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.

Previous Showing 11-17 of 17 results.

A173622 Triangle T(n,m) read by rows: The number of m-Schroeder paths of order n with 2 diagonal steps.

Original entry on oeis.org

1, 6, 21, 30, 180, 546, 140, 1430, 6120, 17710, 630, 10920, 65835, 245700, 695640, 2772, 81396, 690690, 3322704, 11515140, 32212719, 12012, 596904, 7125300, 44170896, 187336380, 619851960, 1721286532, 51480, 4326300, 72624816
Offset: 2

Views

Author

R. J. Mathar, Nov 08 2010

Keywords

Comments

The case with 1 diagonal step is A060543.

Examples

			This is the left-lower portion of the array which starts in row n=2, columns m>=1 as:
1, 2, 3, 4, 5, 6,..
6, 21, 45, 78, 120, 171, 231,.. # A081266
30, 180, 546, 1224, 2310, 3900, 6090, 8976,.. # bisection A055112
140, 1430, 6120, 17710, 40950, 81840, 147630, 246820, 389160,.. # 5-section A034827
630, 10920, 65835, 245700, 695640, 1645020, 3426885, 6497400, ...
2772, 81396, 690690, 3322704, 11515140, 32212719, 77481495, ...
12012, 596904, 7125300, 44170896, 187336380, 619851960, ...

References

  • Chunwei Song, The Generalized Schroeder Theory, El. J. Combin. 12 (2005) #R53 Theorem 2.1.

Formula

T(n,m) = trinomial(m*n+n-2; m*n-2,n-2,2)/(m*n-1) .

A300758 a(n) = 2n*(n+1)*(2n+1).

Original entry on oeis.org

0, 12, 60, 168, 360, 660, 1092, 1680, 2448, 3420, 4620, 6072, 7800, 9828, 12180, 14880, 17952, 21420, 25308, 29640, 34440, 39732, 45540, 51888, 58800, 66300, 74412, 83160, 92568, 102660, 113460, 124992, 137280, 150348, 164220, 178920, 194472, 210900, 228228
Offset: 0

Views

Author

Christopher Purcell, Mar 12 2018

Keywords

Comments

The altitude h(n) = a(n)/A001844(n) of the (A005408(n), A046092(n) and A001844(n)) rectangular triangle is an irreducible fraction. - Ralf Steiner, Feb 25 2020
In this case, area A = a(n)/2 = A055112(n). - Bernard Schott, Feb 27 2020

Links

Crossrefs

Cf. A000330, A002492, A006331, A055112, A059270, A005408, A046092, A001844, A000384, A016754, A060300.
Cf. A027480.

Formula

a(n) = 12*A000330(n).
G.f.: 12*x*(1+x)/(1-x)^4. - Colin Barker, Mar 12 2018
a(n) = 6*A006331(n) = 4*A059270(n) = 3*A002492(n) = 2*A055112(n). - Omar E. Pol, Apr 04 2018
From Ralf Steiner, Feb 27 2020: (Start)
a(n) = 2*n*A000384(n+1).
a(n) = sqrt(A016754(n)*A060300(n)).
(End)
a(n) = A005408(n) * A046092(n). - Bruce J. Nicholson, Apr 24 2020

Extensions

Edited by N. J. A. Sloane, Aug 01 2019

A322170 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = A321768(n, k) * A321769(n, k) / 2.

Original entry on oeis.org

6, 30, 210, 60, 84, 1320, 630, 1560, 7140, 1386, 924, 2340, 210, 180, 4620, 2730, 10920, 45144, 7854, 7980, 23184, 2574, 5016, 63336, 26910, 49476, 242556, 50490, 25200, 57420, 4290, 3570, 34650, 12540, 14490, 79794, 18564, 5610, 10374, 504, 330, 11970, 7956
Offset: 1

Views

Author

Rémy Sigrist, Nov 29 2018

Keywords

Comments

This sequence gives the areas of the primitive Pythagorean triangles corresponding to the primitive Pythagorean triples in the tree described in A321768.
If we order the terms in this sequence and keep duplicates then we obtain A024406.

