A055096 -id:A055096 - cp's OEIS frontend

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.

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

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

A140978 Repeat (n+1)^2 n times.

Original entry on oeis.org

4, 9, 9, 16, 16, 16, 25, 25, 25, 25, 36, 36, 36, 36, 36, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 81, 81, 81, 81, 81, 81, 81, 81, 100, 100, 100, 100, 100, 100, 100, 100, 100, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121

Offset: 1

Paul Curtz, Aug 17 2008

See A093995.
Frenicle writes the entries in the form a(n) = A055096(n)-A133819(n), with the flattened index view of A133819: 4=5-1, 9=10-1, 9=13-4, 16=17-1, 16=20-4, 16=25-9 etc.
Also triangle T(n, k) = (n+1)^2, 1<=k<=n. - Michel Marcus, Feb 03 2013

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
M. de Frenicle, Methode pour trouver la solutions des problemes par les exclusions, in: Divers ouvrages des mathematiques et de physique par messieurs de l'academie royale des sciences, (1693) pp 1-44, table page 11.

Cf. A000290.

a140978 n k = a140978_tabl !! (n-1) !! (k-1)
a140978_row n = a140978_tabl !! (n-1)
a140978_tabl = map snd $ iterate
               (\(i, xs@(x:_)) -> (i + 2, map (+ i) (x:xs))) (5, [4])
-- Reinhard Zumkeller, Mar 23 2013

Table[PadRight[{},n,(n+1)^2],{n,10}]//Flatten (* Harvey P. Dale, Oct 10 2019 *)

from math import isqrt
def A140978(n): return ((m:=isqrt(k:=n<<1))+(k>m*(m+1))+1)**2 # Chai Wah Wu, Nov 07 2024

a(n)=(A003057(n+1))^2. - R. J. Mathar, Aug 25 2008

A056203 Triangle of numbers related to congruum problem: T(n,k)=n^2+2nk-k^2 with n>k>0, starting at T(2,1)=7.

Original entry on oeis.org

7, 14, 17, 23, 28, 31, 34, 41, 46, 49, 47, 56, 63, 68, 71, 62, 73, 82, 89, 94, 97, 79, 92, 103, 112, 119, 124, 127, 98, 113, 126, 137, 146, 153, 158, 161, 119, 136, 151, 164, 175, 184, 191, 196, 199, 142, 161, 178, 193, 206, 217, 226, 233, 238, 241, 167, 188

Offset: 1

Henry Bottomley, Aug 02 2000

The congruum problem is to find h (A057103) such that there are integers x (A055096), y (A057105) and z (A056203) with h=x^2-y^2=z^2-x^2.

Refers to A057102, which had an incorrect description and has been replaced by A256418. As a result the present sequence should be re-checked. - N. J. A. Sloane, Apr 06 2015

			a(1) = T(2,1) = 2^2+2*2*1-1 = 7.

Eric Weisstein's World of Mathematics, Congruum Problem.

Cf. A057102.

a(n) = sqrt(A057103(n)+A055096(n)^2) = sqrt(2*A057103(n)+A057105(n)^2).

A057105 Triangle of numbers (when unsigned) related to congruum problem: T(n,k)=k^2+2nk-n^2 with n>k>0 and starting at T(2,1)=1.

Original entry on oeis.org

1, -2, 7, -7, 4, 17, -14, -1, 14, 31, -23, -8, 9, 28, 49, -34, -17, 2, 23, 46, 71, -47, -28, -7, 16, 41, 68, 97, -62, -41, -18, 7, 34, 63, 94, 127, -79, -56, -31, -4, 25, 56, 89, 124, 161, -98, -73, -46, -17, 14, 47, 82, 119, 158, 199, -119, -92, -63, -32, 1, 36, 73, 112, 153, 196, 241, -142, -113, -82, -49, -14, 23, 62, 103

Offset: 1

Henry Bottomley, Aug 02 2000

Signed values are only relevant for the explicit formula.

Refers to A057102, which had an incorrect description and has been replaced by A256418. As a result the present sequence should be re-checked. - N. J. A. Sloane, Apr 06 2015

			a(1)=T(2,1)=1^2+2*2*1-2^2=1

Eric Weisstein's World of Mathematics, Congruum Problem.

