Craig Ferguson - cp's OEIS frontend

A193228 Truncated octahedron with faces of centered polygons.

1, 39, 185, 511, 1089, 1991, 3289, 5055, 7361, 10279, 13881, 18239, 23425, 29511, 36569, 44671, 53889, 64295, 75961, 88959, 103361, 119239, 136665, 155711, 176449, 198951, 223289, 249535, 277761, 308039, 340441, 375039, 411905, 451111, 492729, 536831, 583489

Offset: 1

Craig Ferguson, Jul 18 2011

The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a truncated octahedron. Each iteration requires the addition of (n-2) edges and (n-1) vertices to complete the centered polygon of each face. [centered squares (A001844) and centered hexagons (A003215)]

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
OEIS, (Centered_polygons) pyramidal numbers
Wikipedia, Tetrahedral number
Wikipedia, Triangular number
Wikipedia, Centered polygonal number
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

(copy and paste the following formula =12*ROW()^3-18*ROW()^2+8*ROW()-1 fill down to desired size.)

[12*n^3-18*n^2+8*n-1: n in [1..50]]; // Vincenzo Librandi, Aug 30 2011

Table[12n^3-18n^2+8n-1,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,39,185,511},40] (* Harvey P. Dale, Aug 27 2011 *)

vector(40, n, 12*n^3 - 18*n^2 + 8*n - 1) \\ G. C. Greubel, Nov 10 2018

a(n) = 12*n^3 - 18*n^2 + 8*n - 1.
G.f.: x*(1+x)*(x^2 + 34*x + 1) / (x-1)^4. - R. J. Mathar, Aug 26 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=39, a(2)=185, a(3)=511. - Harvey P. Dale, Aug 27 2011
E.g.f.: 1 - (1 - 2*x - 18*x^2 - 12*x^3)*exp(x). - G. C. Greubel, Nov 10 2018

A193248 Truncated dodecahedron, and truncated icosahedron with faces of centered polygons.

Original entry on oeis.org

1, 93, 455, 1267, 2709, 4961, 8203, 12615, 18377, 25669, 34671, 45563, 58525, 73737, 91379, 111631, 134673, 160685, 189847, 222339, 258341, 298033, 341595, 389207, 441049, 497301, 558143, 623755, 694317, 770009, 851011, 937503, 1029665, 1127677, 1231719

Offset: 1

Craig Ferguson, Jul 19 2011

The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a truncated dodecahedron or truncated icosahedron. Each iteration requires the addition of (n-2) edges and (n-1) vertices to complete the centered polygon of each face. [centered triangles (A005448) and centered decagons (A062786)] & [centered hexagons (A003215) and centered pentagons (A005891)] respectively.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Wikipedia, Tetrahedral number
Wikipedia, Triangular number
Wikipedia, Centered polygonal number
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A003215, A005448, A005891, A062786.

[30*n^3-45*n^2+17*n-1: n in [1..50]]; // Vincenzo Librandi, Aug 30 2011

Table[30n^3-45n^2+17n-1,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,93,455,1267},40] (* Harvey P. Dale, Aug 28 2011 *)

vector(40, n, 30*n^3 - 45*n^2 + 17*n - 1) \\ G. C. Greubel, Nov 10 2018

a(n) = 30*n^3 - 45*n^2 + 17*n - 1.
G.f.: x*(1+x)*(x^2 + 88*x + 1) / (x-1)^4. - R. J. Mathar, Aug 26 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(1)=1, a(2)=93, a(3)=455, a(4)=1267. - Harvey P. Dale, Aug 28 2011
E.g.f.: 1 - (1 - 2*x - 45*x^2 - 30*x^3)*exp(x). - G. C. Greubel, Nov 10 2018

A193250 Small rhombicuboctahedron with faces of centered polygons.

Original entry on oeis.org

1, 51, 245, 679, 1449, 2651, 4381, 6735, 9809, 13699, 18501, 24311, 31225, 39339, 48749, 59551, 71841, 85715, 101269, 118599, 137801, 158971, 182205, 207599, 235249, 265251, 297701, 332695, 370329, 410699, 453901, 500031, 549185, 601459, 656949, 715751

Offset: 1

Craig Ferguson, Jul 19 2011

The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a small rhombicuboctahedron. Each iteration requires the addition of (n-2) edges and (n-1) vertices to complete the centered polygon of each face. [centered triangles (A005448) and centered squares (A001844)]

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
OEIS, (Centered_polygons) pyramidal numbers
Wikipedia, Tetrahedral number
Wikipedia, Triangular number
Wikipedia, Centered polygonal number
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A079414 (partial sums).

