A027480 -id:A027480 - cp's OEIS frontend

A281381 a(n) = n(n + 1)(4*n + 5)/2.

0, 9, 39, 102, 210, 375, 609, 924, 1332, 1845, 2475, 3234, 4134, 5187, 6405, 7800, 9384, 11169, 13167, 15390, 17850, 20559, 23529, 26772, 30300, 34125, 38259, 42714, 47502, 52635, 58125, 63984, 70224, 76857, 83895, 91350, 99234, 107559, 116337, 125580, 135300, 145509, 156219, 167442, 179190, 191475

Offset: 0

Peter M. Chema, Jan 21 2017

Shares its digital root, zero together with period 9: repeat [3, 3, 3, 6, 6, 6, 9, 9, 9] with A027480.

Final digits cycle a length period 20: repeat [0, 9, 9, 2, 0, 5, 9, 4, 2, 5, 5, 4, 4, 7, 5, 0, 4, 9, 7, 0].

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

Partial sums of A195319.

Cf. A006002, A007742, A016061, A027480, A281258.

[n*(n+1)*(4*n+5)/2 : n in [0..50]]; // Wesley Ivan Hurt, Aug 30 2022

Table[n (n + 1) (4 n + 5)/2, {n, 0, 45}] (* or *)
CoefficientList[Series[3 x (3 + x)/(1 - x)^4, {x, 0, 45}], x] (* Michael De Vlieger, Jan 21 2017 *)

concat(0, Vec(3*x*(3 + x) / (1 - x)^4 + O(x^50))) \\ Colin Barker, Jan 21 2017

a(n) = n*(n + 1)*(4*n + 5)/2 \\ Charles R Greathouse IV, Feb 01 2017

a(n) = 2*n^3 + 9*n^2/2 + 5*n/2.
a(n) = 3*A016061(n).
a(n) = A006002(n+1)*(n) - A006002(n)*(n-1).
a(n) = A007742(n)*(n - 1)/2.
From Colin Barker, Jan 21 2017: (Start)
G.f.: 3*x*(3 + x) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. (End)
From Stefano Spezia, Aug 30 2022: (Start)
E.g.f.: exp(x)*x*(18 + 21*x + 4*x^2)/2.
Sum_{n>0} 1/a(n) = 2*(20*log(8) + 10*Pi - 71)/25 = 0.1603805895595720759728288896228498341201... . (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*sqrt(2)*Pi/5 + 4*(3+sqrt(2))*log(2)/5 - 8*sqrt(2)*log(2-sqrt(2))/5 - 178/25. - Amiram Eldar, Sep 22 2022

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

Christopher Purcell, Mar 12 2018

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

S. P. Borgatti and M. G. Everett, Notions of Position in Social Network Analysis, Sociological Methodology, 22 (1992), 1-35.
C. Purcell and P. Rombach, Role colouring graphs in hereditary classes, arXiv:1802.10180 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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

Cf. A027480.

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

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

A319007 Sum of the next n nonnegative integers repeated (A004526).

Original entry on oeis.org

0, 1, 5, 14, 29, 51, 82, 124, 178, 245, 327, 426, 543, 679, 836, 1016, 1220, 1449, 1705, 1990, 2305, 2651, 3030, 3444, 3894, 4381, 4907, 5474, 6083, 6735, 7432, 8176, 8968, 9809, 10701, 11646, 12645, 13699, 14810, 15980, 17210, 18501, 19855, 21274, 22759, 24311, 25932

Offset: 1

Bruno Berselli, Sep 07 2018

After 29, all terms are composite.

			Next n nonnegative integers repeated:    Sums:
0,  ......................................   0
0, 1,  ...................................   1
1, 2, 2,  ................................   5
3, 3, 4, 4,  .............................  14
5, 5, 6, 6, 7,  ..........................  29
7, 8, 8, 9, 9, 10,  ......................  51, etc.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-7,8,-7,4,-1).

Cf. A004526, A101455, A319006.

Sum of the next n nonnegative integers: A027480.

