A081266 -id:A081266 - cp's OEIS frontend

A144314 a(n) = 3n(6*n + 1).

0, 21, 78, 171, 300, 465, 666, 903, 1176, 1485, 1830, 2211, 2628, 3081, 3570, 4095, 4656, 5253, 5886, 6555, 7260, 8001, 8778, 9591, 10440, 11325, 12246, 13203, 14196, 15225, 16290, 17391, 18528, 19701, 20910, 22155, 23436, 24753, 26106, 27495

Offset: 0

Reinhard Zumkeller, Sep 17 2008

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

Cf. A000217, A014105, A033585, A049453, A081266, A144312.

[18*n^2+3*n: n in [0..50]]; // Vincenzo Librandi, Dec 18 2014

A144314:=n->3*n*(6*n+1): seq(A144314(n), n=0..70); # Wesley Ivan Hurt, Dec 16 2015

Table[3n(6n+1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,21,78},40] (* Harvey P. Dale, Dec 17 2014 *)
CoefficientList[Series[x (21 + 15 x) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 18 2014 *)

a(n)=3*n*(6*n+1) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A000217(6*n) = A014105(3*n) = A081266(2*n).
G.f.: 3*x*(7+5*x)/(1-x)^3. - Vincenzo Librandi, Dec 18 2014
From Wesley Ivan Hurt, Dec 16 2015: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
a(n) = 3 * A049453(n). (End)
E.g.f.: 3*exp(x)*x*(7 + 6*x). - Stefano Spezia, Jun 29 2021
From Amiram Eldar, May 11 2025: (Start)
Sum_{n>=1} 1/a(n) = 2 - Pi/(2*sqrt(3)) - 2*log(2)/3 - log(3)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/3 + log(2)/3 + log(2+sqrt(3))/sqrt(3) - 2. (End)

A193723 Mirror of the fusion triangle A193722.

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 18, 21, 8, 1, 54, 81, 45, 11, 1, 162, 297, 216, 78, 14, 1, 486, 1053, 945, 450, 120, 17, 1, 1458, 3645, 3888, 2295, 810, 171, 20, 1, 4374, 12393, 15309, 10773, 4725, 1323, 231, 23, 1, 13122, 41553, 58320, 47628, 24948, 8694, 2016, 300, 26, 1

Offset: 0

Clark Kimberling, Aug 04 2011

A193723 is obtained by reversing the rows of the triangle A193722.
Triangle T(n,k), read by rows, given by [2,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 04 2011
From Philippe Deléham, Nov 14 2011: (Start)
Riordan array ((1-x)/(1-3x), x/(1-3x)).
Product A200139*A007318 as infinite lower triangular arrays. (End)

			First six rows:
    1;
    2,   1;
    6,   5,   1;
   18,  21,   8,   1;
   54,  81,  45,  11,   1;
  162, 297, 216,  78,  14,   1;

Cf. A084938, A193722, A052924 (antidiagonal sums), Diagonals: A000012, A016789, A081266, Columns: A025192, A081038.

z = 9; a = 1; b = 1; c = 1; d = 2;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193722 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193723 *)

Write w(n,k) for the triangle at A193722.  The triangle at A193723 is then given by w(n,n-k).
T(n,k) = T(n-1,k-1) + 3*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011
From Philippe Deléham, Nov 14 2011: (Start)
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A025192(n), A002001(n), A005054(n), A052934(n), A055272(n), A055274(n), A055275(n), A052268(n), A055276(n), A196731(n) for x=-2,-1,0,1,2,3,4,5,6,7,8,9 respectively.
T(n,k) = Sum_{j>=0} T(n-1-j,k-1)*3^j.
G.f.: (1-x)/(1-(3+y)*x). (End)

A158623 Denominator of the reduced fraction A158620(n)/A158621(n).