Cf. A005448, A001844.

copy and paste the following formula   =16 *ROW()^3-24 *ROW()^2+10*ROW()-1 fill down to desired size.

[16*n^3-24*n^2+10*n-1: n in [1..50]]; // Vincenzo Librandi, Aug 30 2011

Table[16n^3-24n^2+10n-1,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,51,245,679},40] (* Harvey P. Dale, Aug 27 2011 *)

vector(40, n, 16*n^3-24*n^2+10*n-1) \\ G. C. Greubel, Nov 10 2018

a(n) = 16*n^3 - 24*n^2 + 10*n - 1.
G.f.: x*(1+x)*(x^2 + 46*x + 1) / (x-1)^4. - R. J. Mathar, Aug 26 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=51, a(2)=245, a(3)=679. - Harvey P. Dale, Aug 27 2011
E.g.f.: 1 + (-1 + 2*x + 24*x^2 + 16*x^3)*exp(x). - G. C. Greubel, Nov 10 2018

A193251 Small rhombicosidodecahedron with faces of centered polygons.

Original entry on oeis.org

1, 123, 605, 1687, 3609, 6611, 10933, 16815, 24497, 34219, 46221, 60743, 78025, 98307, 121829, 148831, 179553, 214235, 253117, 296439, 344441, 397363, 455445, 518927, 588049, 663051, 744173, 831655, 925737, 1026659, 1134661, 1249983, 1372865, 1503547, 1642269

Offset: 1

Craig Ferguson, Jul 19 2011

The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a small rhombicosidodecahedron. Each iteration requires the addition of (n-2) edges and (n-1) vertices to complete the centered polygon of each face. [centered triangles (A005448), centered squares (A001844) and centered pentagons (A005891)]

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
OEIS Wiki, (Centered_polygons) pyramidal numbers.
Wikipedia, Tetrahedral number.
Wikipedia, Triangular number.
Wikipedia, Centered polygonal number.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A001844, A005448, A005891, A060747, A195317.

[40*n^3-60*n^2+22*n-1: n in [1..50]]; // Vincenzo Librandi, Aug 30 2011

A193251[n_] := (2*n - 1)*(20*(n - 1)*n + 1); Array[A193251, 50] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {1, 123, 605, 1687}, 50] (* Paolo Xausa, Aug 25 2025 *)

a(n)=40*n^3-60*n^2+22*n-1 \\ Charles R Greathouse IV, Oct 19 2022

a(n) = 40*n^3 - 60*n^2 + 22*n - 1.
G.f.: x*(1+x)*(x^2 + 118*x + 1)/(x-1)^4. - R. J. Mathar, Aug 26 2011
From Elmo R. Oliveira, Aug 22 2025: (Start)
E.g.f.: 1 + exp(x)*(-1 + 2*x + 60*x^2 + 40*x^3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.
a(n) = A060747(n)*A195317(n). (End)

A193218 Number of vertices in truncated tetrahedron with faces that are centered polygons.

Original entry on oeis.org

1, 21, 95, 259, 549, 1001, 1651, 2535, 3689, 5149, 6951, 9131, 11725, 14769, 18299, 22351, 26961, 32165, 37999, 44499, 51701, 59641, 68355, 77879, 88249, 99501, 111671, 124795, 138909, 154049, 170251, 187551, 205985, 225589, 246399, 268451, 291781, 316425

