A057813 -id:A057813 - cp's OEIS frontend

A063494 a(n) = (2n - 1)(7n^2 - 7n + 3)/3.

1, 17, 75, 203, 429, 781, 1287, 1975, 2873, 4009, 5411, 7107, 9125, 11493, 14239, 17391, 20977, 25025, 29563, 34619, 40221, 46397, 53175, 60583, 68649, 77401, 86867, 97075, 108053, 119829, 132431, 145887, 160225, 175473, 191659, 208811, 226957, 246125, 266343, 287639

Offset: 1

N. J. A. Sloane, Aug 01 2001

Interpret A176271 as an infinite square array read by antidiagonals, with rows 1,5,11,19,...; 3,9,17,27,... and so on. The sum of the terms in the n X n upper submatrix are s(n) = 1, 18, 93, 296, ... = n^2*(7*n^2-1)/6, and a(n) = s(n) - s(n-1) are the first differences. - J. M. Bergot, Jun 27 2013

Harry J. Smith, Table of n, a(n) for n = 1..1000
T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (10).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

[(2*n - 1)*(7*n^2 - 7*n + 3)/3: n in [1..30]]; // G. C. Greubel, Dec 01 2017

Table[(2*n - 1)*(7*n^2 - 7*n + 3)/3, {n,1,30}] (* or *) LinearRecurrence[{4,-6,4,-1}, {1,17,75,203}, 30] (* G. C. Greubel, Dec 01 2017 *)

a(n) = { (2*n - 1)*(7*n^2 - 7*n + 3)/3 } \\ Harry J. Smith, Aug 23 2009

my(x='x+O('x^30)); Vec(serlaplace((-3+6*x+21*x^2+14*x^3)*exp(x)/3 + 1)) \\ G. C. Greubel, Dec 01 2017

G.f.: x*(1+x)*(1+12*x+x^2)/(1-x)^4. - Colin Barker, Mar 02 2012
E.g.f.: (-3 + 6*x + 21*x^2 + 14*x^3)*exp(x)/3 + 1. - G. C. Greubel, Dec 01 2017
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, May 11 2023

A063491 a(n) = (2n - 1)(3n^2 - 3n + 2)/2.

Original entry on oeis.org

1, 12, 50, 133, 279, 506, 832, 1275, 1853, 2584, 3486, 4577, 5875, 7398, 9164, 11191, 13497, 16100, 19018, 22269, 25871, 29842, 34200, 38963, 44149, 49776, 55862, 62425, 69483, 77054, 85156, 93807, 103025, 112828, 123234, 134261, 145927, 158250, 171248, 184939

Offset: 1

N. J. A. Sloane, Aug 01 2001

A triangle has sides of lengths 6*n-3, 6*n^2-6*n+4, and 6*n^2-6*n+7; for n>2 its area is 6*sqrt(a(n)^2 - 1). - J. M. Bergot, Aug 30 2013

[The source of this is using (n,n+1), (n+1,n+2), and (n+2,n+3) as (a,b) in the creation of three Pythagorean triangles with sides b^2-a^2, 2*a*b, and a^2+b^2. Combine the three respective sides to create a new larger triangle, then find its area. It is not simply working backwards from the sequence. As well, the sequence has this as its first comment to show that the numbers are actually doing something to find a solution.]

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Harry J. Smith, Table of n, a(n) for n = 1..1000
T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (10).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

Cf. A005448, A004188.

[(2*n-1)*(3*n^2 -3*n +2)/2: n in [1..30]]; // G. C. Greubel, Dec 01 2017

LinearRecurrence[{4,-6,4,-1},{1,12,50,133},40] (* Harvey P. Dale, Jun 05 2016 *)
Table[(2*n-1)*(3*n^2 -3*n +2)/2, {n,1,30}] (* G. C. Greubel, Dec 01 2017 *)

a(n) = { (2*n - 1)*(3*n^2 - 3*n + 2)/2 } \\ Harry J. Smith, Aug 23 2009

my(x='x+O('x^30)); Vec(serlaplace((-2 + 4*x + 9*x^2 + 6*x^3)*exp(x)/2 + 1)) \\ G. C. Greubel, Dec 01 2017

a <- c(0, 1, 9, 38, 110)
for(n in (length(a)+1):40)
  a[n] <- +4*a[n-1]-6*a[n-2]+4*a[n-3]-a[n-4]
a [Yosu Yurramendi, Sep 04 2013]