Original entry on oeis.org

9, 18, 10, 45, 63, 28, 108, 135, 55, 198, 234, 91, 315, 360, 136, 459, 513, 190, 630, 693, 253, 828, 900, 325, 1053, 1134, 406, 1305, 1395, 496, 1584, 1683, 595, 1890, 1998, 703, 2223, 2340, 820, 2583, 2709, 946, 2970, 3105, 1081, 3384, 3528, 1225, 3825

Offset: 2

Jonathan Vos Post, Mar 23 2009

A158620(n) = Product_{k=2..n} (k^3-1). A158621(n) = Product_{k=2..n} (k^3+1). A158622(n) is the numerator of the reduced fraction A158620(n)/A158621(n). A158623(n) is the denominator of the reduced fraction A158620(n)/A158621(n). The reduced fractions are 7/9, 13/18, 7/10, 31/45, 43/63, 19/28, 73/108, 91/135, 37/55, 133/198, ...

			a(2) = 9 = denominator of (2^3-1)/2^3+1 = 7/9. a(3) = 18 = denominator of ((2^3-1)*(3^3-1))/((2^3+1)*(3^3+1)) = (7 * 26)/ (9 * 28) = 182/252 = 13/18. a(4) = 10 = denominator of ((2^3-1)*(3^3-1)*(4^3-1))/((2^3+1)*(3^3+1)*(4^3+1)) = (7 * 26 * 63)/(9 * 28 * 65) = 11466/16380 = 7/10. a(5) = 45 = denominator of ((2^3-1)(3^3-1)(4^3-1)(5^3-1))/((2^3+1)(3^3+1)(4^3+1)(5^3+1)) = 1421784/2063880 = 31/45.

Cf. A001093, A016921, A068601, A158620-A158622.

A158623 := proc(n) 2*(n^2+n+1)/3/n/(n+1) ; denom(%) ; end: seq(A158623(n),n=2..100) ; # R. J. Mathar, Mar 27 2009

Denominator of (Product_{k=2..n} (k^3-1)) / Product_{k=2..n} (k^3+1) = denominator of Product_{k=2..n} A068601(k)/A001093(k).
A158620(n)/A158621(n) = 2(n^2+n+1)/(3n(n+1)). Conjecture: a(n) = 3a(n-3) - 3a(n-6) + a(n-9), so trisections are A152996, A060544 and 3*A081266. - R. J. Mathar, Mar 27 2009
Empirical g.f.: -x^2*(x^8 - 2*x^5 + 9*x^4 + 18*x^3 + 10*x^2 + 18*x + 9) / ((x-1)^3*(x^2 + x + 1)^3). - Colin Barker, May 09 2013

More terms from R. J. Mathar, Mar 27 2009

A306368 a(n) = numerator of (n + 3)(n + 4)/((n + 1)(n + 2)).

Original entry on oeis.org

6, 10, 5, 21, 28, 12, 45, 55, 22, 78, 91, 35, 120, 136, 51, 171, 190, 70, 231, 253, 92, 300, 325, 117, 378, 406, 145, 465, 496, 176, 561, 595, 210, 666, 703, 247, 780, 820, 287, 903, 946, 330, 1035, 1081, 376, 1176, 1225, 425, 1326, 1378, 477, 1485, 1540, 532, 1653, 1711, 590

Offset: 0

Peter Bala, Feb 14 2019

If P(x) and Q(x) are coprime integral polynomials such that Q(n) > 0 for n >= 0 then the sequence of numerators of the rational numbers P(n)/Q(n) for n >= 0 and the sequence of denominators of P(n)/Q(n) for n >= 0 are both quasi-polynomial in n. In fact, there exists a purely periodic sequence b(n) such that numerator(P(n)/Q(n)) = P(n)/b(n) and denominator(P(n)/Q(n)) = Q(n)/b(n). Here we take P(n) = (n + 3)*(n + 4) and Q(n) = (n + 1)*(n + 2).

Muniru A Asiru, Table of n, a(n) for n = 0..10000
Peter Bala, A note on the sequence of numerators of a rational function .
Wikipedia, Quasi-polynomial.
Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Cf. A000326, A060544, A081266, A306367.