[Integers()! (n*(n^2-2)+(-(n mod 2))^(n*(n-1)/2))/4: n in [1..50]];

a := n -> (n^3 - 2*n + (-(n mod 2))^binomial(n,2))/4;
seq(a(n), n=1..47); # Peter Luschny, Sep 09 2018

Table[(2 n (n^2 - 2) + (1 - (-1)^n) (-1)^((n-1)/2))/8, {n, 1, 50}]

concat(0, Vec(x^2*(1 + x + x^2)/((1 + x^2)*(1 - x)^4) + O(x^50))) \\ Colin Barker, Sep 10 2018

G.f.: x^2*(1 + x + x^2)/((1 + x^2)*(1 - x)^4).
a(n) = -a(-n) = 4*a(n-1) - 7*a(n-2) + 8*a(n-3) - 7*a(n-4) + 4*a(n-5) - a(n-6).
a(n) = (2*n*(n^2 - 2) + (1 - (-1)^n)*(-1)^((n-1)/2))/8.
a(n) = A319006(n) - n.
a(n) = (n^3 - 2*n + Chi(n))/4 where Chi(n) = A101455(n). - Peter Luschny, Sep 09 2018

A118963 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that have k double rises (n >= 1, k >= 0).

Original entry on oeis.org

2, 3, 3, 4, 12, 4, 5, 30, 30, 5, 6, 60, 120, 60, 6, 7, 105, 350, 350, 105, 7, 8, 168, 840, 1400, 840, 168, 8, 9, 252, 1764, 4410, 4410, 1764, 252, 9, 10, 360, 3360, 11760, 17640, 11760, 3360, 360, 10, 11, 495, 5940, 27720, 58212, 58212, 27720, 5940, 495, 11, 12

Offset: 1

Emeric Deutsch, May 07 2006

A Grand Dyck path of semilength n is a path in the half-plane x >= 0, starting at (0,0), ending at (2n,0) and consisting of steps u = (1,1) and d = (1,-1); a double rise in a Grand Dyck path is an occurrence of uu in the path.
For double rises only above the x-axis see A118964.
This is the triangle of Narayana with row n multiplied by n + 1. - Peter Luschny, May 02 2022

			T(3,2)=4 because we have uuuddd, duuudd, dduuud and ddduuu.
Triangle begins:
  2;
  3,    3;
  4,   12,    4;
  5,   30,   30,    5;
  6,   60,  120,   60,    6;
  7,  105,  350,  350,  105,    7;
  8,  168,  840, 1400,  840,  168,    8;
  9,  252, 1764, 4410, 4410, 1764,  252,    9;

Cf. A000917, A000984 (row sums), A027480, A027789, A118964, A001263.

r:=(1-z-t*z-sqrt(z^2*t^2-2*z^2*t-2*z*t+z^2-2*z+1))/2/t/z: G:=(1+r)^2/(1-t*r^2)-1: Gser:=simplify(series(G,z=0,15)): for n from 1 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=0..n-1) od; # yields sequence in triangular form
for n from 0 to 10 do seq(binomial(n,i)*binomial(n+2,n+1-i), i=0..n ); od; # Zerinvary Lajos, Nov 03 2006

T(n,1) = n(n^2 - 1)/2 (A027480).
T(n,2) = (n+1)n(n-1)^2*(n-2)/12 (A027789).
T(n,k) = ((n+1)/n)*binomial(n,k)*binomial(n,k+1).
Sum_{k>=0} k*T(n,k) = (2n-1)!/(n!(n-2)!) (A000917).
G.f.: G(t,z) = (1+r)^2/(1 - tr^2) - 1, where r = r(t,z) is the Narayana function, defined by (1+r)(1+tr)z = r, r(t,0) = 0. More generally, the g.f. H = H(t,s,u,z), where t,s and u mark double rises above, below and on the x-axis, respectively, is H = (1 + r(s,z))/(1 - z(1 + tr(t,z))(1 + ur(s,z))).
Row n is given by seq(binomial(n, k)*binomial(n+2, n+1-k), k=0..n). - Zerinvary Lajos, Nov 03 2006
T(n,k)/(n+1) = A001263(n,k). - Peter Luschny, May 02 2022

A141535 An eighth of the product of three integers surrounding the (n+1)-st prime, minus half of the product of the 3 numbers surrounding n+1.

