A002414 -id:A002414 - cp's OEIS frontend

A059722 a(n) = n(2n^2 - 2*n + 1).

0, 1, 10, 39, 100, 205, 366, 595, 904, 1305, 1810, 2431, 3180, 4069, 5110, 6315, 7696, 9265, 11034, 13015, 15220, 17661, 20350, 23299, 26520, 30025, 33826, 37935, 42364, 47125, 52230, 57691, 63520, 69729, 76330, 83335, 90756, 98605, 106894, 115635, 124840

Offset: 0

Henry Bottomley, Feb 07 2001

Mean of the first four nonnegative powers of 2n+1, i.e., ((2n+1)^0 + (2n+1)^1 + (2n+1)^2 + (2n+1)^3)/4. E.g., a(2) = (1 + 3 + 9 + 27)/4 = 10.
Equatorial structured meta-diamond numbers, the n-th number from an equatorial structured n-gonal diamond number sequence. There are no 1- or 2-gonal diamonds, so 1 and (n+2) are used as the first and second terms since all the sequences begin as such. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Starting with offset 1 = row sums of triangle A143803. - Gary W. Adamson, Sep 01 2008
Form an array from the antidiagonals containing the terms in A002061 to give antidiagonals 1; 3,3; 7,4,7; 13,8,8,13; 21,14,9,14,21; and so on. The difference between the sum of the terms in n+1 X n+1 matrices and those in n X n matrices is a(n) for n>0. - J. M. Bergot, Jul 08 2013
Sum of the numbers from (n-1)^2 to n^2. - Wesley Ivan Hurt, Sep 08 2014

Harry J. Smith, Table of n, a(n) for n = 0..1000
Bruno Berselli, Table of sequences with the formula m*(m+1)^2 - (k+2)*m^2 (table includes this sequence for k = 2-m, m >= 0).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A005900, A081436, A100178, A100179, A059722: "equatorial" structured diamonds; A000447: "polar" structured meta-diamond; A006484 for other structured meta numbers; and A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Cf. A000217, A000290, A001844, A002061, A002414, A053698, A059721, A059723, A136392, A143803.

[n*(2*n^2 - 2*n + 1) : n in [0..50]]; // Wesley Ivan Hurt, Sep 08 2014

A059722:=n->n*(2*n^2 - 2*n + 1): seq(A059722(n), n=0..50); # Wesley Ivan Hurt, Sep 08 2014

Table[n (2 n^2 - 2 n + 1), {n, 0, 50}] (* Wesley Ivan Hurt, Sep 08 2014 *)

a(n) = { n*(2*n^2 - 2*n + 1) } \\ Harry J. Smith, Jun 28 2009

a(n) = A053698(2*n-1)/4.
a(n) = Sum_{j=1..n} ((n+j-1)^2-j^2+1). - Zerinvary Lajos, Sep 13 2006
From R. J. Mathar, Sep 02 2008: (Start)
G.f.: x*(1 + x)*(1 + 5*x)/(1 - x)^4.
a(n) = A002414(n-1) + A002414(n).
a(n+1) - a(n) = A136392(n+1). (End)
a(n) = (A000290(n) + A000290(n+1)) * (A000217(n+1) - A000217(n)). - J. M. Bergot, Nov 02 2012
a(n) = n * A001844(n-1). - Doug Bell, Aug 18 2015
a(n) = A000217(n^2) - A000217(n^2-2*n). - Bruno Berselli, Jun 26 2018
E.g.f.: exp(x)*x*(1 + 4*x + 2*x^2). - Stefano Spezia, Jun 20 2021

Edited with new definition by N. J. A. Sloane, Aug 29 2008

A144823 Square array A(n,k), n>=1, k>=1, read by antidiagonals, with A(1,k)=1 and sequence a_k of column k shifts left when Dirichlet convolution with a_k (DC:(b,a_k)->a) applied k times.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 9, 1, 1, 5, 16, 30, 18, 1, 1, 6, 25, 70, 90, 40, 1, 1, 7, 36, 135, 280, 288, 80, 1, 1, 8, 49, 231, 675, 1168, 864, 168, 1, 1, 9, 64, 364, 1386, 3475, 4672, 2647, 340, 1, 1, 10, 81, 540, 2548, 8496, 17375, 18884, 7968, 698, 1, 1, 11, 100

Offset: 1

Alois P. Heinz, Sep 21 2008

			Square array A(n,k) begins:
   1,   1,    1,     1,     1,      1,      1,      1, ...
   1,   1,    1,     1,     1,      1,      1,      1, ...
   2,   3,    4,     5,     6,      7,      8,      9, ...
   4,   9,   16,    25,    36,     49,     64,     81, ...
   9,  30,   70,   135,   231,    364,    540,    765, ...
  18,  90,  280,   675,  1386,   2548,   4320,   6885, ...
  40, 288, 1168,  3475,  8496,  18130,  35008,  62613, ...
  80, 864, 4672, 17375, 50976, 126910, 280064, 563517, ...