List([0..100],n->NumeratorRat((n+3)*(n+4)/((n+1)*(n+2)))); # Muniru A Asiru, Feb 25 2019

seq((n + 3)*(n + 4)/gcd((n + 3)*(n + 4), (n + 1)*(n + 2)), n = 0..100);

Table[((n+3)(n+4))/((n+1)(n+2)),{n,0,60}]//Numerator (* or *) LinearRecurrence[{0,0,3,0,0,-3,0,0,1},{6,10,5,21,28,12,45,55,22},60] (* Harvey P. Dale, Mar 28 2020 *)

a(n) = numerator((n + 3)*(n + 4)/((n + 1)*(n + 2))); \\ Michel Marcus, Feb 26 2019

O.g.f.: (x^8 + x^7 - 3*x^5 - 2*x^4 + 3*x^3 + 5*x^2 + 10*x + 6)/((1 - x)^3*(x^2 + x + 1)^3).
a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) for n >= 9.
a(n) = (n + 3)*(n + 4)/b(n), where (b(n))n>=0 is the purely periodic sequence [2, 2, 6, 2, 2, 6, ...] with period 3.
a(n) = (n + 3)*(n + 4)/gcd((n + 3)*(n + 4), (n + 1)*(n + 2)).
a(3*n)   = (3*n + 3)*(3*n + 4)/2 = A081266(n+1).
a(3*n+1) = (3*n + 4)*(3*n + 5)/2 = A060544(n+2).
a(3*n+2) = (n + 2)*(3*n + 5)/2   = A000326(n+2).
Sum_{n>=0} 1/a(n) = 2*log(3) - 2*Pi/(3*sqrt(3)). - Amiram Eldar, Aug 11 2022

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

Original entry on oeis.org

2, 4, 12, 18, 31, 41, 59, 73, 96, 114, 142, 164, 197, 223, 261, 291, 334, 368, 416, 454, 507, 549, 607, 653, 716, 766, 834, 888, 961, 1019, 1097, 1159, 1242, 1308, 1396, 1466, 1559, 1633, 1731, 1809, 1912, 1994, 2102, 2188, 2301, 2391, 2509, 2603, 2726, 2824, 2952

Offset: 0

Bruno Berselli, Oct 25 2011

For an origin of this sequence, see the triangular spiral illustrated in the Links section.

First bisection gives A117625 (without the initial term).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A152832 (by Superseeker).

Cf. sequences related to the triangular spiral: A022266, A022267, A027468, A038764, A045946, A051682, A062708, A062725, A062728, A062741, A064225, A064226, A081266-A081268, A081270-A081272, A081275 [incomplete list].

[(6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1: n in [0..50]];

LinearRecurrence[{1,2,-2,-1,1},{2,4,12,18,31},60] (* Harvey P. Dale, Jun 15 2022 *)

for(n=0, 50, print1((6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1", "));

G.f.: (2+2*x+4*x^2+2*x^3-x^4)/((1+x)^2*(1-x)^3).
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
a(n)-a(-n-1) = A168329(n+1).
a(n)+a(n-1) = A102214(n).
a(2n)-a(2n-1) = A016885(n).
a(2n+1)-a(2n) = A016825(n).

A268351 a(n) = 3n(9*n - 1)/2.

Original entry on oeis.org

0, 12, 51, 117, 210, 330, 477, 651, 852, 1080, 1335, 1617, 1926, 2262, 2625, 3015, 3432, 3876, 4347, 4845, 5370, 5922, 6501, 7107, 7740, 8400, 9087, 9801, 10542, 11310, 12105, 12927, 13776, 14652, 15555, 16485, 17442, 18426, 19437, 20475, 21540, 22632, 23751, 24897, 26070, 27270

Offset: 0

Ilya Gutkovskiy, Feb 02 2016

First trisection of pentagonal numbers (A000326).

More generally, the ordinary generating function for the first trisection of k-gonal numbers is 3*x*(k - 1 + (2*k - 5)*x)/(1 - x)^3.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pentagonal Number.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A001318, A016766, A022266, A022284, A081266, A188623, A211538.

[3*n*(9*n-1)/2: n in [0..50]]; // Vincenzo Librandi, Feb 04 2016

Table[3 n (9 n - 1)/2, {n, 0, 45}]
Table[Binomial[9 n, 2]/3, {n, 0, 45}]
LinearRecurrence[{3, -3, 1}, {0, 12, 51}, 45]