Original entry on oeis.org

0, 3, 12, 105, 168, 444, 603, 1158, 2550, 3060, 5469, 7518, 8568, 11292, 16563, 23217, 25458, 34167, 40740, 43998, 56307, 65391, 81210, 106272, 120000, 126750, 142155, 149685, 166863, 241152

Offset: 1

Roger L. Bagula, Aug 12 2008

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A127917 := proc(n) p := ithprime(n) ; (p-1)*p*(p+1) ; end: A027480 := proc(n) n*(n+1)*(n+2)/2 ; end: A := proc(n) A127917(n+1)/8-A027480(n) ; end: for n from 1 to 40 do printf("%d,",A(n)) ; od: # R. J. Mathar, Aug 20 2008

a[n_] = (Prime[n + 1] - 1)*Prime[n + 1]*(Prime[n + 1] + 1)/8 - n*(n + 1)*(n + 2)/2; Table[a[n], {n, 1, 30}]
f[n_]:=Module[{pr=Prime[n+1],n1=n+1},(pr(pr^2-1))/8-(n1(n1^2-1))/2]; Array[f,30] (* Harvey P. Dale, May 24 2012 *)

a(n) = A127917(n+1)/8-A027480(n) = {p(n+1)-1}*p(n+1)*{p(n+1)+1}/8-n(n+1)(n+2)/2.

Edited by N. J. A. Sloane, Aug 23 2008

A147656 The arithmetic mean of the n-th and (n+1)-st cubes, rounded down.

Original entry on oeis.org

0, 4, 17, 45, 94, 170, 279, 427, 620, 864, 1165, 1529, 1962, 2470, 3059, 3735, 4504, 5372, 6345, 7429, 8630, 9954, 11407, 12995, 14724, 16600, 18629, 20817, 23170, 25694, 28395, 31279, 34352, 37620, 41089, 44765, 48654, 52762, 57095, 61659

Offset: 0

Alexander R. Povolotsky, Nov 09 2008

The terms of this sequence relate to intervals between cubes in the same fashion as terms of A002378 are related to intervals between squares.

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

Cf. A000578.

Cf. A027480, A006003.

I:=[0, 4, 17, 45]; [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..40]]; // Vincenzo Librandi, May 06 2012

seq(coeff(series(x*(x^2+x+4)/(1-x)^4,x,n+1), x, n), n = 0 .. 40); # Muniru A Asiru, Sep 11 2018

Table[(n^3+(n+1)^3-1)/2,{n,0,70}] (* Vladimir Joseph Stephan Orlovsky, May 04 2011 *)

j=[];for (n=0,40,j=concat(j,n^3+floor(((n+1)^3 - n^3)/2)));j

a(n) = n*(2*n^2+3*n+3)/2; \\ Altug Alkan, Sep 20 2018

a(n) = floor((A000578(n) + A000578(n+1))/2).
From R. J. Mathar, Nov 11 2008: (Start)
a(n) = A000578(n) + A045943(n) = n*(2n^2+3n+3)/2.
G.f.: x*(4+x+x^2)/(1-x)^4. (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, May 06 2012
a(n) = A027480(n) + A006003(n). - Bruce J. Nicholson, Jun 03 2018
From A.H.M. Smeets, Sep 10 2018: (Start)
a(n) = Sum_{k=0..n-1} (n+1)^2-k for n >= 0 with empty domain of summation for n = 0.
a(n) = n*(n+1)^2 - n*(n-1)/2 for n >= 0.
Lim_{n -> inf} a(n-1)/n^3 = 1. (End)
E.g.f.: exp(x)*(8*x + 9*x^2 + 2*x^3)/2. - Stefano Spezia, Sep 12 2018
a(n) = A081435(n)-1. - R. J. Mathar, Sep 14 2018

A157024 a(0)=1, a(n) = (3n-1)3n(3n+1)/2 for n>0.

Original entry on oeis.org

1, 12, 105, 360, 858, 1680, 2907, 4620, 6900, 9828, 13485, 17952, 23310, 29640, 37023, 45540, 55272, 66300, 78705, 92568, 107970, 124992, 143715, 164220, 186588, 210900, 237237, 265680, 296310, 329208, 364455, 402132, 442320, 485100, 530553, 578760, 629802

Offset: 0

Jaume Oliver Lafont, Feb 21 2009

nonn

Cf. A002391, A027480, A058962, A097321, A154920.

nxt[{n_,a_}]:={n+1,((3n)(3n-1)(3n+1))/2}; NestList[nxt,{0,1},40][[All,2]]/.(0->Nothing) (* Harvey P. Dale, Sep 24 2016 *)

Sum_{n>=0} 1/a(n) = log(3).
G.f.: (1+8x+63x^2+8x^3+x^4)/(1-x)^4.
a(n) = A027480(3n-1), n>0. - R. J. Mathar, Apr 07 2009
Sum_{n>=0} (-1)^n/a(n) = 4*log(2)/3. - Amiram Eldar, Feb 27 2022

A180013 Triangular array read by rows: T(n,k) = number of fixed points in the permutations of {1,2,...,n} that have exactly k cycles; n>=1, 1<=k<=n.