Alois P. Heinz, Antidiagonals n = 1..100, flattened
N. J. A. Sloane, Transforms

Columns 1-9 give: A038044, A144817, A144316, A144818, A144819, A144820, A144317, A144821, A144822.

Rows 1+2, 3-4 give: A000012, A000027, A000290, A002414.

with(numtheory): dc:= proc(b,c) proc(n) option remember; add(b(d) *c(n/d), d=`if`(n<0,{},divisors(n))) end end: A:= proc(n, k) local a, b, t; b[1]:= dc(a,a); for t from 2 to k do b[t]:= dc(b[t-1],a) od: a:= n-> `if`(n=1, 1, b[k](n-1)); a(n) end: seq(seq(A(n, 1+d-n), n=1..d), d=1..12);

dc[b_, c_] := Module[{proc}, proc[n_] := proc[n] = Sum [b[d] *c[n/d], {d, If[n < 0, {}, Divisors[n]]}]; proc]; A [n_, k_] := Module[{a, b, t}, b[1] = dc[a, a]; For[t = 2, t <= k, t++, b[t] = dc[b[t-1], a]]; a = Function[m, If[m == 1, 1, b[k][m-1]]]; a[n]]; Table[Table [A[n, 1+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Dec 20 2013, translated from Maple *)

A081422 Triangle read by rows in which row n consists of the first n+1 n-gonal numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 10, 1, 4, 9, 16, 25, 1, 5, 12, 22, 35, 51, 1, 6, 15, 28, 45, 66, 91, 1, 7, 18, 34, 55, 81, 112, 148, 1, 8, 21, 40, 65, 96, 133, 176, 225, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451

Offset: 0

Paul Barry, Mar 21 2003

			The array starts
  1  1  3 10 ...
  1  2  6 16 ...
  1  3  9 22 ...
  1  4 12 28 ...
The triangle starts
  1;
  1,  1;
  1,  2,  3;
  1,  3,  6, 10;
  1,  4,  9, 16, 25;
  ...

T. D. Noe, Rows n = 0..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Polygonal Number

Rows include A060354, A064808, A006000, A006003, A002411.
Diagonals include A001093, A053698, A069778, A000578, A002414, A081423, A081435, A081436, A081437, A081438, A081441.
Antidiagonals are composed of n-gonal numbers.

Flat(List([0..10], n-> List([1..n+1], k-> k*((n-2)*k-(n-4))/2 ))); # G. C. Greubel, Aug 14 2019

[[k*((n-2)*k-(n-4))/2: k in [1..n+1]]: n in [0..10]]; // G. C. Greubel, Oct 13 2018

Table[PolygonalNumber[n,i],{n,0,10},{i,n+1}]//Flatten (* Requires Mathematica version 10.4 or later *) (* Harvey P. Dale, Aug 27 2016 *)

tabl(nn) = {for (n=0, nn, for (k=1, n+1, print1(k*((n-2)*k-(n-4))/2, ", ");); print(););} \\ Michel Marcus, Jun 22 2015

[[k*((n-2)*k -(n-4))/2 for k in (1..n+1)] for n in (0..10)] # G. C. Greubel, Aug 14 2019

Array of coefficients of x in the expansions of T(k, x) = (1 + k*x -(k-2)*x^2)/(1-x)^4, k > -4.

T(n, k) = k*((n-2)*k -(n-4))/2 (see MathWorld link). - Michel Marcus, Jun 22 2015

A152618 a(n) = (n-1)^2*(n+1).