a(n)=3*n*(9*n-1)/2 \\ Charles R Greathouse IV, Jul 26 2016

G.f.: 3*x*(4 + 5*x)/(1 - x)^3.
a(n) = binomial(9*n,2)/3.
a(n) = A000326(3*n) = 3*A022266(n).
a(n) = A211538(6*n+2).
a(n) = A001318(6*n-1), with A001318(-1)=0.
a(n) = A188623(9*n-2), with A188623(-2)=0.
Sum_{n>=1} 1/a(n) = 0.132848490245209886617568... = (-Pi*cot(Pi/9) + 5*log(3) + 4*cos(Pi/9)*log(cos(Pi/18)) - 4*cos(2*Pi/9)*log(sin(Pi/9)) - 4*log(sin(2*Pi/9))*sin(Pi/18))/3. [Corrected by Vaclav Kotesovec, Feb 25 2016]
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: 3*exp(x)*x*(8 + 9*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = A022284(n) - n. (End)

Edited by Bruno Berselli, Feb 03 2016

A342719 Array read by ascending antidiagonals: T(k, n) is the sum of the consecutive positive integers from 1 to (n - 1)*k placed along the perimeter of an n-th order perimeter-magic k-gon.

Original entry on oeis.org

21, 36, 45, 55, 78, 78, 78, 120, 136, 120, 105, 171, 210, 210, 171, 136, 231, 300, 325, 300, 231, 171, 300, 406, 465, 465, 406, 300, 210, 378, 528, 630, 666, 630, 528, 378, 253, 465, 666, 820, 903, 903, 820, 666, 465, 300, 561, 820, 1035, 1176, 1225, 1176, 1035, 820, 561

Offset: 3

Stefano Spezia, Mar 19 2021

			The array begins:
k\n|   3    4    5    6    7 ...
---+------------------------
3  |  21   45   78  120  171 ...
4  |  36   78  136  210  300 ...
5  |  55  120  210  325  465 ...
6  |  78  171  300  465  666 ...
7  | 105  231  406  630  903 ...
...

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equation 3).

Cf. A014105 (n = 3), A033585 (n = 5), A037270 (1st superdiagonal), A081266 (n = 4), A083374 (1st subdiagonal), A110450 (diagonal), A144312 (n = 6), A144314 (n = 7), A342757, A342758.

T[k_,n_]:=(n-1)k((n-1)k+1)/2; Table[T[k+3-n,n],{k,3,12},{n,3,k}]//Flatten

O.g.f.: (x^2 - 3*x^2*y + x*y^2 + 3*x^2*y^2)/((1 - x)^3*(1 - y)^3).
E.g.f.: exp(x+y)*x*(x - x*y + y^2 + x*y^2)/2.
T(k, n) = (n - 1)*k*((n - 1)*k + 1)/2.

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

R. J. Mathar, Nov 08 2010

The case with 1 diagonal step is A060543.

			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, ...

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

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

A143942 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in a linear chain of n squares joined at vertices (i.e., joined like <><><>...<>; here <> is a square!); 1 <= k <= 2n.

Original entry on oeis.org

4, 2, 8, 8, 4, 1, 12, 14, 8, 6, 4, 1, 16, 20, 12, 11, 8, 6, 4, 1, 20, 26, 16, 16, 12, 11, 8, 6, 4, 1, 24, 32, 20, 21, 16, 16, 12, 11, 8, 6, 4, 1, 28, 38, 24, 26, 20, 21, 16, 16, 12, 11, 8, 6, 4, 1, 32, 44, 28, 31, 24, 26, 20, 21, 16, 16, 12, 11, 8, 6, 4, 1, 36, 50, 32, 36, 28, 31, 24, 26

Offset: 1

Emeric Deutsch, Sep 06 2008

Row n has 2n entries.
The entries in row n are the coefficients of the Wiener polynomial of the linear chain of n squares.
Sum of entries in row n = 3n(3n+1)/2 = A081266(n).
Sum_{k=1..n} k*T(n,k) = the Wiener index of a linear chain of n squares joined at vertices (like <><><>...) = A143943(n).

			T(2,1)=8 because the chain of 2 squares (<><>) has 8 edges.
Triangle starts:
   4,  2;
   8,  8,  4,  1;
  12, 14,  8,  6,  4,  1;
  16, 20, 12, 11,  8,  6,  4,  1;
  20, 26, 16, 16, 12, 11,  8,  6,  4,  1;

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Cf. A081266, A143943.

