A003991 -id:A003991 - cp's OEIS frontend

A001283 Triangle read by rows, in which row n consists of n(n+m) for m = 1 .. n-1.

6, 12, 15, 20, 24, 28, 30, 35, 40, 45, 42, 48, 54, 60, 66, 56, 63, 70, 77, 84, 91, 72, 80, 88, 96, 104, 112, 120, 90, 99, 108, 117, 126, 135, 144, 153, 110, 120, 130, 140, 150, 160, 170, 180, 190, 132, 143, 154, 165, 176, 187, 198, 209, 220, 231, 156, 168, 180

Offset: 2

N. J. A. Sloane

With a different offset: triangle read by rows: t(n, m) = T(n+1, m) = (n+1)(n+m+1) = radius of C-excircle of Pythagorean triangle with sides a=(n+1)^2-m^2, b=2*(n+1)*m and c=(n+1)^2+m^2. - Floor van Lamoen, Aug 21 2001

			The triangle T(n, m) begins:
n\m   1   2   3   4   5   6   7   8   9  10  11  12  13  14 ...
2:    6
3:   12  15
4:   20  24  28
5:   30  35  40  45
6:   42  48  54  60  66
7:   56  63  70  77  84  91
8:   72  80  88  96 104 112 120
9:   90  99 108 117 126 135 144 153
10: 110 120 130 140 150 160 170 180 190
11: 132 143 154 165 176 187 198 209 220 231
12: 156 168 180 192 204 216 228 240 252 264 276
13: 182 195 208 221 234 247 260 273 286 299 312 325
14: 210 224 238 252 266 280 294 308 322 336 350 364 378
15: 240 255 270 285 300 315 330 345 360 375 390 405 420 435
...
[Reformatted and extended by _Wolfdieter Lang_, Dec 02 2014]
----------------------------------------------------------------

T. D. Noe, Rows n = 2..100, flattened

Cf. A003991, A063929, A063930.

Row sums are in A085788. Central column is A033581.

Flatten[Table[n*(n+m), {n, 2, 10}, {m, n-1}]] (* T. D. Noe, Jun 27 2012 *)

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

Edited comment by Wolfdieter Lang, Dec 02 2014

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

A102310 Square array read by antidiagonals: Fibonacci(k*n).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 3, 8, 8, 3, 5, 21, 34, 21, 5, 8, 55, 144, 144, 55, 8, 13, 144, 610, 987, 610, 144, 13, 21, 377, 2584, 6765, 6765, 2584, 377, 21, 34, 987, 10946, 46368, 75025, 46368, 10946, 987, 34, 55, 2584, 46368, 317811, 832040, 832040, 317811, 46368, 2584, 55

Offset: 1

Ralf Stephan, Jan 06 2005

			1,  1,   2,    3,     5, ...
1,  3,   8,   21,    55, ...
2,  8,  34,  144,   610, ...
3, 21, 144,  987,  6765, ...
5, 55, 610, 6765, 75025, ...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. 2nd Edition. Addison-Wesley, Reading, MA, 1994, p. 294.

Freddy Barrera, Antidiagonals n = 1..50, flattened

Equals A000045(A003991(k, n)).
Columns include A000045, A001906, A014445, A033888, A102312.
Main diagonal is in A054783. Antidiagonal sums are in A102311.

/* As triangle */ [[Fibonacci(k*(n-k+1)): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jul 04 2019

Table[Fibonacci[k*(n-k+1)], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 10 2017 *)

F = fibonacci # A000045
def A(n, k):
    return F((n-1)*k)*F(k+1) + F((n-1)*k - 1)*F(k)
[A(n, k) for d in (1..10) for n, k in zip((d..1, step=-1), (1..d))] # Freddy Barrera, Jun 24 2019

For prime p, the formula holds: Fibonacci(k*p) = Fibonacci(p) * Sum_{i=0..floor((k-1)/2)} C(k-i-1, i)*(-1)^(i*p+i)*Lucas(p)^(k-2i-1).

A(n, k) = F((n-1)*k)*F(k+1) + F((n-1)*k-1)*F(k), where F(n) = A000045(n). - Freddy Barrera, Jun 24 2019

A286102 Square array A(n,k) read by antidiagonals: A(n,k) = T(lcm(n,k), gcd(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.