Original entry on oeis.org

1, 0, 3, 16, 45, 96, 175, 288, 441, 640, 891, 1200, 1573, 2016, 2535, 3136, 3825, 4608, 5491, 6480, 7581, 8800, 10143, 11616, 13225, 14976, 16875, 18928, 21141, 23520, 26071, 28800, 31713, 34816, 38115, 41616, 45325, 49248, 53391, 57760, 62361

Offset: 0

Philippe Deléham, Dec 09 2008

For n>0 this is the same under substitution of variables as d(d-2)^2, the number of connected components in Bertrand et al.: "We construct a polynomial of degree d in two variables whose Hessian curve has (d-2)^2 connected components using Viro patchworking. In particular, this implies the existence of a smooth real algebraic surface of degree d in RP^3 whose parabolic curve is smooth and has d(d-2)^2 connected components." - Jonathan Vos Post, Apr 30 2009

For n>0 a(n) is twice the area of the trapezoid created by plotting the four points (n-1,n), (n,n-1), (n*(n-1)/2,n*(n+1)/2), (n*(n+1)/2,(n-1)*n/2). - J. M. Bergot, Mar 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Benoît Bertand and Erwan Brugallé, On the number of connected components of the parabolic curve, Comptes Rendus Mathématique, Vol. 348, No. 5-6 (2010), pp. 287-289; arXiv preprint, arXiv:0904.4652 [math.AG], Apr 29 2009. - _Jonathan Vos Post_, Apr 30 2009
Jim Propp and Adam Propp-Gubin, Counting Triangles in Triangles, arXiv:2409.17117 [math.CO], 2024. See p. 9.

[(n-1)^2*(n+1): n in [0..50]]; // Vincenzo Librandi, Jun 25 2013

A152618:=n->(n-1)^2*(n+1); seq(A152618(k), k=0..100); # Wesley Ivan Hurt, Oct 06 2013

f[n_]:=(n-1)^2*(n+1);f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011*)
CoefficientList[Series[(9 x^2 - 4 x + 1)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 25 2013 *)

a(n)=(n+1)*(n-1)^2 \\ Charles R Greathouse IV, Mar 21 2014

a(n) = n^3 - n^2 - n + 1 = A083074(n) + 2. - Jeremy Gardiner, Jun 23 2013
G.f.: (9*x^2 - 4*x + 1)/(1-x)^4. - Vincenzo Librandi, Jun 25 2013
a(n+1) = A005449(n) + A002414(n), n > 0. - Wesley Ivan Hurt, Oct 06 2013
Sum_{n>1} 1/a(n) = (1/24) * (2*Pi^2 - 9). - Enrique Pérez Herrero, May 31 2015
Sum_{n>=2} (-1)^n/a(n) = (Pi^2 - 3)/24. - Amiram Eldar, Jan 13 2021
E.g.f.: exp(x)*(x^3+2*x^2-x+1). - Nikolaos Pantelidis, Feb 06 2023

A157704 G.f.s of the z^p coefficients of the polynomials in the GF3 denominators of A156927.

Original entry on oeis.org

1, 1, 5, 32, 186, 132, 10, 56, 2814, 17834, 27324, 11364, 1078, 10, 48, 17988, 494720, 3324209, 7526484, 6382271, 2004296, 203799, 4580, 5, 16, 72210, 7108338, 146595355, 1025458635, 2957655028, 3828236468

Offset: 0

Johannes W. Meijer, Mar 07 2009

The formula for the PDGF3(z;n) polynomials in the GF3 denominators of A156927 can be found below.
The general structure of the GFKT3(z;p) that generate the z^p coefficients of the PDGF3(z; n) polynomials can also be found below. The KT3(z;p) polynomials in the numerators of the GFKT3(z; p) have a nice symmetrical structure.
The sequence of the number of terms of the first few KT3(z;p) polynomials is 1, 2, 4, 7, 10, 13, 14, 17, 20, 23, 26, 29, 32, 34, 36, 39, 42. The differences of this sequence and that of the number of terms of the KT4(z;p), see A157705, follow a simple pattern.
A Maple algorithm that generates relevant GFKT3(z;p) information can be found below.

