A100147 -id:A100147 - cp's OEIS frontend

A100145 Structured great rhombicosidodecahedral numbers.

1, 120, 579, 1600, 3405, 6216, 10255, 15744, 22905, 31960, 43131, 56640, 72709, 91560, 113415, 138496, 167025, 199224, 235315, 275520, 320061, 369160, 423039, 481920, 546025, 615576, 690795, 771904, 859125, 952680, 1052791, 1159680

Offset: 1

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

Structured polyhedral numbers are a type of figurate polyhedral numbers. Structurate polyhedra differ from regular figurate polyhedra by having appropriate figurate polygonal faces at any iteration, i.e., a regular truncated octahedron, n=2, would have 7 points on its hexagonal faces, whereas a structured truncated octahedron, n=2, would have 6 points - just as a hexagon, n=2, would have. Like regular figurate polygons, structured polyhedra seem to originate at a vertex and since many polyhedra have different vertices (a pentagonal diamond has 2 "polar" vertices with 5 adjacent vertices and 5 "equatorial" vertices with 4 adjacent vertices), these polyhedra have multiple structured number sequences, dependent on the "vertex structures" which are each equal to the one vertex itself plus its adjacent vertices. For polystructurate polyhedra the notation, structured polyhedra (vertex structure x) is used to differentiate between alternate vertices, where VS stands for vertex structure.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A051673, A100146 through A100156 - structured Archimedean solids; A100157 through A100175 - structured Catalan solids; A100147 - structured prisms; A000447 - structured diamonds; A100185 - structured anti-prisms; and A100188 - structured anti-diamonds.

[(1+(n-1))*(1+22*(n-1)+37*(n-1)^2): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011

LinearRecurrence[{4,-6,4,-1},{1,120,579,1600},40] (* Harvey P. Dale, Mar 27 2019 *)

a(n)=37*n^3 - 52*n^2 + 16*n \\ Charles R Greathouse IV, Nov 07 2016

a(n) = (1/6)*(222*n^3 - 312*n^2 + 96*n).
From Jaume Oliver Lafont, Sep 08 2009: (Start)
a(n) = (1+(n-1))*(1+22*(n-1)+37*(n-1)^2);
G.f.: x*(1+116*x+105*x^2)/(1-x)^4. (End)
E.g.f.: exp(x)*x*(1 + 59*x + 37*x^2). - Stefano Spezia, Jun 06 2025

Corrected by T. D. Noe, Oct 25 2006

A079588 a(n) = (n+1)(2n+1)(4n+1).

Original entry on oeis.org

1, 30, 135, 364, 765, 1386, 2275, 3480, 5049, 7030, 9471, 12420, 15925, 20034, 24795, 30256, 36465, 43470, 51319, 60060, 69741, 80410, 92115, 104904, 118825, 133926, 150255, 167860, 186789, 207090, 228811, 252000, 276705, 302974, 330855, 360396, 391645

Offset: 0

N. J. A. Sloane, Jan 26 2003

Apart from offset, same as A100147.

R. Tijdeman, Some applications of Diophantine approximation, pp. 261-284 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A100147.

Cf. A258721 (first differences), A011199.

a079588 n = product $ map ((+ 1) . (* n)) [1, 2, 4]
-- Reinhard Zumkeller, Jun 08 2015

Table[(n + 1)*(2*n + 1)*(4*n + 1), {n, 0, 40}] (* Amiram Eldar, Jan 13 2021 *)
LinearRecurrence[{4,-6,4,-1},{1,30,135,364},40] (* Harvey P. Dale, Aug 01 2022 *)

Sum_{n>=0} 1/a(n) = Pi/3 (cf. Tijdeman).
G.f.: (1+26*x+21*x^2)/(1-x)^4. - L. Edson Jeffery, Mar 25 2013
Sum_{n>=0} a(n)/2^n = 308; Sum_{n>=0} (-1)^n*a(n)/2^n = -4/3. - L. Edson Jeffery, Mar 25 2013
a(n) = 8*n^3 + 14*n^2 + 7*n + 1. - Reinhard Zumkeller, Jun 08 2015
Sum_{n>=0} (-1)^n/a(n) = log(2)/3 - Pi/2 + sqrt(2)*Pi/3 + 2*sqrt(2)*arcsin(1)/3. - Amiram Eldar, Jan 13 2021
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Wesley Ivan Hurt, Jun 23 2021

A155103 Triangle read by rows: Matrix inverse of A155102.