Original entry on oeis.org

1, 3, 3, 6, 5, 6, 10, 21, 21, 10, 15, 14, 13, 14, 15, 21, 55, 78, 78, 55, 21, 28, 27, 120, 25, 120, 27, 28, 36, 105, 34, 210, 210, 34, 105, 36, 45, 44, 231, 90, 41, 90, 231, 44, 45, 55, 171, 300, 406, 465, 465, 406, 300, 171, 55, 66, 65, 64, 63, 630, 61, 630, 63, 64, 65, 66, 78, 253, 465, 666, 820, 903, 903, 820, 666, 465, 253, 78, 91, 90, 561, 230, 1035, 324, 85, 324, 1035, 230, 561, 90, 91

Offset: 1

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

			The top left 12 X 12 corner of the array:
   1,   3,   6,  10,   15,   21,   28,   36,   45,   55,   66,   78
   3,   5,  21,  14,   55,   27,  105,   44,  171,   65,  253,   90
   6,  21,  13,  78,  120,   34,  231,  300,   64,  465,  561,  103
  10,  14,  78,  25,  210,   90,  406,   63,  666,  230,  990,  117
  15,  55, 120, 210,   41,  465,  630,  820, 1035,  101, 1540, 1830
  21,  27,  34,  90,  465,   61,  903,  324,  208,  495, 2211,  148
  28, 105, 231, 406,  630,  903,   85, 1596, 2016, 2485, 3003, 3570
  36,  44, 300,  63,  820,  324, 1596,  113, 2628,  860, 3916,  375
  45, 171,  64, 666, 1035,  208, 2016, 2628,  145, 4095, 4950,  739
  55,  65, 465, 230,  101,  495, 2485,  860, 4095,  181, 6105, 1890
  66, 253, 561, 990, 1540, 2211, 3003, 3916, 4950, 6105,  221, 8778
  78,  90, 103, 117, 1830,  148, 3570,  375,  739, 1890, 8778,  265

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array
MathWorld, Pairing Function

Cf. A000027, A003989, A003990, A003991, A038722, A285722, A285732, A286098, A286099, A286101, A285724.

Cf. A000217 (row 1 and column 1), A001844 (main diagonal).

(define (A286102 n) (A286102bi (A002260 n) (A004736 n)))
(define (A286102bi row col) (A000027bi (lcm row col) (gcd row col)))
(define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))

A(n,k) = T(lcm(n,k), gcd(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table, that is, as a pairing function from N x N to N.

A(n,k) = A(k,n), or equivalently, a(A038722(n)) = a(n). [Array is symmetric.]

A353109 Array read by antidiagonals: A(n, k) is the digital root of n*k with n >= 0 and k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 1, 3, 3, 1, 6, 0, 0, 7, 3, 6, 7, 6, 3, 7, 0, 0, 8, 5, 9, 2, 2, 9, 5, 8, 0, 0, 9, 7, 3, 6, 7, 6, 3, 7, 9, 0, 0, 1, 9, 6, 1, 3, 3, 1, 6, 9, 1, 0, 0, 2, 2, 9, 5, 8, 9, 8, 5, 9, 2, 2, 0

Offset: 0

Stefano Spezia, Apr 24 2022

			The array begins:
    0, 0, 0, 0, 0, 0, 0, 0, ...
    0, 1, 2, 3, 4, 5, 6, 7, ...
    0, 2, 4, 6, 8, 1, 3, 5, ...
    0, 3, 6, 9, 3, 6, 9, 3, ...
    0, 4, 8, 3, 7, 2, 6, 1, ...
    0, 5, 1, 6, 2, 7, 3, 8, ...
    0, 6, 3, 9, 6, 3, 9, 6, ...
    0, 7, 5, 3, 1, 8, 6, 4, ...
    ...

Cf. A003991, A004247, A010888, A056992 (diagonal), A073636, A139413, A180592, A180593, A180594, A180595, A180596, A180597, A180598, A180599, A303296, A336225, A353128 (antidiagonal sums), A353933, A353974 (partial sum of the main diagonal).

A[i_,j_]:=If[i*j==0,0,1+Mod[i*j-1,9]];Flatten[Table[A[n-k,k],{n,0,12},{k,0,n}]]

T(n,k) = if (n && k, (n*k-1)%9+1, 0); \\ Michel Marcus, May 12 2022

A(n, k) = A010888(A004247(n, k)).

A(n, k) = A010888(A003991(n, k)) for n*k > 0.

A353933 a(n) is the permanent of the n X n symmetric matrix M(n) whose generic element M[i,j] is equal to the digital root of i*j.

Original entry on oeis.org

1, 1, 8, 216, 7344, 168183, 7226091, 295259094, 11801772252, 1673511251940, 65568867621336, 2710049208604776, 202103867012027328, 12881755844526953376, 736186737257150962752, 70484099228399057425344, 5507570249593121504026368, 434305172863416192470350848, 122043063804581668929348667392

Offset: 0

Stefano Spezia, May 11 2022

The matrix M(n) is nonsingular only for n = 1, 5 and 6 with determinant equal respectively to 1, 6561 and 59049.