G.f.: x*(1+x)*(1+7*x+x^2)/(1-x)^4. - Colin Barker, Apr 20 2012
a(n) = +4*a(n-1) -6*a(n-2) +4*a(n-3) -1*a(n-4) n > 3, a(1)=1, a(2)=12, a(3)=50, a(4)=133. - Yosu Yurramendi, Sep 04 2013
E.g.f.: (-2 + 4*x + 9*x^2 + 6*x^3)*exp(x)/2 + 1. - G. C. Greubel, Dec 01 2017
From Bruce J. Nicholson, Jun 17 2020: (Start)
a(n) = A005448(n) * A005408(n-1).
a(n) = A004188(n) + A004188(n-1). (End)

A063492 a(n) = (2n - 1)(11n^2 - 11n + 6)/6.

Original entry on oeis.org

1, 14, 60, 161, 339, 616, 1014, 1555, 2261, 3154, 4256, 5589, 7175, 9036, 11194, 13671, 16489, 19670, 23236, 27209, 31611, 36464, 41790, 47611, 53949, 60826, 68264, 76285, 84911, 94164, 104066, 114639, 125905, 137886, 150604, 164081, 178339, 193400, 209286, 226019

Offset: 1

N. J. A. Sloane, Aug 01 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (10).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/12*t*(2*n^3 - 3*n^2 + n) + 2*n - 1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

[(2*n-1)*(11*n^2-11*n+6)/6: n in [1..40]]; // Vincenzo Librandi, Dec 16 2015

A063492:=n->(2*n - 1)*(11*n^2 - 11*n + 6)/6: seq(A063492(n), n=1..50); # Wesley Ivan Hurt, Dec 16 2015

Table[(2*n-1)*(11*n^2-11*n+6)/6, {n, 5!}] (* Vladimir Joseph Stephan Orlovsky, Sep 18 2008 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 14, 60, 161}, 40] (* Vincenzo Librandi, Dec 16 2015 *)

a(n) = { (2*n - 1)*(11*n^2 - 11*n + 6)/6 } \\ Harry J. Smith, Aug 23 2009

Vec(x*(1+x)*(1+9*x+x^2)/(1-x)^4 + O(x^100)) \\ Altug Alkan, Dec 16 2015

A063492_list, m = [], [22, -11, 2, 1]
for _ in range(10**2):
    A063492_list.append(m[-1])
    for i in range(3):
        m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015

G.f.: x*(1+x)*(1 + 9*x + x^2)/(1-x)^4. - Colin Barker, Apr 24 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4. - Wesley Ivan Hurt, Dec 16 2015
E.g.f.: (-6 + 12*x + 33*x^2 + 22*x^3)*exp(x)/6 + 1. - G. C. Greubel, Dec 01 2017

A063493 a(n) = (2n-1)(13n^2-13n+6)/6.

Original entry on oeis.org

1, 16, 70, 189, 399, 726, 1196, 1835, 2669, 3724, 5026, 6601, 8475, 10674, 13224, 16151, 19481, 23240, 27454, 32149, 37351, 43086, 49380, 56259, 63749, 71876, 80666, 90145, 100339, 111274, 122976, 135471, 148785, 162944, 177974, 193901, 210751, 228550, 247324, 267099

Offset: 1

N. J. A. Sloane, Aug 01 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (10).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

[(2*n-1)*(13*n^2-13*n+6)/6: n in [1..40]]; // Vincenzo Librandi, Dec 16 2015

Table[(2 n - 1) (13 n^2 - 13 n + 6)/6, {n, 1, 40}] (* Bruno Berselli, Dec 16 2015 *)
LinearRecurrence[{4,-6,4,-1}, {1,16,70,189}, 30] (* G. C. Greubel, Dec 01 2017 *)

a(n) = { (2*n - 1)*(13*n^2 - 13*n + 6)/6 } \\ Harry J. Smith, Aug 23 2009

my(x='x+O('x^30)); Vec(serlaplace((-6+12*x+39*x^2+26*x^3)*exp(x)/6 + 1)) \\ G. C. Greubel, Dec 01 2017