Offset: 1

Craig Ferguson, Jul 18 2011

The sequence starts with a central vertex and expands outward with (n-1) centered polygonal pyramids producing a truncated tetrahedron. Each iteration requires the addition of (n-2) edges and (n-1) vertices to complete the centered polygon in each face. For centered triangles see A005448 and centered hexagons A003215.
This sequence is the 18th in the series (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 and t = 36. While adjusting for offsets, the beginning sequence A049480 is generated by adding the square pyramidal numbers A000330 to the odd numbers A005408 and each subsequent sequence is found by adding another set of square pyramidals A000330. (T/2) * A000330(n) + A005408(n). At 30 * A000330 + A005408 = centered dodecahedral numbers, 36 * A000330 + A005408 = A193228 truncated octahedron and 90 * A000330 + A005408 = A193248 = truncated icosahedron and dodecahedron. All five of the "Centered Platonic Solids" numbers sequences are in this series of sequences. Also 4 out of five of the "truncated" platonic solid number sequences are in this series. - Bruce J. Nicholson, Jul 06 2018
It would be good to have a detailed description of how the sequence is constructed. Maybe in the Examples section? - N. J. A. Sloane, Sep 07 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
OEIS, (Centered_polygons) pyramidal numbers
Wikipedia, Tetrahedral number
Wikipedia, Triangular number
Wikipedia, Centered polygonal number
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A260810 (partial sums).

[6*n^3-9*n^2+5*n-1: n in [1..40]]; // Vincenzo Librandi, Aug 30 2011

Table[6 n^3 - 9 n^2 + 5 n - 1, {n, 35}] (* Alonso del Arte, Jul 18 2011 *)
CoefficientList[Series[(1+x)*(x^2+16*x+1)/(1-x)^4, {x, 0, 50}], x] (* Stefano Spezia, Sep 04 2018 *)

a(n) = 6*n^3 - 9*n^2 + 5*n - 1.
G.f.: x*(1+x)*(x^2+16*x+1) / (1-x)^4. - R. J. Mathar, Aug 26 2011
a(n) = 18 * A000330(n-1) + A005408(n-1) = A063496(n) + A006331(n-1). - Bruce J. Nicholson, Jul 06 2018

A193253 Great rhombicosidodecahedron with faces of centered polygons.

Original entry on oeis.org

1, 183, 905, 2527, 5409, 9911, 16393, 25215, 36737, 51319, 69321, 91103, 117025, 147447, 182729, 223231, 269313, 321335, 379657, 444639, 516641, 596023, 683145, 778367, 882049, 994551, 1116233, 1247455, 1388577, 1539959, 1701961, 1874943, 2059265, 2255287

Offset: 1

Craig Ferguson, Jul 19 2011

The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a great rhombicosidodecahedron. Each iteration requires the addition of n-2 edges and n-1 vertices to complete the centered polygon of each face.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000.
OEIS Wiki, (Centered_polygons) pyramidal numbers.
Eric W. Weisstein, MathWorld: Great Rhombicosidodecahedron.
Wikipedia, Tetrahedral number.
Wikipedia, Triangular number.
Wikipedia, Centered polygonal number.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A001844 (centered squares), A062786 (centered decagons), and A003215 (centered hexagons).

=60*ROW()^3-90*ROW()^2+32*ROW()-1 fill down  to desired size.

[60*n^3-90*n^2+32*n-1: n in [1..40]] // Vincenzo Librandi, Feb 18 2012

LinearRecurrence[{4, -6, 4, -1}, {1, 183, 905, 2527}, 50] (* Vincenzo Librandi, Feb 18 2012 *)
a[n_]:=60*n^3 - 90*n^2 + 32*n - 1 ; Array[a, 50] (* or *)
CoefficientList[Series[(1 + x)*(1 + 178*x + x^2)/(1 - x)^4 , {x, 0, 50}], x] (* Stefano Spezia, Sep 02 2018 *)

a(n)=60*n^3-90*n^2+32*n-1 \\ Charles R Greathouse IV, Feb 12 2012

a(n) = 60*n^3 - 90*n^2 + 32*n - 1.

G.f.: x*(1 + 179*x + 179*x^2 + x^3)/(1-x)^4 = x*(1+x)*(1 + 178*x + x^2)/(1-x)^4. - Colin Barker, Feb 12 2012

A193249 Snub dodecahedron with faces of centered polygons.

