTo[ans, {m, u, v, w}], {u, lhsD[rhs]}];];];];
   ans = SortBy[ans, #[[2]] &];
   Do[Print["Solution ", i, ": ", ans[[i]], " (", m, "^", p, " + ",
     ans[[i, 2]], "^", q, " = ", ans[[i, 3]], "^", p, " + ",
     ans[[i, 4]], "^", q, " = ", m^p + ans[[i, 2]]^q, ")"], {i,
     Length[ans]}]; ans];
solns = quartals[2, 1, 3, 2000]  (* solutions restricted to v<2000 *)
Grid[solns]
u1 = Map[#[[2]] &, solns]  (*u, A003057 *)
v1 = Map[#[[3]] &, solns]  (*v, A386215 *)
w1 = Map[#[[4]] &, solns]  (*w, A004736 *)
(* Peter J. C. Moses, Jun 20 2025 *)

a(n) = 2 + u^3 - (u*(u-1)/2 + 1 - n)^3 where u = floor((3+sqrt(8*n-7))/2). - Robert Israel, Jul 27 2025

A063929 Radius of A-excircle of Pythagorean triangle with a = (n+1)^2 - m^2, b = 2(n+1)m and c = (n+1)^2 + m^2.

Original entry on oeis.org

2, 6, 3, 12, 8, 4, 20, 15, 10, 5, 30, 24, 18, 12, 6, 42, 35, 28, 21, 14, 7, 56, 48, 40, 32, 24, 16, 8, 72, 63, 54, 45, 36, 27, 18, 9, 90, 80, 70, 60, 50, 40, 30, 20, 10, 110, 99, 88, 77, 66, 55, 44, 33, 22, 11, 132, 120, 108, 96, 84, 72, 60, 48, 36, 24, 12, 156, 143, 130, 117

Offset: 1

Floor van Lamoen, Aug 21 2001

From Wolfdieter Lang, Dec 03 2014: (Start)
For excircles and their radii see the Eric W. Weisstein links. Here the circle radius with center J_A is considered.
Note that not all Pythagorean triangles are covered, e.g., the nonprimitive one (9, 12, 15) does not appear. However, the nonprimitive one (8, 6, 10) does appear as (n, m) = (2, 1). (End)
This triangle T appears also in the problem of finding all positive integer solutions for a and b of the general Fibonacci sequence F(a,b;k+1) = a*F(a,b;k) + b*F(a,b;k-1) (with some inputs F(a,b;0) and F(a,b;1)) such that the limit r = r(a,b) = F(a,b;k+1)/F(a,b;k) for k -> infinity becomes a positive integer r = (a + sqrt(a^2 + 4*b))/2. Namely, for any a = m >= 1 there are infinitely many b solutions b = T(n,m) = (n+1)*(n+1-m) for n >= m. The limit is r(a,b) = n+1 for a = m = 1..n, which is A003057 read as a triangle with offset 1. This entry was motivated by A249973 and A249974 by Kerry Mitchell concerned with real values of r. - Wolfdieter Lang, Jan 11 2015

			The triangle T(n, m) begins:
n\m   1   2   3   4   5   6   7   8   9 10 11 12 13 14 15 ...
1:    2
2:    6   3
3:   12   8   4
4:   20  15  10  5
5:   30  24  18  12   6
6:   42  35  28  21  14   7
7:   56  48  40  32  24  16   8
8:   72  63  54  45  36  27  18   9
9:   90  80  70  60  50  40  30  20  10
10: 110  99  88  77  66  55  44  33  22 11
11: 132 120 108  96  84  72  60  48  36 24 12
12: 156 143 130 117 104  91  78  65  52 39 26 13
13: 182 168 154 140 126 112  98  84  70 56 42 28 14
14: 210 195 180 165 150 135 120 105  90 75 60 45 30 15
15: 240 224 208 192 176 160 144 128 112 96 80 64 48 32  1
... Formatted and extended by _Wolfdieter Lang_, Dec 02 2014
--------------------------------------------------------------
Example of general (a,b)-Fibonacci sequence positive integer limits r(a,b) (see the Jan 11 2015 comment above):
T(3, 2) = 8, that is a = m = 2 has a solution b = T(3, 2) = 8 with r = r(2,8) = n+1 = 4 = (2 + sqrt(4 + 4*8))/2. The other two solutions with r = 4 appear for b = T(3, m) with m = a = 1 and 3. In general, row n has n times the value n+1 for r, namely r(a=m,b=T(n,m)) = n+1, for m = 1..n. - _Wolfdieter Lang_, Jan 11 2015

Eric Weisstein's World of Mathematics, Excircles
Eric Weisstein's World of Mathematics, Exradius

Cf. A003991 (incircle radius), A063930 (B-excircle radius), A001283 (C-excircle radius), A055096 (circumcircle diameter).

T(n, m) = (n+1)(n-m+1), n >= m >= 1.

T(n, m) = rho_A = sqrt(s*(s-b)*(s-c)/(s-a)) with the semiperimeter s = (a + b + c)/2 and the substituted a, b, c values as given in the name. - Wolfdieter Lang, Dec 02 2014

Edited: Crossreferences commented and A055096 added by Wolfdieter Lang, Dec 02 2014

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cp's OEIS Frontend

A014410 Elements in Pascal's triangle (by row) that are not 1.

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A119741 A008279, with the first and last of each row removed.

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A349082 The number of two-term Egyptian fractions of rational numbers, x/y, 0 < x/y < 1, ordered as below. The sequence is the number of (p,q) pairs such that x/y = 1/p + 1/q where p and q are integers with p < q.

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A350361 2-tone chromatic number of a tree with maximum degree n.

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A351120 Pair chromatic number of a cycle graph with n vertices.

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A349090 Where zeros occur in A349082. These correspond to rationals, 0 < p/q < 1, that have no solution p/q = 1/x + 1/y, 0 < x < y.

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A350715 2-tone chromatic number of a wheel graph with n vertices.

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A385882 Values of v in the (1,3)-quartals (m,u,v,w) having m=1; i.e., values of v for solutions to m^1 + u^3 = v^1 + w^3, in positive integers, with m

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A063929 Radius of A-excircle of Pythagorean triangle with a = (n+1)^2 - m^2, b = 2(n+1)m and c = (n+1)^2 + m^2.