A063493_list, m = [], [26, -13, 2, 1]
for _ in range(10**2):
    A063493_list.append(m[-1])
    for i in range(3):
        m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015

G.f.: x*(1+x)*(1+11*x+x^2)/(1-x)^4. - Colin Barker, Apr 20 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, Dec 16 2015
E.g.f.: (-6 + 12*x + 39*x^2 + 26*x^3)*exp(x)/6 + 1. - G. C. Greubel, Dec 01 2017

A063495 a(n) = (2n-1)(5n^2-5n+2)/2.

Original entry on oeis.org

1, 18, 80, 217, 459, 836, 1378, 2115, 3077, 4294, 5796, 7613, 9775, 12312, 15254, 18631, 22473, 26810, 31672, 37089, 43091, 49708, 56970, 64907, 73549, 82926, 93068, 104005, 115767, 128384, 141886, 156303, 171665, 188002, 205344, 223721, 243163, 263700, 285362

Offset: 1

N. J. A. Sloane, Aug 01 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000
T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (10).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

[(2*n-1)*(5*n^2-5*n+2)/2: n in [1..30]]; // G. C. Greubel, Dec 01 2017

Table[(2n-1)(5n^2-5n+2)/2,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,18,80,217},40] (* Harvey P. Dale, Dec 18 2011 *)

a(n) = (2*n - 1)*(5*n^2 - 5*n + 2)/2 \\ Harry J. Smith, Aug 23 2009

my(x='x+O('x^30)); Vec(serlaplace((-2+4*x+15*x^2+10*x^3)*exp(x)/2 + 1)) \\ G. C. Greubel, Dec 01 2017

From Harvey P. Dale, Dec 18 2011: (Start)
a(1)=1, a(2)=18, a(3)=80, a(4)=217, a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) - a(n-4).
G.f.: (x^3+14*x^2+14*x+1)/(1-x)^4. (End)
E.g.f.: (-2 + 4*x + 15*x^2 + 10*x^3)*exp(x)/2 + 1. - G. C. Greubel, Dec 01 2017

A049480 a(n) = (2n-1)(n^2 -n +6)/6.

Original entry on oeis.org

1, 4, 10, 21, 39, 66, 104, 155, 221, 304, 406, 529, 675, 846, 1044, 1271, 1529, 1820, 2146, 2509, 2911, 3354, 3840, 4371, 4949, 5576, 6254, 6985, 7771, 8614, 9516, 10479, 11505, 12596, 13754, 14981, 16279, 17650, 19096, 20619, 22221

Offset: 1

N. J. A. Sloane, Aug 01 2001

Harvey P. Dale, Table of n, a(n) for n = 1..1000
T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (10).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

[(2*n-1)*(n^2-n+6)/6: n in [1..30]]; // G. C. Greubel, Dec 01 2017

Table[(2n-1)(n^2-n+6)/6,{n,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,4,10,21},50] (* Harvey P. Dale, Jan 01 2012 *)

a(n)=(2*n-1)*(n^2-n+6)/6 \\ Charles R Greathouse IV, Sep 24 2015

x='x+O('x^30); Vec(serlaplace((-6 + 12*x + 3*x^2 + 2*x^3)*exp(x)/6 + 1)) \\ G. C. Greubel, Dec 01 2017

From Harvey P. Dale, Jan 01 2012: (Start)
G.f.: x*(x^3 + 1)/(x-1)^4.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4); a(1)=1, a(2)=4, a(3)=10, a(4)=21. (End)
E.g.f.: (-6 + 12*x + 3*x^2 + 2*x^3)*exp(x)/6 + 1. - G. C. Greubel, Dec 01 2017

A132366 Partial sum of centered tetrahedral numbers A005894.