Cf. A057102. The congruum problem is about finding solutions for h (A057103) where there are integers x (A055096), y (A057105 unsigned) and z (A056203) such that h=x^2-y^2=z^2-x^2.

Unsigned: a(n) =sqrt(A055096(n)^2-A057103(n)) =sqrt(A056203(n)^2-2*A057103(n)).

A294774 a(n) = 2n^2 + 2n + 5.

Original entry on oeis.org

5, 9, 17, 29, 45, 65, 89, 117, 149, 185, 225, 269, 317, 369, 425, 485, 549, 617, 689, 765, 845, 929, 1017, 1109, 1205, 1305, 1409, 1517, 1629, 1745, 1865, 1989, 2117, 2249, 2385, 2525, 2669, 2817, 2969, 3125, 3285, 3449, 3617, 3789, 3965, 4145, 4329, 4517, 4709, 4905

Offset: 0

Bruno Berselli, Nov 08 2017

This is the case k = 9 of 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 (similar sequences are listed in Crossrefs section). Note that:

2*( 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 ) - k = ( 2*n + (1-(-1)^k)/2 )^2. From this follows an alternative definition for the sequence: Numbers h such that 2*h - 9 is a square. Therefore, if a(n) is a square then its base is a term of A075841.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

1st diagonal of A154631, 3rd diagonal of A055096, 4th diagonal of A070216.
Second column of Mathar's array in A016813 (Comments section).
Subsequence of A001481, A001983, A004766, A020668, A046711 and A057653 (because a(n) = (n+2)^2 + (n-1)^2); A097268 (because it is also a(n) = (n^2+n+3)^2 - (n^2+n+2)^2); A047270; A243182 (for y=1).
Similar sequences (see the first comment): A161532 (k=-14), A181510 (k=-13), A152811 (k=-12), A222182 (k=-11), A271625 (k=-10), A139570 (k=-9), (-1)*A147973 (k=-8), A059993 (k=-7), A268581 (k=-6), A090288 (k=-5), A054000 (k=-4), A142463 or A132209 (k=-3), A056220 (k=-2), A046092 (k=-1), A001105 (k=0), A001844 (k=1), A058331 (k=2), A051890 (k=3), A271624 (k=4), A097080 (k=5), A093328 (k=6), A271649 (k=7), A255843 (k=8), this sequence (k=9).

seq(2*n^2 + 2*n + 5, n=0..100); # Robert Israel, Nov 10 2017

Table[2n^2+2n+5,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{5,9,17},50] (* Harvey P. Dale, Sep 18 2023 *)

Vec((5 - 6*x + 5*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Nov 13 2017

O.g.f.: (5 - 6*x + 5*x^2)/(1 - x)^3.
E.g.f.: (5 + 4*x + 2*x^2)*exp(x).
a(n) = a(-1-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 5*A000217(n+1) - 6*A000217(n) + 5*A000217(n-1).
n*a(n) - Sum_{j=0..n-1} a(j) = A002492(n) for n>0.
a(n) = Integral_{x=0..2n+4} |3-x| dx. - Pedro Caceres, Dec 29 2020

A008224 Coordination sequence T6 for Zeolite Code PAU.

Original entry on oeis.org

1, 4, 9, 17, 29, 45, 65, 89, 117, 149, 185, 225, 268, 314, 364, 418, 476, 538, 605, 676, 750, 827, 907, 991, 1078, 1169, 1264, 1362, 1464, 1570, 1680, 1794, 1913, 2036, 2161, 2289, 2421, 2557, 2697, 2842, 2992, 3145, 3301, 3460, 3622, 3788, 3958, 4132, 4309

Offset: 0

Ralf W. Grosse-Kunstleve

nonn

For n = 3 to n = 12, a(n) is the sum of two squares separated by 3 (3rd diagonal of A055096). - Avi Friedlich, Apr 01 2015