Original entry on oeis.org

1, 0, 2, 0, 3, 3, 0, 8, 12, 4, 0, 30, 55, 30, 5, 0, 144, 300, 210, 60, 6, 0, 840, 1918, 1575, 595, 105, 7, 0, 5760, 14112, 12992, 5880, 1400, 168, 8, 0, 45360, 117612, 118188, 60921, 17640, 2898, 252, 9, 0, 403200, 1095840, 1181240, 672840, 224490, 45360, 5460, 360, 10

Offset: 1

Geoffrey Critzer, Jan 13 2011

Row sums = n! which is the number of fixed points in all the permutations of {1,2,...,n}.
It appears that column k = 2 is A001048 (with different offset).
From Olivier Gérard, Oct 23 2012: (Start)
This is a multiple of the triangle of Stirling numbers of the first kind, A180013(n,k) = (n)*A132393(n-1,k).
Another interpretation is : T(n,n-k) is the total number of ways to insert the symbol n among the cycles of permutations of [n-1] with (n+1-k) cycles to form a canonical cycle representation of a permutation of [n]. For each cycle of length c, there are c places to insert a symbol, and for each permutation there is the possibility to create a new cycle (a fixed point).
(End)

			T(4,3)= 12 because there are 12 fixed points in the permutations of 4 that have 3 cycles: (1)(2)(4,3); (1)(3,2)(4); (1)(4,2)(3); (2,1)(3)(4); (3,1)(2)(4); (4,1)(2)(3) where the permutations are represented in their cycle notation.
1
0   2
0   3    3
0   8   12    4
0  30   55   30   5
0 144  300  210  60    6
0 840 1918 1575 595  105   7

G. C. Greubel, Table of n, a(n) for n = 1..5050

Cf. A000142, A001048. Diagonal, lower diagonal give: A000027, A027480(n+1).

egf:= k-> x * (log(1/(1-x)))^(k-1) / (k-1)!:
T:= (n,k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1..n), n=1..10); # Alois P. Heinz, Jan 16 2011
# As coefficients of polynomials:
with(PolynomialTools): with(ListTools): A180013_row := proc(n)
`if`(n=0, 1,(n+1)!*hypergeom([-n,1-x],[1],1)); CoefficientList(simplify(%),x) end: FlattenOnce([seq(A180013_row(n), n=0..9)]); # Peter Luschny, Jan 28 2016

Flatten[Table[Table[(n + 1) Abs[StirlingS1[n, k]], {k, 0, n}], {n, 0, 9}],1] (* Olivier Gérard, Oct 23 2012 *)

E.g.f.: for column k: x*(log(1/(1-x)))^(k-1)/(k-1)!.

T(n, k) = [x^k] (n+1)!*hypergeom([-n,1-x],[1],1) for n>0. - Peter Luschny, Jan 28 2016

More terms from Alois P. Heinz, Jan 16 2011

A189913 Triangle read by rows: T(n,k) = binomial(n, k) * k! / (floor(k/2)! * floor((k+2)/2)!).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 3, 1, 4, 6, 12, 2, 1, 5, 10, 30, 10, 10, 1, 6, 15, 60, 30, 60, 5, 1, 7, 21, 105, 70, 210, 35, 35, 1, 8, 28, 168, 140, 560, 140, 280, 14, 1, 9, 36, 252, 252, 1260, 420, 1260, 126, 126, 1, 10, 45, 360, 420, 2520, 1050, 4200, 630, 1260, 42

Offset: 0

Peter Luschny, May 24 2011

The triangle may be regarded a generalization of the triangle A097610:
A097610(n,k) = binomial(n,k)*(2*k)$/(k+1);
T(n,k) = binomial(n,k)*(k)$/(floor(k/2)+1).
Here n$ denotes the swinging factorial A056040(n). As A097610 is a decomposition of the Motzkin numbers A001006, a combinatorial interpretation of T(n,k) in terms of lattice paths can be expected.
T(n,n) = A057977(n) which can be seen as extended Catalan numbers.