The rank of M(n) is 1 for 1 <= n <= 3, 3 for n = 4, 5 for n = 5, 6 for 6 <= n <= 8, and 7 for n >= 9. - Jianing Song, Sep 28 2022

			a(7) = 7226091:
     1, 2, 3, 4, 5, 6, 7
     2, 4, 6, 8, 1, 3, 5
     3, 6, 9, 3, 6, 9, 3
     4, 8, 3, 7, 2, 6, 1
     5, 1, 6, 2, 7, 3, 8
     6, 3, 9, 6, 3, 9, 6
     7, 5, 3, 1, 8, 6, 4

Cf. A003991, A010888, A353109, A353128, A353974 (trace of the matrix M(n)).

M[i_, j_]:=If[i*j==0, 0, 1+Mod[i*j-1, 9]]; Join[{1},Table[Permanent[Table[M[i, j], {i,  n}, {j, n}]],{n,18}]]

a(n) = matpermanent(matrix(n, n, i, j, (i*j-1)%9+1)); \\ Michel Marcus, May 12 2022

Sum_{i=1..n} M[n-i+1,i] = A353128(n+1).

A360851 Array read by antidiagonals: T(m,n) is the number of induced paths in the rook graph K_m X K_n.

Original entry on oeis.org

0, 1, 1, 3, 8, 3, 6, 27, 27, 6, 10, 64, 126, 64, 10, 15, 125, 426, 426, 125, 15, 21, 216, 1125, 2208, 1125, 216, 21, 28, 343, 2493, 8830, 8830, 2493, 343, 28, 36, 512, 4872, 27456, 55700, 27456, 4872, 512, 36, 45, 729, 8676, 70434, 265635, 265635, 70434, 8676, 729, 45

Offset: 1

Andrew Howroyd, Feb 24 2023

Paths of length zero are not counted here.

			Array begins:
===================================================
m\n|  1   2    3     4      5        6        7 ...
---+-----------------------------------------------
1  |  0   1    3     6     10       15       21 ...
2  |  1   8   27    64    125      216      343 ...
3  |  3  27  126   426   1125     2493     4872 ...
4  |  6  64  426  2208   8830    27456    70434 ...
5  | 10 125 1125  8830  55700   265635   961975 ...
6  | 15 216 2493 27456 265635  2006280 11158161 ...
7  | 21 343 4872 70434 961975 11158161 98309778 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Eric Weisstein's World of Mathematics, Rook Graph.
Wikipedia, Induced path.

Main diagonal is A360852.
Rows 1..2 are A000217(n-1), A000578.
Cf. A003991, A360199, A360850.

T(m,n) = sum(j=1, min(m,n), j!^2*binomial(m,j)*binomial(n,j)*(1 + (m+n)/2 - j)) - m*n

T(m,n) = A360850(m,n) - A003991(m,n).
T(m,n) = -m*n + Sum_{j=1..min(m,n)} j!^2*binomial(m,j)*binomial(n,j)*(1 + (m+n)/2 - j).
T(m,n) = T(n,m).

A059036 In a triangle of numbers (such as that in A059032, A059033, A059034) how many entries lie above position (n,k)? Answer: T(n,k) = (n+1)*(k+1)-1 (n >= 0, k >= 0).

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 5, 5, 3, 4, 7, 8, 7, 4, 5, 9, 11, 11, 9, 5, 6, 11, 14, 15, 14, 11, 6, 7, 13, 17, 19, 19, 17, 13, 7, 8, 15, 20, 23, 24, 23, 20, 15, 8, 9, 17, 23, 27, 29, 29, 27, 23, 17, 9, 10, 19, 26, 31, 34, 35, 34, 31, 26, 19, 10, 11, 21, 29, 35, 39, 41

Offset: 0

N. J. A. Sloane, Feb 13 2001

			As an infinite triangular array:
  0
  1   1
  2   3   2
  3   5   5   3
  4   7   8   7   4
  5   9  11  11   9   5
As an infinite square array (matrix):
  0   1   2   3   4   5
  1   3   5   7   9  11
  2   5   8  11  14  17
  3   7  11  15  19  23
  4   9  14  19  24  29
  5  11  17  23  29  35

T(n, k) = A003991(n, k) - 1.

Cf. A000005, A005581.