			Some PDGF3 (z;n) are:
  PDGF3(z;n=3) = (1-z)*(1-2*z)^4*(1-3*z)^7*(1-4*z)^10
  PDGF3(z;n=4) = (1-z)*(1-2*z)^4*(1-3*z)^7*(1-4*z)^10*(1-5*z)^13
The first few GFKT3's are:
  GFKT3(z;p=0) = 1/(1-z)
  GFKT3(z;p=1) = -(5*z+1)/(1-z)^4
  GFKT3(z;p=2) = z*(32+186*z+132*z^2+10*z^3)/(1-z)^7
Some KT3(z,p) polynomials are:
  KT3(z;p=2) = 32+186*z+132*z^2+10*z^3
  KT3(z;p=3) = 56+2814*z+17834*z^2+27324*z^3+11364*z^4+1078*z^5+10*z^6

Originator sequence A156927.
See A002414 for the z^1 coefficients and A157707 for the z^2 coefficients divided by 2.
Row sums equal A064350 and those of A157705.
Cf. A157702, A157703, A157705.

p:=2; fn:=sum((-1)^(n1+1)*binomial(3*p+1,n1) *a(n-n1),n1=1..3*p+1): fk:=rsolve(a(n) = fn,a(k)): for n2 from 0 to 3*p+1 do fz(n2):=product((1-(k+1)*z)^(1+3*k), k=0..n2): a(n2):= coeff(fz(n2),z,p); end do: b:=n-> a(n): seq(b(n), n=0..3*p+1); a(n)=fn; a(k)=sort(simplify(fk)); GFKT3(p):=sum((fk)*z^k, k=0..infinity); q3:=ldegree((numer(GFKT3(p)))): KT3(p):=sort((-1)^(p)*simplify((GFKT3(p)*(1-z)^(3*p+1))/z^q3),z, ascending);

PDGF3(z;n) = Product_{k=0..n} (1-(k+1)*z)^(1+3*k) with n = 1, 2, 3, ...
GFKT3(z;p) = (-1)^(p)*(z^q3)*KT3(z, p)/(1-z)^(3*p+1) with p = 0, 1, 2, ...
The recurrence relation for the z^p coefficients a(n) is a(n) = Sum_{k=1..3*p+1} (-1)^(k+1)*binomial(3*p + 1, k)*a(n-k) with p = 0, 1, 2, ... .

A213820 Principal diagonal of the convolution array A213819.

Original entry on oeis.org

2, 18, 60, 140, 270, 462, 728, 1080, 1530, 2090, 2772, 3588, 4550, 5670, 6960, 8432, 10098, 11970, 14060, 16380, 18942, 21758, 24840, 28200, 31850, 35802, 40068, 44660, 49590, 54870, 60512, 66528, 72930, 79730, 86940, 94572, 102638, 111150, 120120, 129560, 139482

Offset: 1

Clark Kimberling, Jul 04 2012

Every term is even: a(n) = 2*A002414(n).

a(n) is the first Zagreb index of the graph obtained by joining one vertex of a complete graph K[n] with each vertex of a second complete graph K[n]. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph. - Emeric Deutsch, Nov 07 2016

Clark Kimberling, Table of n, a(n) for n = 1..1000
Ivan Gutman and Kinkar C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002414, A213819, A328910.

(See A213819.)
a[n_] := 2*n^3 + n^2 - n; Array[a, 50] (* Amiram Eldar, Mar 12 2023 *)

a(n) = -n + n^2 + 2*n^3.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: f(x)/g(x), where f(x) = 2*x*(1 + 5*x) and g(x) = (1-x)^4.
From Amiram Eldar, Mar 12 2023: (Start)
Sum_{n>=1} 1/a(n) = (4*log(2) - 1)/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi - 4*log(2) + 1)/3. (End)

A220084 a(n) = (n + 1)(20n^2 + 19*n + 6)/6.

Original entry on oeis.org

1, 15, 62, 162, 335, 601, 980, 1492, 2157, 2995, 4026, 5270, 6747, 8477, 10480, 12776, 15385, 18327, 21622, 25290, 29351, 33825, 38732, 44092, 49925, 56251, 63090, 70462, 78387, 86885, 95976, 105680, 116017, 127007, 138670, 151026, 164095, 177897, 192452

Offset: 0

Bruno Berselli, Dec 11 2012

Sequence related to heptagonal pyramidal numbers (A002413) by a(n) = n*A002413(n) - (n-1)*A002413(n-1).
Other sequences of numbers of the form m*P(k,m)-(m-1)*P(k,m-1), where P(k,m) is the m-th k-gonal pyramidal number:
k=3, A002412(m) = m*A000292(m)-(m-1)*A000292(m-1);
k=4, A051662(m) = (m+1)*A000330(m+1)-m*A000330(m);
k=5, A213772(m) = m*A002411(m)-(m-1)*A002411(m-1);
k=6, A213837(m) = m*A002412(m)-(m-1)*A002412(m-1);
k=7, this sequence;
k=8, A130748(m) = m*A002414(m)-(m-1)*A002414(m-1).
Also, first bisection of A212983.
Binomial transform of (1, 14, 33, 20, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2015

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A000330, A000566, A002411, A002412, A002413, A002414, A051662, A130748, A212983, A213772, A213837.

