cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A100155 Structured truncated octahedral numbers.

Original entry on oeis.org

1, 24, 103, 272, 565, 1016, 1659, 2528, 3657, 5080, 6831, 8944, 11453, 14392, 17795, 21696, 26129, 31128, 36727, 42960, 49861, 57464, 65803, 74912, 84825, 95576, 107199, 119728, 133197, 147640, 163091, 179584, 197153, 215832, 235655, 256656, 278869, 302328
Offset: 1

Views

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

Links

Crossrefs

Cf. A100154, A100156 for adjacent structured Archimedean solids; A100145 for more on structured polyhedral numbers. Similar to truncated octahedral numbers A005910.

Programs

  • Magma
    [(1/6)*(34*n^3-36*n^2+8*n): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011
  • Mathematica
    LinearRecurrence[{4, -6, 4, -1}, {1, 24, 103, 272}, 50] (* Paolo Xausa, Aug 06 2025 *)

Formula

a(n) = (1/3)*n*(17*n^2 - 18*n + 4).
G.f.: x*(1 + 20*x + 13*x^2)/(1-x)^4. - Colin Barker, Feb 12 2012
From Elmo R. Oliveira, Aug 05 2025: (Start)
E.g.f.: exp(x)*x*(17*x^2 + 33*x + 3)/3.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4. (End)

A100156 Structured truncated tetrahedral numbers.

Original entry on oeis.org

1, 12, 44, 108, 215, 376, 602, 904, 1293, 1780, 2376, 3092, 3939, 4928, 6070, 7376, 8857, 10524, 12388, 14460, 16751, 19272, 22034, 25048, 28325, 31876, 35712, 39844, 44283, 49040, 54126, 59552, 65329, 71468, 77980, 84876, 92167, 99864, 107978, 116520, 125501, 134932
Offset: 1

Views

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

Links

Crossrefs

Cf. A100155, A100157 for adjacent structured Archimedean solids; A100145 for more on structured polyhedral numbers. Similar to truncated tetrahedral numbers A005906.

Programs

  • Magma
    [(1/6)*(11*n^3-3*n^2-2*n): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011
  • Mathematica
    Table[(11n^3-3n^2-2n)/6,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,12,44,108},40] (* Harvey P. Dale, Sep 28 2011 *)
  • PARI
    vector(50, n, (11*n^3 - 3*n^2 - 2*n)/6) \\ G. C. Greubel, Oct 18 2018

Formula

a(n) = (1/6)*(11*n^3 - 3*n^2 - 2*n).
From Harvey P. Dale, Sep 28 2011: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=12, a(2)=44, a(3)=108.
G.f.: x*(2*x*(x+4)+1)/(x-1)^4. (End)
E.g.f.: x*(6 + 30*x + 11*x^2)*exp(x)/6. - G. C. Greubel, Oct 18 2018

A100160 Structured disdyakis triacontahedral numbers (vertex structure 5).

Original entry on oeis.org

1, 62, 299, 828, 1765, 3226, 5327, 8184, 11913, 16630, 22451, 29492, 37869, 47698, 59095, 72176, 87057, 103854, 122683, 143660, 166901, 192522, 220639, 251368, 284825, 321126, 360387, 402724, 448253, 497090, 549351, 605152
Offset: 1

Views

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

Links

Crossrefs

Cf. A100158, A100159 = alternate vertices; A100145 for more on structured polyhedral numbers.

Programs

  • Magma
    [(1/6)*(116*n^3-168*n^2+58*n): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011

Formula

a(n) = (1/6)*(116*n^3 - 168*n^2 + 58*n).
G.f.: x*(1+x)*(1+57*x)/(1-x)^4. [Colin Barker, Apr 16 2012]

A100161 Structured disdyakis dodecahedral numbers (vertex structure 9).

Original entry on oeis.org

1, 26, 115, 308, 645, 1166, 1911, 2920, 4233, 5890, 7931, 10396, 13325, 16758, 20735, 25296, 30481, 36330, 42883, 50180, 58261, 67166, 76935, 87608, 99225, 111826, 125451, 140140, 155933, 172870, 190991, 210336
Offset: 1

Views

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

Comments

Also structured deltoidal icositetrahedral numbers (vertex structure 9) (cf. A100162 = alternate vertex).