Examples

			The first rows are:
   6
   30, 210, 60
   84, 1320, 630, 1560, 7140, 1386, 924, 2340, 210
T(1,1) corresponds to the area of the triangle with sides 3, 4, 5; hence T(1, 1) = 3 * 4 / 2 = 6.

Links

Crossrefs

Cf. A024406, A029549, A055112, A069072, A321768, A321769.

Programs

  • PARI
    M = [[1, -2, 2; 2, -1, 2; 2, -2, 3], [1, 2, 2; 2, 1, 2; 2, 2, 3], [-1, 2, 2; -2, 1, 2; -2, 2, 3]];
T(n, k) = my (t=[3; 4; 5], d=digits(3^(n-1)+k-1, 3)); for (i=2, #d, t = M[d[i]+1] * t); return (t[1, 1] * t[2, 1] / 2)

Formula

Empirically:
- T(n, 1) = A055112(n),
- T(n, (3^(n-1) + 1)/2) = A029549(n),
- T(n, 3^(n-1)) = A069072(n-1).

A181773 Molecular topological indices of the cocktail party graphs.

Original entry on oeis.org

0, 48, 240, 672, 1440, 2640, 4368, 6720, 9792, 13680, 18480, 24288, 31200, 39312, 48720, 59520, 71808, 85680, 101232, 118560, 137760, 158928, 182160, 207552, 235200, 265200, 297648, 332640, 370272, 410640
Offset: 1

Views

Author

Eric W. Weisstein, Jul 10 2011

Keywords

Comments

a(n) is the number of 2 X 2 matrices (all four elements distinct) having entries in {-n,...,0,...,n} with determinant equal to the permanent. - Indranil Ghosh, Dec 25 2016

Links

Crossrefs

Cf. A280059 (2 X 2 matrices, elements can be repeated).

Programs

Formula

a(n) = 8*(n-1)*n*(2n-1).
a(n) = 16*A059270(n-1).
G.f.: 48*x^2*(x+1)/(x-1)^4. - Colin Barker, Oct 17 2012
a(n) = 48*A000330(n-1). - R. J. Mathar, Jan 04 2017
From Omar E. Pol, Jan 05 2017: (Start)
a(n) = 24*A006331(n-1) = 12*A002492(n-1) = 8*A055112(n-1).
a(n) = 2*A069074(n-2), n >= 2. (End)

A385029 a(n) = Sum_{-n <= a, b, c <= n} (b^2 - 4*a*c).

Original entry on oeis.org

18, 250, 1372, 4860, 13310, 30758, 63000, 117912, 205770, 339570, 535348, 812500, 1194102, 1707230, 2383280, 3258288, 4373250, 5774442, 7513740, 9648940, 12244078, 15369750, 19103432, 23529800, 28741050, 34837218, 41926500, 50125572, 59559910, 70364110, 82682208, 96668000
Offset: 1

Views

Author

Darío Clavijo, Jun 15 2025

Keywords

Comments

There are (2*n + 1)^3 combinations of a, b, c.

Links

Crossrefs

Cf. A000384, A002378, A014105, A016755, A016754, A055112, A384666.

Programs

  • Mathematica
    A385029[n_] := (n*(n + 1)*(2*n + 1)^3)/3;
Array[A385029, 50] (* Paolo Xausa, Jun 18 2025 *)
  • Python
    a = lambda n: ((n*n+n)*((n << 1)+1)**3)//3
print([a(n) for n in range(1, 11)])

Formula

a(n) = (n*(n+1)*(2*n+1)^3)/3.
a(n) = (A055112(n)*A016754(n))/3.
a(n) = (A002378(n)*A016755(n))/3.
G.f.: 2*x*(9 + 71*x + 71*x^2 + 9*x^3)/(1 - x)^6. - Stefano Spezia, Jun 15 2025
From Amiram Eldar, Jun 18 2025; (Start)
Sum_{n>=1} 1/a(n) = 21*(1 - zeta(3)/2) - 12*log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi^3/8 + 3*Pi - 21. (End)

A237576 Smallest integer areas of integer-sided triangles such that the perimeter equals n times the smallest side.