[(n+1)*(20*n^2+19*n+6)/6: n in [0..40]]; // Bruno Berselli, Jun 28 2016

/* By first comment: */  A002413:=func; [n*A002413(n)-(n-1)*A002413(n-1): n in [1..40]];

I:=[1,15,62,162]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

Table[(n + 1) (20 n^2 + 19 n + 6)/6, {n, 0, 40}]
LinearRecurrence[{4,-6,4,-1},{1,15,62,162},40] (* Harvey P. Dale, Dec 23 2012 *)
CoefficientList[Series[(1 + 11 x + 8 x^2) / (1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

makelist((n+1)*(20*n^2+19*n+6)/6, n, 0, 20); /* Martin Ettl, Dec 12 2012 */

a(n)=(n+1)*(20*n^2+19*n+6)/6 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1+11*x+8*x^2)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), for n>3, a(0)=1, a(1)=15, a(2)=62, a(3)=162. - Harvey P. Dale, Dec 23 2012
a(n) = (n+1)*A000566(n+1) + Sum_{i=0..n} A000566(i). - Bruno Berselli, Dec 18 2013
E.g.f.: exp(x)*(6 + 84*x + 99*x^2 + 20*x^3)/6. - Elmo R. Oliveira, Aug 06 2025

A279219 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(k+1)(2*k-1)/2).

Original entry on oeis.org

1, 1, 10, 40, 155, 560, 2051, 7080, 24064, 79370, 257067, 815593, 2545201, 7812699, 23639459, 70551216, 207932549, 605611061, 1744513262, 4973116444, 14038641287, 39263308551, 108849552289, 299248060986, 816159923366, 2209102273109, 5936069692320, 15840122529455, 41987363787469, 110584436073149

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Euler transform of the octagonal pyramidal numbers (A002414).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index to sequences related to pyramidal numbers

Cf. A000335, A002414, A279215, A279216, A279217, A279218.

nmax=29; CoefficientList[Series[Product[1/(1 - x^k)^(k (k + 1) (2 k - 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(2*k-1)/2).

a(n) ~ exp(-Zeta'(-1)/2 - Zeta(3)/(8*Pi^2) - Pi^16/(671846400000*Zeta(5)^3) - Pi^8*Zeta(3)/(5184000*Zeta(5)^2) - Zeta(3)^2/(240*Zeta(5)) + Zeta'(-3) + (Pi^12/(388800000*2^(3/5)*3^(1/5)*Zeta(5)^(11/5)) + Pi^4*Zeta(3)/(3600*2^(3/5) * 3^(1/5)*Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8/(432000*2^(1/5)*3^(2/5)*Zeta(5)^(7/5)) - Zeta(3)/(2^(11/5)*(3*Zeta(5))^(2/5))) * n^(2/5) + (Pi^4/(180*2^(4/5)*(3*Zeta(5))^(3/5))) * n^(3/5) + ((5*(3*Zeta(5))^(1/5))/(2^(7/5))) * n^(4/5)) * (3*Zeta(5))^(9/100) / (2^(23/100) * sqrt(5*Pi) * n^(59/100)). - Vaclav Kotesovec, Dec 08 2016

A322637 Numbers that are sums of consecutive octagonal numbers (A000567).

Original entry on oeis.org

0, 1, 8, 9, 21, 29, 30, 40, 61, 65, 69, 70, 96, 105, 126, 133, 134, 135, 161, 176, 201, 222, 225, 229, 230, 231, 280, 294, 309, 334, 341, 355, 363, 364, 401, 405, 408, 470, 481, 505, 510, 531, 534, 539, 540, 560, 621, 630, 645, 681, 695, 735, 736, 749, 756, 764, 765, 814, 833, 846

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Octagonal Number

Cf. A000567, A002414, A034705, A034706, A319184, A319185, A322636.