Links

Crossrefs

Cf. A100162, A100163 = alternate vertices; A100145 for more on structured polyhedral numbers.

Programs

  • Magma
    [(1/6)*(40*n^3-48*n^2+14*n): n in [1..40]]; // Vincenzo Librandi, Jul 24 2011
  • Mathematica
    Table[(40n^3-48n^2+14n)/6,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,26,115,308},40] (* Harvey P. Dale, Sep 23 2016 *)

Formula

a(n) = (1/6)*(40*n^3 - 48*n^2 + 14*n).
G.f.: x*(1 + 22*x + 17*x^2)/(1-x)^4. - Colin Barker, Jan 19 2012

A100163 Structured disdyakis dodecahedral numbers (vertex structure 5).

Original entry on oeis.org

1, 26, 119, 324, 685, 1246, 2051, 3144, 4569, 6370, 8591, 11276, 14469, 18214, 22555, 27536, 33201, 39594, 46759, 54740, 63581, 73326, 84019, 95704, 108425, 122226, 137151, 153244, 170549, 189110, 208971, 230176
Offset: 1

Views

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

Links

Crossrefs

Cf. A100161, A100162 = alternate vertices; A100145 for more on structured polyhedral numbers.

Programs

  • Magma
    [(1/6)*(44*n^3-60*n^2+22*n): n in [1..40]]; // Vincenzo Librandi, Jul 25 2011

Formula

a(n) = (1/6)*(44*n^3 - 60*n^2 + 22*n).
G.f.: x*(1 + 22*x + 21*x^2)/(1-x)^4. - Colin Barker, Jan 19 2012

A100164 Structured rhombic triacontahedral numbers (vertex structure 11).

Original entry on oeis.org

1, 32, 143, 384, 805, 1456, 2387, 3648, 5289, 7360, 9911, 12992, 16653, 20944, 25915, 31616, 38097, 45408, 53599, 62720, 72821, 83952, 96163, 109504, 124025, 139776, 156807, 175168, 194909, 216080, 238731, 262912, 288673, 316064, 345135, 375936, 408517, 442928
Offset: 1

Views

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

Comments

Also structured triakis icosahedral numbers (vertex structure 11) (cf. A100172 = alternate vertex).

Links

Crossrefs

Cf. A100165 (alternate vertex), A100145 for more on structured polyhedral numbers.
Cf. A001622, A100172.

Programs

  • Magma
    [(1/6)*(50*n^3-60*n^2+16*n): n in [1..40]]; // Vincenzo Librandi, Jul 25 2011
  • Mathematica
    a[n_] := (n*(5*n - 2)*(5*n - 4))/3; Array[a, 30] (* Amiram Eldar, Sep 20 2022 *)

Formula

a(n) = (1/6)*(50*n^3 - 60*n^2 + 16*n) = (1/3)*n*(5*n-2)*(5*n-4).
From Jaume Oliver Lafont, Sep 08 2009: (Start)
a(n) = (5*(n-1) + 1)*(5*(n-1) + 3)*(5*(n-1) + 5)/15.
G.f.: x*(1 + 28*x + 21*x^2)/(1-x)^4. (End)
Sum_{n>=1} 1/a(n) = 3*sqrt((25-2*sqrt(5))/5)*Pi/16 + 9*sqrt(5)*log(phi)/16 - 15*log(5)/32, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 20 2022

A100166 Structured deltoidal hexacontahedral numbers (vertex structure 9).

Original entry on oeis.org

1, 62, 295, 812, 1725, 3146, 5187, 7960, 11577, 16150, 21791, 28612, 36725, 46242, 57275, 69936, 84337, 100590, 118807, 139100, 161581, 186362, 213555, 243272, 275625, 310726, 348687, 389620, 433637, 480850, 531371, 585312
Offset: 1

Views

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

Links

Crossrefs

Cf. A100158, A100159 = alternate vertices; A100145 for more on structured numbers.

Programs

  • Magma
    [(1/6)*(112*n^3-156*n^2+50*n): n in [1..40]]; // Vincenzo Librandi, Jul 26 2011
  • Mathematica
    LinearRecurrence[{4,-6,4,-1},{1,62,295,812},40] (* Harvey P. Dale, Nov 15 2022 *)

Formula

a(n) = (1/6)*(112*n^3 - 156*n^2 + 50*n).
G.f.: x*(1 + 58*x + 53*x^2)/(1-x)^4. - Colin Barker, Jan 19 2012

A100171 Structured triakis octahedral numbers (vertex structure 4).