W. M. Meier, D. H. Olson and Ch. Baerlocher, Atlas of Zeolite Structure Types, 4th Ed., Elsevier, 1996

R. W. Grosse-Kunstleve, Table of n, a(n) for n = 0..1000
R. W. Grosse-Kunstleve, Coordination Sequences and Encyclopedia of Integer Sequences
R. W. Grosse-Kunstleve, G. O. Brunner and N. J. A. Sloane, Algebraic Description of Coordination Sequences and Exact Topological Densities for Zeolites, Acta Cryst., A52 (1996), pp. 879-889.
Sean A. Irvine, Generating Functions for Coordination Sequences of Zeolites, after Grosse-Kunstleve, Brunner, and Sloane
International Zeolite Association, Database of Zeolite Structures

A255848 a(n) = 2*n^2 + 18.

Original entry on oeis.org

18, 20, 26, 36, 50, 68, 90, 116, 146, 180, 218, 260, 306, 356, 410, 468, 530, 596, 666, 740, 818, 900, 986, 1076, 1170, 1268, 1370, 1476, 1586, 1700, 1818, 1940, 2066, 2196, 2330, 2468, 2610, 2756, 2906, 3060, 3218, 3380, 3546, 3716, 3890, 4068, 4250, 4436

Offset: 0

Avi Friedlich, Mar 08 2015

For n>3, the sequence gives the 6th diagonal of triangle in A055096.
Also, this is the case k=9 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. It is noted that a(n)*n = (n + sqrt(3))^3 + (n - sqrt(3))^3.
Equivalently, numbers m such that 2*m-36 is a square.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016825 (first differences), A055096, A189834.
Subsequence of A047463.
Cf. similar sequences listed in A255843.

[2*n^2+18: n in [0..50]]; // Vincenzo Librandi, Mar 08 2015

f[n_] := 2 n^2 + 18; Array[f, 50, 0] (* Robert G. Wilson v, Mar 08 2015 *)
CoefficientList[Series[(18 - 34 x + 20 x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 08 2015 *)
LinearRecurrence[{3,-3,1},{18,20,26},50] (* Harvey P. Dale, Aug 20 2021 *)

vector(50, n, 2*n^2+18) \\ Derek Orr, Mar 09 2015

[2*n^2+18 for n in (0..50)] # Bruno Berselli, Mar 11 2015

a(n) = 2*A189834(n).
From Vincenzo Librandi, Mar 08 2015: (Start)
G.f.: 2*(9 - 17*x + 10*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + 3*Pi*coth(3*Pi))/36.
Sum_{n>=0} (-1)^n/a(n) = (1 + 3*Pi*cosech(3*Pi))/36. (End)
E.g.f.: 2*exp(x)*(9 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Edited by Bruno Berselli, Mar 11 2015

A270835 a(n) = smallest k such that A004431(n) +/- k are both positive squares.

Original entry on oeis.org

4, 6, 12, 8, 16, 24, 10, 20, 30, 12, 24, 40, 36, 14, 48, 28, 42, 60, 16, 32, 48, 70, 64, 18, 36, 80, 54, 72, 96, 20, 40, 90, 60, 112, 80, 108, 22, 44, 66, 120, 88, 24, 110, 48, 140, 72, 132, 96, 160, 120, 26, 52, 78, 144, 180

Offset: 1

Bob Selcoe, Mar 23 2016

nonn

There can be more than one value k such that A004431(n) +/- k are both positive squares; i.e., when there are multiple ways to express A004431(n) as the sum of positive squares. These are the terms which appear more than once in A055096. For example A004431(19) = 65 = {(1^2 + 8^2), (4^2 + 7^2)}: 65 +/- 16 = {7^2, 9^2} and 65 +/- 56 = {3^2, 11^2}. So a(19) = 16 rather than 56.

Sequence contains every even number >=4 and no odd numbers.

			a(11)=24 because A004431(11) = 40; 40+24 = 8^2 and 40-24 = 4^2.