Original entry on oeis.org

0, 0, 0, 6, 60, 30, 210, 24, 84, 60, 198, 330, 1716, 546, 2730, 252, 4080, 36, 5814, 210, 7980, 2310, 10626, 924, 1380, 1248, 90, 4914, 4176, 6090, 26970, 480, 32736, 1224, 39270, 1938, 46620, 2394, 54834, 4560, 63960, 4620, 74046, 19866, 85140, 22770, 97290
Offset: 1

Views

Author

Michel Lagneau, Feb 09 2014

Keywords

Comments

The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2.
The sequence a(n) is the union of four subsequences A, B, C and D where:
A is the subsequence with areas 60, 210, 1716, 2730, 4080, 5814, 7980, 10626, ... where n is odd, and the corresponding sides are of the form (4k, 4k^2-1, 4k^2+1) with areas 2k(4k^2-1) for k = 2, 3, 6, 7, 8, 9, 11, ... These areas are in the sequence A069072 (areas of primitive Pythagorean triangles whose odd sides differ by 2).
B is the subsequence with areas 6, 30, 84, 330, 546, 2310, 4914, 6090, ... where n is even, and the corresponding sides are of the form (2k+1, 2k(k+1), 2k(k+1)+1) with areas k(k+1)(2k+1) for k = 1, 2, 3, 5, 6, 7, 10, 13, 14, ... These areas are in the sequence A055112 (Areas of Pythagorean triangles (a, b, c) with c = b+1).
C is the subsequence with areas 84, 198, 1380, 4176, ... where n is odd but the areas are not Pythagorean triangles.
D is the subsequence with areas 24, 60, 210, 924, 1248, 480, 1224, 1938, ... where n is even but the areas are not Pythagorean triangles.
The triangles with the same areas are not unique; for example:
(8, 15, 17) and (6, 25, 29) => A = 60; the first is a Pythagorean triangle, the second is not.
(12, 35, 37) and (7, 65, 68) => A = 210; the first is a Pythagorean triangle, the second is not.
The following table gives the first values (n, A, p, a, b, c) where A is the area of the triangles, p is the perimeter and a, b, c are the sides.
+----+------+-------------+----+-----+-----+
| n | A | p | a | b | c |
+----+------+-------------+----+-----+-----+
| 4 | 6 | 12 = 4*3 | 3 | 4 | 5 |
| 5 | 60 | 40 = 5*8 | 8 | 15 | 17 |
| 6 | 30 | 30 = 6*5 | 5 | 12 | 13 |
| 7 | 210 | 84 = 7*12 | 12 | 35 | 37 |
| 8 | 24 | 32 = 8*4 | 4 | 13 | 15 |
| 9 | 84 | 72 = 9*8 | 8 | 29 | 35 |
| 10 | 60 | 60 = 10*6 | 6 | 25 | 29 |
| 11 | 198 | 132 = 11*12 | 12 | 55 | 65 |
| 12 | 330 | 132 = 12*11 | 11 | 60 | 61 |
| 13 | 1716 | 312 = 13*24 | 24 | 143 | 145 |
| 14 | 546 | 182 = 14*13 | 13 | 84 | 85 |
| 15 | 2730 | 420 = 15*28 | 28 | 195 | 197 |
+----+------+-------------+----+-----+-----+

Crossrefs

Cf. A188158.