Original entry on oeis.org

1, 14, 60, 160, 335, 606, 994, 1520, 2205, 3070, 4136, 5424, 6955, 8750, 10830, 13216, 15929, 18990, 22420, 26240, 30471, 35134, 40250, 45840, 51925, 58526, 65664, 73360, 81635, 90510, 100006, 110144
Offset: 1

Views

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

Links

Crossrefs

Cf. A100157 = alternate vertex; A100145 for more on structured numbers.

Programs

  • Magma
    [(1/6)*(21*n^3-27*n^2+12*n): n in [1..40]]; // Vincenzo Librandi, Jul 27 2011
  • Mathematica
    Table[(21n^3-27n^2+12n)/6,{n,40}] (* or *) LinearRecurrence[ {4,-6,4,-1},{1,14,60,160},40] (* Harvey P. Dale, Jun 28 2011 *)

Formula

a(n)=(1/6)*(21*n^3-27*n^2+12*n).
a(0)=1, a(1)=14, a(2)=60, a(3)=160, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Jun 28 2011
G.f.: (10*x^2+10*x+1)/(x-1)^4. - Harvey P. Dale, Jun 28 2011

A100174 Structured tetrakis hexahedral numbers (vertex structure 5).

Original entry on oeis.org

1, 14, 59, 156, 325, 586, 959, 1464, 2121, 2950, 3971, 5204, 6669, 8386, 10375, 12656, 15249, 18174, 21451, 25100, 29141, 33594, 38479, 43816, 49625, 55926, 62739, 70084, 77981, 86450, 95511, 105184, 115489, 126446, 138075, 150396, 163429, 177194, 191711, 207000
Offset: 1

Views

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

Links

Crossrefs

Cf. A005915 = alternate vertex; cf. A100145 for more on structured numbers.

Programs

  • Magma
    [(1/6)*(20*n^3-24*n^2+10*n): n in [1..50] ]; // Vincenzo Librandi, Aug 02 2011

Formula

a(n) = (1/6)*(20*n^3 - 24*n^2 + 10*n).
G.f.: x*(1+x)*(1+9*x)/(1-x)^4. - Colin Barker, May 29 2012
From Elmo R. Oliveira, Aug 04 2025: (Start)
E.g.f.: exp(x)*x*(10*x^2 + 18*x + 3)/3.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4. (End)

Extensions

Formula corrected by Helena Verrill (verrill(AT)math.lsu.edu), Mar 23 2007

A100176 Structured octagonal prism numbers.

Original entry on oeis.org

1, 16, 63, 160, 325, 576, 931, 1408, 2025, 2800, 3751, 4896, 6253, 7840, 9675, 11776, 14161, 16848, 19855, 23200, 26901, 30976, 35443, 40320, 45625, 51376, 57591, 64288, 71485, 79200, 87451, 96256, 105633, 115600, 126175, 137376, 149221, 161728, 174915, 188800
Offset: 1

Views

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

Comments

Number of divisors of 120^(n-1). - J. Lowell, Aug 30 2008
Partial sums of A214675. - J. M. Bergot, Jul 08 2013

Links

Crossrefs

Cf. A100177 (structured prisms), A100145 (for more on structured numbers).
Cf. similar sequences, with the formula (k*n - k + 2)*n^2/2, listed in A262000.
Cf. A000567, A016777, A214675.

Programs

Formula

a(n) = 3*n^3 - 2*n^2.
G.f.: x*(1+12*x+5*x^2)/(1-x)^4. - Colin Barker, Jun 08 2012
a(n) = Sum_{i=0..n-1} n*(6*i+1). - Bruno Berselli, Sep 08 2015
Sum_{n>=1} 1/a(n) = sqrt(3)*Pi/8 - Pi^2/12 + 9*log(3)/8 = 1.0936465529153418... . - Vaclav Kotesovec, Oct 04 2016
a(n) = n*A000567(n) = n^2 * A016777(n-1). - Bruce J. Nicholson, Aug 10 2017
From Elmo R. Oliveira, Aug 06 2025: (Start)
E.g.f.: exp(x)*x*(1 + 7*x + 3*x^2).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

Extensions

More terms from Zerinvary Lajos, Jun 29 2006
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