Original entry on oeis.org

1, 6, 21, 56, 125, 246, 441, 736, 1161, 1750, 2541, 3576, 4901, 6566, 8625, 11136, 14161, 17766, 22021, 27000, 32781, 39446, 47081, 55776, 65625, 76726, 89181, 103096, 118581, 135750, 154721, 175616, 198561, 223686, 251125, 281016, 313501, 348726, 386841

Offset: 0

Jonathan Vos Post, Nov 09 2007

From Robert A. Russell, Oct 09 2020: (Start)
a(n-1) is the number of achiral colorings of the 5 tetrahedral facets (or vertices) of a regular 4-dimensional simplex using n or fewer colors. An achiral arrangement is identical to its reflection. The 4-dimensional simplex is also called a 5-cell or pentachoron. Its Schläfli symbol is {3,3,3}.
There are 60 elements in the automorphism group of the 4-dimensional simplex that are not in its rotation group. Each is an odd permutation of the vertices and can be associated with a partition of 5 based on the conjugacy class of the permutation. The first formula for a(n-1) is obtained by averaging their cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Partition  Count  Odd Cycle Indices
  41         30     x_1x_4^1
  32         20     x_2^1x_3^1
  2111       10     x_1^3x_2^1 (End)

Nathaniel Johnston, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000292, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.
Cf. A337895 (oriented), A000389(n+4) (unoriented), A000389 (chiral), A331353 (5-cell edges, faces), A337955 (8-cell vertices, 16-cell facets), A337958 (16-cell vertices, 8-cell facets), A338951 (24-cell), A338967 (120-cell, 600-cell).
a(n-1) = A325001(4,n).

Do[Print[n, " ", (n^4 + 4 n^3 + 11 n^2 + 14 n + 6)/6 ], {n, 0, 10000}]
Accumulate[Table[(2n+1)(n^2+n+3)/3,{n,0,40}]] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,6,21,56,125},40] (* Harvey P. Dale, Feb 26 2020 *)

a(n) = (n^4 + 4*n^3 + 11*n^2 + 14*n + 6)/6 = (n^2+2*n+6)*(n+1)^2/6.
G.f.: -(x+1)*(x^2+1) / (x-1)^5. - Colin Barker, May 04 2013
From Robert A. Russell, Oct 09 2020: (Start)
a(n-1) = n^2 * (5 + n^2) / 6.
a(n-1) = binomial(n+4,5) - binomial(n,5) = A000389(n+4) - A000389(n).
a(n-1) = 1*C(n,1) + 4*C(n,2) + 6*C(n,3) + 4*C(n,4), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n-1) = 2*A000389(n+4) - A337895(n) = A337895(n) - 2*A000389(n) .
G.f. for a(n-1): x * (x+1) * (x^2+1) / (1-x)^5. (End)
From Amiram Eldar, Feb 14 2023: (Start)
Sum_{n>=0} 1/a(n) = Pi^2/5 + 3/25 - 3*Pi*coth(sqrt(5)*Pi)/(5*sqrt(5)).
Sum_{n>=0} (-1)^n/a(n) = Pi^2/10 - 3/25 + 3*Pi*cosech(sqrt(5)*Pi)/(5*sqrt(5)). (End)
a(n) = A006007(n) + A006007(n+1) = A002415(n) + A002415(n+2). - R. J. Mathar, Jun 05 2025

Corrected offset, Mathematica program by Tomas J. Bulka (tbulka(AT)rodincoil.com), Sep 02 2009

A167875 One third of product plus sum of three consecutive nonnegative integers; a(n)=(n+1)(n^2+2n+3)/3.