Programs

  • Maple
    with(numtheory):nn:=600:for n from 4 to 50 do: ii:=0:for a from 1
  to nn while(ii=0) do: for b from a to nn while(ii=0) do: for c from b to nn while(ii=0) do: p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c): if x>0 then x0:= sqrt(x):else fi:if x0=floor(x0) and 2*p=n*a then ii:=1:printf ( "%d %d %d %d %d \n",n,x0,a,b,c):else fi:od:od:od:od:
  • Mathematica
    nn=600;lst={};Do[k=0;Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s (s-a) (s-b) (s-c);If[0

A337841 Triangle read by rows, T(n, k) = binomial(2*n-1, 2*k-1) * binomial(2*n-2*k, n-k) * (k+1) / binomial(n+k+1, n-k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, 0, 3, 3, 0, 6, 10, 4, 0, 14, 30, 21, 5, 0, 36, 90, 84, 36, 6, 0, 99, 275, 308, 180, 55, 7, 0, 286, 858, 1092, 780, 330, 78, 8, 0, 858, 2730, 3822, 3150, 1650, 546, 105, 9, 0, 2652, 8840, 13328, 12240, 7480, 3094, 840, 136, 10, 0, 8398, 29070, 46512, 46512, 31977, 15561, 5320, 1224, 171, 11
Offset: 0

Views

Author

Werner Schulte, Oct 30 2020

Keywords

Examples

			The triangle T(n,k) for 0 <= k <= n starts:
  n\k:  0     1      2      3      4      5      6     7     8    9  10
=======================================================================
   0 :  1
   1 :  0     2
   2 :  0     3      3
   3 :  0     6     10      4
   4 :  0    14     30     21      5
   5 :  0    36     90     84     36      6
   6 :  0    99    275    308    180     55      7
   7 :  0   286    858   1092    780    330     78     8
   8 :  0   858   2730   3822   3150   1650    546   105     9
   9 :  0  2652   8840  13328  12240   7480   3094   840   136   10
  10 :  0  8398  29070  46512  46512  31977  15561  5320  1224  171  11
etc.

Crossrefs

Cf. Row sums: A000984, main diagonal: A000027, 1st subdiagonal: A014105, 2nd subdiagonal: A055112, column 0: A000007, column 1: A007054.

Programs

  • Maple
    T := proc(n, k) option remember; if k = n then n+1 else
T(n-1, k)*(2*n-2)*(2*n-1)/((n-1)*(n+2)-(k-1)*(k+2)) fi end:
for n from 0 to 9 do seq(T(n, k), k=0..n) od; # Peter Luschny, Nov 02 2020

Formula

T(n,k) = binomial(2*n-1,n-k) * k * (2*k+1) * (2*k+2) / ((n+k)*(n+k+1)) for 1 <= k <= n, and T(n,0) = 0^n for n >= 0.
T(n,n) = n+1 for n >= 0; T(n,n-1) = (n-1) * (2*n-1) for n > 0; T(n,n-2) = (n-1) * (n-2) * (2*n-3) for n > 1.
T(n,k) = T(n-1,k) * (2*n-2) * (2*n-1) / ((n-1) * (n+2) - (k-1) * (k+2)) for 0 <= k < n with initial values T(n,n) = n+1 for n >= 0.
Row sums are A000984(n) for n >= 0.
Alternating row sums are 0 for n > 1.
Sum_{k=0..n} (-1)^k * T(n,k) * (k*(k+1)/2)^m = 0 for 0 <= m <= n-2.
T(n,1) = 12 * binomial(2*n-1,n-1)/((n+1)*(n+2)) = A007054(n) for n > 0.
T(n,k) = T(n,1)*(k*(k+1)*(2*k+1)/6)*binomial(n-1,k-1)/binomial(n+1+k,k-1) for 1 <= k <= n.
From Werner Schulte, Nov 09 2020: (Start)
T(n,k) = A128899(n,k) * (k+1) * (2*k+1) / (n+k+1) for 0 <= k <= n.
T(n,0) + Sum_{k=1..n} T(n,k) / (k*(k+1)) = A000108(n) for n >= 0. (End)
Previous Showing 11-17 of 17 results.

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