Cf. A001844, A004431, A055096.

nn = 80; s = Select[Range[4 nn], Length[PowersRepresentations[#, 2,
2] /. {{0, } -> Nothing, {a, b_} /; a == b -> Nothing}] > 0 &]; Table[SelectFirst[Range[10 nn], And[IntegerQ@ Sqrt[s[[n]] + #], IntegerQ@ Sqrt[s[[n]] - #]] &], {n, nn}] (* Michael De Vlieger, Mar 24 2016, Version 10 *)

issum(n)=if(n<5, return(0)); my(f=factor(n)%4); if(vecmin(f[, 1])>1, return(0)); for(i=1, #f[, 1], if(f[i, 1]==3 && f[i, 2]%2, return(0))); 1; \\ after A004431
findk(n) = {for (k=1, n, if (issquare(n+k) && issquare(n-k), return (k)););}
lista(nn) = {for (n=1, nn, if (issum(n), print1(findk(n), ", ");););} \\ Michel Marcus, Mar 31 2016

A354968 Triangle read by rows: T(n, k) = nk(n+k)*(n-k)/6.

Original entry on oeis.org

1, 4, 5, 10, 16, 14, 20, 35, 40, 30, 35, 64, 81, 80, 55, 56, 105, 140, 154, 140, 91, 84, 160, 220, 256, 260, 224, 140, 120, 231, 324, 390, 420, 405, 336, 204, 165, 320, 455, 560, 625, 640, 595, 480, 285, 220, 429, 616, 770, 880, 935, 924, 836, 660, 385, 286, 560, 810, 1024

Offset: 2

Ali Sada and Yifan Xie, Jun 14 2022

Given a Pythagorean triple (a,b,c), define S = c^4 - a^4 - b^4. Using Euclid's parameterization (a = 2*n*k, b = n^2 - k^2, c = n^2 + k^2), substituting to get S in terms of n and k gives S = 8*n^2*k^2*((n^2 - k^2))^2, which is a multiple of 288; T(n, k) = sqrt(S/288) = n*k*(n^2 - k^2)/6 = n*k*(n+k)*(n-k)/6.

			Triangle begins:
  n/k   1    2    3    4    5    6    7
  2     1;
  3     4,   5;
  4    10,  16,  14;
  5    20,  35,  40,  30;
  6    35,  64,  81,  80,  55;
  7    56, 105, 140, 154, 140,  91;
  8    84, 160, 220, 256, 260, 224, 140;
  ...
For n = 3, k = 2, a = 5, b = 12, c = 13. T(3, 2) = sqrt((13^4 - 5^4 - 12^4)/288) = 5.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Page 72.

M. F. Hasler, Table of n, a(n) for n = 2..1000, May 08 2025
Wikipedia, Pythagorean Triple.
Index entries related to Pythagorean Triples.

Cf. A120070 (b leg), A055096 (c hypotenuse).

Cf. A006414 (row sums), A000292 (column 1), A077414 (column 2), A000330 (diagonal), A107984 (transpose), A210440 (diagonal which begins with 4).

T[n_,k_]:=n*k(n^2-k^2)/6; Table[T[n,k],{n,2,11},{k,n-1}]//Flatten (* Stefano Spezia, Jul 11 2025 *)

apply( {A354968(n, k=0)=k|| k=n-1-(1-n=ceil(sqrt(8*n-7)/2+.5))*(2-n)\2; k*(n-k)*n*(n+k)\6}, [2..66]) \\ M. F. Hasler, May 08 2025

G.f.: x^2*y*(1 + x*y - 4*x^2*y + x^3*y + x^4*y^2)/((1 - x)^4*(1 - x*y)^4). - Stefano Spezia, Jul 11 2025

cp's OEIS Frontend

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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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.

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A140978 Repeat (n+1)^2 n times.

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A056203 Triangle of numbers related to congruum problem: T(n,k)=n^2+2nk-k^2 with n>k>0, starting at T(2,1)=7.

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A057105 Triangle of numbers (when unsigned) related to congruum problem: T(n,k)=k^2+2nk-n^2 with n>k>0 and starting at T(2,1)=1.

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A294774 a(n) = 2*n^2 + 2*n + 5.

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Maple

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A008224 Coordination sequence T6 for Zeolite Code PAU.

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A255848 a(n) = 2*n^2 + 18.

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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.

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.

A294774 a(n) = 2n^2 + 2n + 5.

A354968 Triangle read by rows: T(n, k) = nk(n+k)*(n-k)/6.