Original entry on oeis.org

1, 4, 11, 24, 45, 76, 119, 176, 249, 340, 451, 584, 741, 924, 1135, 1376, 1649, 1956, 2299, 2680, 3101, 3564, 4071, 4624, 5225, 5876, 6579, 7336, 8149, 9020, 9951, 10944, 12001, 13124, 14315, 15576, 16909, 18316, 19799, 21360, 23001, 24724, 26531

Offset: 0

Klaus Brockhaus, Nov 14 2009

a(n) = ((n*(n+1)*(n+2))+(n+(n+1)+(n+2)))/3, n >= 0.
Equals A006527 without initial term 0: a(n) = A006527(n+1).
Binomial transform of A167876.
Inverse binomial transform of A080930.
a(n) = A007290(n+2)+n+1.
a(n) = A014820(n)/(n+1) for n > 0.
a(n) = A116731(n+2)-1.
a(n) = A033547(n+1)-n.
a(n) = A054602(n)/3.
a(n) = A086514(n+3)-2.
a(n) = A002061(n+1)+a(n-1) for n > 0.
a(n) = A005894(n)-a(n-1) for n > 0.
First bisection is A057813.
Second differences are in A004277.
a(n) = A177342(n)*(-1)+a(n-1)*5 with n>0. For n=8, a(8)=-A177342(8)+a(7)*5=-631+176*5=249. - Bruno Berselli, May 18 2010

			a(0) = (0*1*2+0+1+2)/3 = (0+3)/3 = 1.
a(1) = (1*2*3+1+2+3)/3 = (6+6)/3 = 4.
a(6)-4*a(5)+6*a(4)-4*a(3)+a(2) = 119-4*76+6*45-4*24+11 = 0. - _Bruno Berselli_, May 26 2010

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. A001477 (nonnegative integers),
A006527 ((n^3+2*n)/3),
A167876 (1, 3, 4, 2, 0, 0, 0, 0, ...),
A080930,
A007290 (2*C(n, 3)),
A014820 ((1/3)*(n^2+2*n+3)*(n+1)^2),
A116731,
A033547 (n*(n^2+5)/3),
A054602 (Sum_{d|3} phi(d)*n^(3/d)),
A086514 ((n^3-6*n^2+14*n-6)/3),
A002061 (n^2-n+1),
A005894 (centered tetrahedral numbers),
A057813 ((2*n+1)*(4*n^2+4*n+3)/3),
A004277 (1 and the positive even numbers),
A028387 (n+(n+1)^2),
A166941, A166942, A166943.

[ (&*s + &+s)/3 where s is [n..n+2]: n in [0..42] ];

Select[Table[(n*(n+1)*(n+2)+n+(n+1)+(n+2))/3,{n,0,5!}],IntegerQ[#]&] (* Vladimir Joseph Stephan Orlovsky, Dec 04 2010 *)
(Times@@#+Total[#])/3&/@Partition[Range[0,65],3,1]  (* Harvey P. Dale, Mar 14 2011 *)

a(n)=(n+1)*(n^2+2*n+3)/3 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (n^3+3*n^2+5*n+3)/3.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+2 for n > 3; a(0)=1, a(1)=4, a(2)=11, a(3)=24.
G.f.: (1+x^2)/(1-x)^4.
a(n) = SUM(A109613(k)*A005408(n-k): 0<=k<=n). - Reinhard Zumkeller, Dec 05 2009
a(n)-4*a(n-1)+6*a(n-2)-4*a(n-3)+a(n-4)=0 for n>3. - Bruno Berselli, May 26 2010

A019560 Coordination sequence for C_4 lattice.

Original entry on oeis.org

1, 32, 192, 608, 1408, 2720, 4672, 7392, 11008, 15648, 21440, 28512, 36992, 47008, 58688, 72160, 87552, 104992, 124608, 146528, 170880, 197792, 227392, 259808, 295168, 333600, 375232, 420192, 468608

Offset: 0

mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de (Michael Baake)

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A103884 (row 4). For coordination sequences of other C_n lattices see A022144 (C_2), A010006 (C3), A019560 - A019564 (C_4 through C_8), A035746 - A035787 (C_9 through C_50).

Cf. A008412, A137513.

[1] cat [(32/3)*n*(1 + 2*n^2): n in [1..40]]; // Vincenzo Librandi, Apr 10 2017

Join[{1}, Table[(32/3) n (1 + 2 n^2), {n, 30}]] (* Vincenzo Librandi, Apr 10 2017 *)

a(n) = (32/3)*n*(1 + 2*n^2) for n>0.
G.f.: (1 + 28*x + 70*x^2 + 28*x^3 + x^4)/(1 - x)^4.
G.f. for sequence with interpolated zeros: cosh(8*arctanh(x)) = 1/2*(((1 + x)/(1 - x))^4 + ((1 - x)/(1 + x))^4) = 1 + 32*x^2 + 192*x^4 + 608*x^6 + .... Cf. A057813. - Peter Bala, Apr 09 2017
a(n) = A008412(2*n). - Seiichi Manyama, Jun 08 2018

A336266 Decimal expansion of (3/16)*Pi.