T:=proc(n,k) if 2*n < k then 0 elif k = 1 then 4*n elif k = 2 then 6*n-4 elif `mod`(k,2)=1 then 4*n-2*k+2 elif `mod`(k,2)=0 then 5*n-(5/2)*k+1 else 0 end if end proc: for n to 10 do seq(T(n,k),k=1..2*n) end do; # yields sequence in triangular form

T[n_, k_] := Which[2n < k, 0, k == 1, 4n, k == 2, 6n - 4, OddQ[k], 4n - 2k + 2, EvenQ[k], 5n - (5/2) k + 1, True, 0];
Table[T[n, k], {n, 1, 10}, {k, 1, 2n}] // Flatten (* Jean-François Alcover, Aug 23 2024, after Maple program *)

T(n,1) = 4n; T(n,2) = 6n-4; T(n,2p+1) = 4(n-p); T(n,2p) = 5(n-p)+1.

G.f. = G(q,z) = qz/(4+2q+4qz-q^3*z)/((1-q^2*z)*(1-z)^2).

A206294 Riordan array (1, x/(1-x)^3).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 6, 6, 1, 0, 10, 21, 9, 1, 0, 15, 56, 45, 12, 1, 0, 21, 126, 165, 78, 15, 1, 0, 28, 252, 495, 364, 120, 18, 1, 0, 36, 462, 1287, 1365, 680, 171, 21, 1, 0, 45, 792, 3003, 4368, 3060, 1140, 231, 24, 1

Offset: 0

Philippe Deléham, Feb 05 2012

The convolution triangle of the triangular numbers A000217. - Peter Luschny, Oct 07 2022

			Triangle begins:
1
0, 1
0, 3, 1
0, 6, 6, 1
0, 10, 21, 9, 1
0, 15, 56, 45, 12, 1
0, 21, 126, 165, 78, 15, 1
0, 28, 252, 495, 364, 120, 18, 1
0, 36, 462, 1287, 1365, 680, 171, 21, 1
0, 45, 792, 3003, 4368, 3060, 1140, 231, 24, 1
0, 55, 1287, 6435, 12376, 11628, 5985, 1771, 300, 27, 1
0, 66, 2002, 12870, 31824, 38760, 26324, 10626, 2600, 378, 30, 1

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. Columns: A000007, A000217 (triangular numbers), A000389, A000581, A001288, A010967..(+3)..A011000, A017714..(+3)..A017762.
Row sums are A052529.
Cf. A127893.

# Uses function PMatrix from A357368.
PMatrix(10, n -> n * (n + 1) / 2); # Peter Luschny, Oct 07 2022

Table[If[n == 0 && k == 0 , 1, Binomial[n - 1 + 2 k, n - k]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 25 2017 *)

{T(n,k)=polcoeff(1/(1-x+x*O(x^(n-k)))^(3*k),n-k)}

{T(n,k)=polcoeff(polcoeff((1-x)^3/((1-x)^3-y*x +x*O(x^n)),n,x),k,y)}
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

Triangle T(n,k), read by rows, given by (0, 3, -1, 2/3, -1/6, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
T(n,0) = 0^n, T(n,k) = C(n-1+2k, n-k) for k > 0.
T(n,n) = 1, T(k+1,k) = 3*k = A008585(k), T(k+2,k) = A081266(k).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A052529(n), A052910(n) for x = 0, 1, 2 respectively.
G.f.: (1-x)^3/((1-x)^3-y*x).

cp's OEIS Frontend

A144314 a(n) = 3*n*(6*n + 1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A193723 Mirror of the fusion triangle A193722.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A158623 Denominator of the reduced fraction A158620(n)/A158621(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A306368 a(n) = numerator of (n + 3)*(n + 4)/((n + 1)*(n + 2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Formula

A198392 a(n) = (6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16 + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A268351 a(n) = 3*n*(9*n - 1)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A342719 Array read by ascending antidiagonals: T(k, n) is the sum of the consecutive positive integers from 1 to (n - 1)*k placed along the perimeter of an n-th order perimeter-magic k-gon.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

A144314 a(n) = 3n(6*n + 1).

A306368 a(n) = numerator of (n + 3)(n + 4)/((n + 1)(n + 2)).

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

A268351 a(n) = 3n(9*n - 1)/2.