N:= 1000: # for terms up to N
Octa:= [seq(n*(3*n-2),n=0..floor((1+sqrt(1+3*N))/3))]:
PS:= ListTools:-PartialSums(Octa):
S:= select(`<=`,{0,seq(seq(PS[i]-PS[j],j=1..i-1),i=1..nops(PS))},N):
sort(convert(S,list)); # Robert Israel, May 22 2025

terms = 60;
nmax = 17; kmax = 9; (* empirical *)
T = Table[n(3n-2), {n, 0, nmax}];
Union[T, Table[k MovingAverage[T, k], {k, 2, kmax}]//Flatten][[1 ;; terms]] (* Jean-François Alcover, Dec 26 2018 *)

A130748 Place n points on each of the three sides of a triangle, 3n points in all; a(n) = number of nondegenerate triangles that can be constructed using these points (plus the 3 original vertices) as vertices.

Original entry on oeis.org

17, 72, 190, 395, 711, 1162, 1772, 2565, 3565, 4796, 6282, 8047, 10115, 12510, 15256, 18377, 21897, 25840, 30230, 35091, 40447, 46322, 52740, 59725, 67301, 75492, 84322, 93815, 103995, 114886, 126512, 138897, 152065, 166040, 180846, 196507, 213047, 230490

Offset: 1

Denis Borris, Jul 12 2007

Define b(1)=1 and b(n) = a(n-1) for n>1. Then (b(n)) is the principal diagonal of the convolution array A213833. - Clark Kimberling, Jul 04 2012

			5 points are put on each side of a triangle (n = 5); we then have 18 vertices to construct with: 5 * 3 + 3 originals. The number of total arrangements = combi(18,3) : combi[3(n+1),3]. But these include degenerates along the 3 sides: 7 points on each side, so combi(7,3) on each side : 3 * combi[n+2, 3] combi[18,3] - 3 * combi[7,3] = 816 - 105 = 711.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Jim Propp and Adam Propp-Gubin, Counting Triangles in Triangles, arXiv:2409.17117 [math.CO], 2024. See p. 9.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002414, A213833, A220084 (for a list of numbers of the form n*P(k,n)-(n-1)*P(k,n-1), where P(k,n) is the n-th k-gonal pyramidal number).

Cf. A260260 (comment). [Bruno Berselli, Jul 22 2015]

A130748:=n->(8*n^3 + 15*n^2 + 9*n + 2)/2; seq(A130748(n), n=1..100); # Wesley Ivan Hurt, Jan 28 2014

Table[(8 n^3 + 15 n^2 + 9 n + 2)/2, {n, 100}] (* Wesley Ivan Hurt, Jan 28 2014 *)

a(n)=4*n^3+n*(15*n+9)/2+1 \\ Charles R Greathouse IV, Feb 14 2013

a(n) = C(3*(n+1),3) - 3*C(n+2,3) where n>0.
a(n) = (n+1)*A002414(n+1) - n*A002414(n). - Bruno Berselli, Dec 11 2012
a(n) = (8*n^3 + 15*n^2 + 9*n + 2)/2. - Charles R Greathouse IV, Feb 14 2013
From Elmo R. Oliveira, Aug 25 2025: (Start)
G.f.: x*(17 + 4*x + 4*x^2 - x^3)/(x-1)^4.
E.g.f.: -1 + exp(x)*(2 + 32*x + 39*x^2 + 8*x^3)/2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4. (End)

More terms from Wesley Ivan Hurt, Jan 28 2014

cp's OEIS Frontend

A059722 a(n) = n*(2*n^2 - 2*n + 1).

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A144823 Square array A(n,k), n>=1, k>=1, read by antidiagonals, with A(1,k)=1 and sequence a_k of column k shifts left when Dirichlet convolution with a_k (DC:(b,a_k)->a) applied k times.

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A081422 Triangle read by rows in which row n consists of the first n+1 n-gonal numbers.

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A152618 a(n) = (n-1)^2*(n+1).

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A157704 G.f.s of the z^p coefficients of the polynomials in the GF3 denominators of A156927.

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A213820 Principal diagonal of the convolution array A213819.

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A220084 a(n) = (n + 1)*(20*n^2 + 19*n + 6)/6.

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A059722 a(n) = n(2n^2 - 2*n + 1).

A220084 a(n) = (n + 1)(20n^2 + 19*n + 6)/6.

A279219 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(k+1)(2*k-1)/2).