Original entry on oeis.org

5, 8, 9, 0, 4, 8, 6, 2, 2, 5, 4, 8, 0, 8, 6, 2, 3, 2, 2, 1, 1, 7, 4, 5, 6, 3, 4, 3, 6, 4, 9, 0, 6, 7, 9, 0, 7, 8, 6, 9, 6, 9, 2, 6, 2, 3, 8, 2, 8, 3, 2, 3, 4, 1, 4, 3, 2, 8, 0, 2, 1, 1, 1, 0, 5, 7, 7, 1, 5, 5, 7, 6, 1, 7, 8, 6, 6, 4, 1, 8, 7, 2, 4, 2, 7, 5, 6, 5

Offset: 0

Bernard Schott, Jul 15 2020

Area of one egg of the "double egg" whose polar equation is r(t) = a * cos(t)^2 and a Cartesian equation is (x^2+y^2)^3 = a^2*x^4 is equal to (3/16)*Pi * a^2. See the curve at the Mathcurve link.

			0.58904862254808623221174563436490679078696926...

L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 586.

Peter Bala, A note on A336266
Robert Ferréol, Double egg, Mathcurve.
Index to sequences related to curves.
Index entries for transcendental numbers

Cf. A259830 (length of an egg).

Cf. A019692 (2*Pi for deltoid), A122952 (3*Pi for cycloid and nephroid), A180434 (2-Pi/2 for Newton strophoid), A197723 (3*Pi/2 for cardioid), A336308.

Maple
```
evalf(3*Pi/16,140);
```

RealDigits[3*Pi/16, 10, 100][[1]] (* Amiram Eldar, Jul 15 2020 *)

PARI
```
3*Pi/16 \\ Michel Marcus, Jul 15 2020
```

Equals Integral_{t=0..Pi} (1/2) * cos(t)^4 * dt.
Equals Integral_{x=0..oo} 1/(x^2 + 1)^3 dx. - Amiram Eldar, Aug 13 2020
From Peter Bala, Mar 21 2024: (Start)
Equals 1/2 + Sum_{n >= 0} (-1)^n/(u(n)*u(-n)), where the polynomial u(n) = (2*n - 1)*(4*n^2 - 4*n + 3)/3 = A057813(n-1) has its zeros on the vertical line Re(z) = 1/2 in the complex plane. Cf. A336308.
Equals 1/2 + 1/(11 + 3/(12 + 15/(12 + 35/(12 + ... + (4*n^2 - 1)/(12 + ... ))))). See Lorentzen and Waadeland, p. 586, equation 4.7.10 with n = 2. (End)

cp's OEIS Frontend

A063494 a(n) = (2*n - 1)*(7*n^2 - 7*n + 3)/3.

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A063491 a(n) = (2*n - 1)*(3*n^2 - 3*n + 2)/2.

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A063492 a(n) = (2*n - 1)*(11*n^2 - 11*n + 6)/6.

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A063493 a(n) = (2*n-1)*(13*n^2-13*n+6)/6.

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A063495 a(n) = (2*n-1)*(5*n^2-5*n+2)/2.

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A049480 a(n) = (2*n-1)*(n^2 -n +6)/6.

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A063494 a(n) = (2n - 1)(7n^2 - 7n + 3)/3.

A063491 a(n) = (2n - 1)(3n^2 - 3n + 2)/2.

A063492 a(n) = (2n - 1)(11n^2 - 11n + 6)/6.

A063493 a(n) = (2n-1)(13n^2-13n+6)/6.

A063495 a(n) = (2n-1)(5n^2-5n+2)/2.

A049480 a(n) = (2n-1)(n^2 -n +6)/6.