{T(n, k) = n + k + n*k}; /* Michael Somos, Jul 28 2015 */

T(n, k) = max(T(n-1, k-1), T(n-1, k)) + min(k, n-k+1). - Jon Perry, Aug 05 2004
E.g.f.: exp(x+y)(x+y+xy) (as a square array read by antidiagonals). - Paul Barry, Sep 24 2004
From Michael Somos, Jul 28 2015: (Start)
Row sums = Sum_{k=0..n} T(n-k, k) = A005581(n+1).
T(n, k) = T(k, n) = T(-2-n, -2-k) for all n, k in Z.
Sum_{n, k >= 0} x^T(n, k) = f(x) / x where f() is the g.f. for A000005. (End)

A063930 Radius of B-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

3, 4, 10, 5, 12, 21, 6, 14, 24, 36, 7, 16, 27, 40, 55, 8, 18, 30, 44, 60, 78, 9, 20, 33, 48, 65, 84, 105, 10, 22, 36, 52, 70, 90, 112, 136, 11, 24, 39, 56, 75, 96, 119, 144, 171, 12, 26, 42, 60, 80, 102, 126, 152, 180, 210, 13, 28, 45, 64, 85, 108, 133, 160, 189, 220

Offset: 1

Floor van Lamoen, Aug 21 2001

See a comment for excircle and exradius on A063929, also for links.

			The triangle T(n, m) begins:
n\m  1  2  3  4   5   6   7   8   9  10  11  12  13  14  15 ...
1:   3
2:   4 10
3:   5 12 21
4:   6 14 24 36
5:   7 16 27 40  55
6:   8 18 30 44  60  78
7:   9 20 33 48  65  84 105
8:  10 22 36 52  70  90 112 136
9:  11 24 39 56  75  96 119 144 171
10: 12 26 42 60  80 102 126 152 180 210
11: 13 28 45 64  85 108 133 160 189 220 253
12: 14 30 48 68  90 114 140 168 198 230 264 300
13: 15 32 51 72  95 120 147 176 207 240 275 312 351
14: 16 34 54 76 100 126 154 184 216 250 286 324 364 406
15: 17 36 57 80 105 132 161 192 225 260 297 336 377 420 465
...
[Formatted and extended by _Wolfdieter Lang_, Dec 02 2014]

Cf. A003991 (inradius), A063929 (A-exradius), A001283 (C-exradius), A055096 (circumradius diameter).

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

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

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

A075374 a(n+2) = n*a(n+1) - a(n), with a(1)=1, a(2)=2.

Original entry on oeis.org

1, 2, 1, 0, -1, -4, -19, -110, -751, -5898, -52331, -517412, -5639201, -67153000, -867349799, -12075744186, -180268812991, -2872225263670, -48647560669399, -872783866785512, -16534245908255329, -329812134298321068, -6909520574356487099, -151679640501544395110

Offset: 1

Amarnath Murthy, Sep 20 2002

sign

Starting with offset 5 unsigned: (1, 4, 19, 110, 751, ...) = eigensequence of triangle A003991. - Gary W. Adamson, May 17 2010

Seiichi Manyama, Table of n, a(n) for n = 1..451

Cf. A003991. - Gary W. Adamson, May 17 2010

[n le 2 select n else (n-2)*Self(n-1) - Self(n-2): n in [1..50]]; // G. C. Greubel, Mar 04 2022

a[1] := 1:a[2] := 2:for n from 1 to 45 do a[n+2] := n*a[n+1]-a[n]:od:seq(a[i],i=1..45);

a[n_]:= a[n]= If[n<3, n, (n-2)*a[n-1] -a[n-2]];
Table[a[n], {n,50}] (* G. C. Greubel, Mar 04 2022 *)

@CachedFunction
def a(n): return n if (n<3) else (n-2)*a(n-1) - a(n-2) # A075374
[a(n) for n in (1..50)] # G. C. Greubel, Mar 04 2022

a(n+1) = (a(n) + a(n+2))/n with a(1) = 1, a(2) = 2.

More terms from Sascha Kurz, Jan 30 2003

cp's OEIS Frontend

A001283 Triangle read by rows, in which row n consists of n(n+m) for m = 1 .. n-1.

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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.

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A102310 Square array read by antidiagonals: Fibonacci(k*n).

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A286102 Square array A(n,k) read by antidiagonals: A(n,k) = T(lcm(n,k), gcd(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.

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A353109 Array read by antidiagonals: A(n, k) is the digital root of n*k with n >= 0 and k >= 0.

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A353933 a(n) is the permanent of the n X n symmetric matrix M(n) whose generic element M[i,j] is equal to the digital root of i*j.

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A360851 Array read by antidiagonals: T(m,n) is the number of induced paths in the rook graph K_m X K_n.

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Formula

A059036 In a triangle of numbers (such as that in A059032, A059033, A059034) how many entries lie above position (n,k)? Answer: T(n,k) = (n+1)*(k+1)-1 (n >= 0, k >= 0).

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.

A063930 Radius of B-excircle of Pythagorean triangle with a=(n+1)^2-m^2, b=2(n+1)m and c=(n+1)^2+m^2.