			[0]  1
[1]  1, 1
[2]  1, 2,  1
[3]  1, 3,  3,   3
[4]  1, 4,  6,  12,  2
[5]  1, 5, 10,  30, 10,  10
[6]  1, 6, 15,  60, 30,  60,  5
[7]  1, 7, 21, 105, 70, 210, 35, 35

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
Peter Luschny, The lost Catalan numbers.

Row sums are A189912.

Cf. A097610, A057977, A001006.

/* As triangle */ [[Binomial(n,k)*Factorial(k)/(Factorial(Floor(k/2))*Factorial(Floor((k + 2)/2))): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jan 13 2018

A189913 := (n,k) -> binomial(n,k)*(k!/iquo(k,2)!^2)/(iquo(k,2)+1):
seq(print(seq(A189913(n,k),k=0..n)),n=0..7);

T[n_, k_] := Binomial[n, k]*k!/((Floor[k/2])!*(Floor[(k + 2)/2])!); Table[T[n, k], {n, 0, 10}, {k, 0, n}]// Flatten (* G. C. Greubel, Jan 13 2018 *)

{T(n,k) = binomial(n,k)*k!/((floor(k/2))!*(floor((k+2)/2))!) };
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jan 13 2018

From R. J. Mathar, Jun 07 2011: (Start)
T(n,1) = n.
T(n,2) = A000217(n-1).
T(n,3) = A027480(n-2).
T(n,4) = A034827(n). (End)

A206492 Sums of rows of the sequence of triangles with nonnegative integers and row widths defined by A004738.

Original entry on oeis.org

0, 3, 3, 9, 21, 19, 11, 25, 45, 74, 66, 49, 26, 55, 90, 134, 190, 170, 138, 97, 50, 103, 162, 230, 310, 405, 365, 310, 243, 167, 85, 173, 267, 370, 485, 615, 763, 693, 605, 502, 387, 263, 133, 269, 411, 562, 725, 903, 1099, 1316, 1204, 1071, 920, 754, 576, 389

Offset: 1

Alex Ratushnyak, Jun 28 2012

Row widths: A004738(n): 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5...
Pits: A051925(n+1): 0, 3, 11, 26, 50, 85, 133, 196, 276, 375, 495, 638...
Peak tops: A007290(n+3): 2, 8, 20, 40, 70, 112, 168, 240, 330, 440, 572...
Peak bases: A084990(n+1): 1, 6, 17, 36, 65, 106, 161, 232, 321, 430, 561...

			The sequence of triangles begins:
0
1 2
3
4 5
6 7 8
9 10
11
12 13
14 15 16
17 18 19 20
21 22 23
24 25
26
27 28
29 30 31
32 33 34 35
36 37 38 39 40
41 42 43 44
45 46 47
48 49
50
51 52

Cf. A004738, A051925, A007290, A084990.

Cf. A027480: sums of rows of a triangle with increasing row widths: 0; 1,2; 3,4,5; 6,7,8,9; ...

curSign=-1
curLength=sum=0
rowLength=topLength=1
for n in range(1232):
    sum += n
    curLength += 1
    if curLength==rowLength:
        print(sum, end=',')
        curLength = sum = 0
        if rowLength==1 or rowLength==topLength:
            curSign = -curSign
        topLength += (rowLength==1)
        rowLength += curSign

cp's OEIS Frontend

A281381 a(n) = n*(n + 1)*(4*n + 5)/2.

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

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A319007 Sum of the next n nonnegative integers repeated (A004526).

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A118963 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that have k double rises (n >= 1, k >= 0).

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A141535 An eighth of the product of three integers surrounding the (n+1)-st prime, minus half of the product of the 3 numbers surrounding n+1.

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A147656 The arithmetic mean of the n-th and (n+1)-st cubes, rounded down.

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A157024 a(0)=1, a(n) = (3n-1)*3n*(3n+1)/2 for n>0.

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A180013 Triangular array read by rows: T(n,k) = number of fixed points in the permutations of {1,2,...,n} that have exactly k cycles; n>=1, 1<=k<=n.

A281381 a(n) = n(n + 1)(4*n + 5)/2.

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

A157024 a(0)=1, a(n) = (3n-1)3n(3n+1)/2 for n>0.