Original entry on oeis.org

1, 153, 755, 2107, 4509, 8261, 13663, 21015, 30617, 42769, 57771, 75923, 97525, 122877, 152279, 186031, 224433, 267785, 316387, 370539, 430541, 496693, 569295, 648647, 735049, 828801, 930203, 1039555, 1157157, 1283309, 1418311, 1562463, 1716065, 1879417

Offset: 1

Craig Ferguson, Jul 19 2011

The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a snub dodecahedron. Each iteration requires the addition of (n-2) edges and (n-1) vertices to complete the centered polygon of each face. [centered triangles (A005448) and centered pentagons (A005891)]

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

=50*ROW()^3-75*ROW()^2+27*ROW()-1 fill down to desired size.

[50*n^3-75*n^2+27*n-1: n in [1..34]];  // Bruno Berselli, Jul 22 2011

A193249:=n->(2*n-1)*(25*n^2-25*n+1); seq(A193249(n), n=1..50); # Wesley Ivan Hurt, Apr 30 2014

Table[(2 n - 1) (25 n^2 - 25 n + 1), {n, 50}] (* Wesley Ivan Hurt, Apr 30 2014 *)

for(n=1, 34, print1(50*n^3-75*n^2+27*n-1", "));  \\ Bruno Berselli, Jul 21 2011

a(n) = 50*n^3-75*n^2+27*n-1 = (2*n-1)*(25*n^2-25*n+1).

G.f.: x*(1+x)*(1+148*x+x^2)/(1-x)^4. - Bruno Berselli, Jul 22 2011

A193252 Great rhombicuboctahedron with faces of centered polygons.

Original entry on oeis.org

1, 75, 365, 1015, 2169, 3971, 6565, 10095, 14705, 20539, 27741, 36455, 46825, 58995, 73109, 89311, 107745, 128555, 151885, 177879, 206681, 238435, 273285, 311375, 352849, 397851, 446525, 499015, 555465, 616019, 680821, 750015, 823745, 902155, 985389, 1073591

Offset: 1

Craig Ferguson, Jul 19 2011

The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a great rhombicosidodecahedron. Each iteration requires the addition of (n-2) edge units and (n-1) vertices to complete the centered polygon of each face: centered squares, centered octagons and centered hexagons.

Bruno Berselli, Table of n, a(n) for n = 1..1000
OEIS Wiki, (Centered polygons) pyramidal numbers
Wikipedia, Tetrahedral number
Wikipedia, Triangular number
Wikipedia, Centered polygonal number
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First differences in 2*A158591.

Cf. A001844 (centered square numbers), A016754 (centered octagonal numbers), A003215 (centered hexagonal numbers).

Excel
```
=24*ROW()^3-36*ROW()^2+14*ROW()-1
```

List([1..40], n-> 24*n^3 -36*n^2 +14*n -1); # G. C. Greubel, Feb 26 2019

A069190:=func; [(2*n-1)*A069190(n): n in [1..40]];  // Bruno Berselli, Jul 21 2011

Table[24n^3-36n^2+14n-1,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,75,365,1015},40] (* Harvey P. Dale, Jul 27 2011 *)

for(n=1,40, print1(24*n^3-36*n^2+14*n-1", "));  \\ Bruno Berselli, Jul 21 2011

[24*n^3 -36*n^2 +14*n -1 for n in (1..40)] # G. C. Greubel, Feb 26 2019

a(n) = 24*n^3 - 36*n^2 + 14*n - 1.
G.f.: x*(1+x)*(1+70*x+x^2)/(1-x)^4; a(n) = (2*n-1)*A069190(n). - Bruno Berselli, Jul 21 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=75, a(2)=365, a(3)=1015. - Harvey P. Dale, Jul 27 2011
a(n) = 72 * A000330(n-1) + A005408(n-1). - Bruce J. Nicholson, Feb 23 2019
E.g.f.: 1 + (-1 + 2*x + 36*x^2 + 24*x^3)*exp(x). - G. C. Greubel, Feb 26 2019

cp's OEIS Frontend

User: Craig Ferguson

A193228 Truncated octahedron with faces of centered polygons.

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A193248 Truncated dodecahedron, and truncated icosahedron with faces of centered polygons.

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A193250 Small rhombicuboctahedron with faces of centered polygons.

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A193251 Small rhombicosidodecahedron with faces of centered polygons.

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A193218 Number of vertices in truncated tetrahedron with faces that are centered polygons.

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A193253 Great rhombicosidodecahedron with faces of centered polygons.

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A193249 Snub dodecahedron with faces of centered polygons.

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