Original entry on oeis.org

1, 2, 1, 0, 0, 1, 6, 3, 0, 1, 0, 0, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 30, 15, 0, 5, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 6, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 28, 0, 0, 7, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0

Offset: 1

Mats Granvik, Jan 20 2009

A028361 appears in the first column at A036987 positions. A028362 appears in the second column, A155105 in the third and A155104 in the fourth. A000384 appears as the third ray from zero and A100147 as the fourth.

			Table begins:
1,
2,1,
0,0,1,
6,3,0,1,
0,0,0,0,1,
0,0,4,0,0,1,
0,0,0,0,0,0,1,
30,15,0,5,0,0,0,1,

Cf. A028361, A036987, A028362, A155105, A155104. A000384, A100147.

m = 14; t = Inverse[ Table[ Which[n == k, 1, n == 2*k, -k - 1, True, 0], {n, 1, m}, {k, 1, m}]]; Flatten[ Table[t[[n, k]], {n, 1, m}, {k, 1, n}]] (* Jean-François Alcover, Jul 19 2012 *)

A100146 Structured great rhombicubeoctahedral numbers.

Original entry on oeis.org

1, 48, 221, 600, 1265, 2296, 3773, 5776, 8385, 11680, 15741, 20648, 26481, 33320, 41245, 50336, 60673, 72336, 85405, 99960, 116081, 133848, 153341, 174640, 197825, 222976, 250173, 279496, 311025, 344840, 381021, 419648, 460801, 504560, 551005, 600216, 652273

Offset: 1

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

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A100145, A100147 for adjacent structured Archimedean solids; and A100145 for more on structured polyhedral numbers.

[((n-1)+1)*(5*(n-1)+3)*(8*(n-1)+1)/3: n in [1..40]]; // Vincenzo Librandi, Jul 19 2011

a(n) = (1/6)*(80*n^3 - 102*n^2 + 28*n).
From Jaume Oliver Lafont, Sep 08 2009: (Start)
a(n) = ((n-1)+1)*(5*(n-1)+3)*(8*(n-1)+1)/3.
G.f.: x*(1 + 44*x + 35*x^2)/(1-x)^4. (End)
From Elmo R. Oliveira, Aug 05 2025: (Start)
E.g.f.: exp(x)*x*(40*x^2 + 69*x + 3)/3.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4. (End)

A100148 Structured small rhombicosidodecahedral numbers.

Original entry on oeis.org

1, 60, 285, 784, 1665, 3036, 5005, 7680, 11169, 15580, 21021, 27600, 35425, 44604, 55245, 67456, 81345, 97020, 114589, 134160, 155841, 179740, 205965, 234624, 265825, 299676, 336285, 375760, 418209, 463740, 512461, 564480

Offset: 1

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

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A100147, A100149 for adjacent structured Archimedean solids; and A100145 for more on structured polyhedral numbers.

[(1/6)*(108*n^3-150*n^2+48*n): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011

Table[(108n^3-150n^2+48n)/6,{n,40}] (* or *) LinearRecurrence[ {4,-6,4,-1},{1,60,285,784},40](* Harvey P. Dale, Oct 10 2011 *)

vector(50, n, (108*n^3 - 150*n^2 + 48*n)/6) \\ G. C. Greubel, Oct 18 2018

a(n) = (1/6)*(108*n^3 - 150*n^2 + 48*n).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(1)=1, a(2)=60, a(3)=285, a(4)=784. - Harvey P. Dale, Oct 10 2011
G.f.: x*(x*(51*x+56)+1)/(x-1)^4. - Harvey P. Dale, Oct 10 2011
E.g.f.: x*(1 + 29*x + 18*x^2)*exp(x). - G. C. Greubel, Oct 18 2018

A143254 Triangle read by rows, T(n,k) = (4n-3)*(4k-3); 1<=k<=n.

Original entry on oeis.org

1, 5, 25, 9, 45, 81, 13, 65, 117, 169, 17, 85, 153, 221, 289, 21, 105, 189, 273, 357, 441, 25, 125, 225, 325, 425, 525, 625, 29, 145, 261, 377, 493, 609, 725, 841, 33, 165, 297, 429, 561, 693, 825, 957, 1089, 37, 185, 333, 481, 629, 777, 925, 1073, 1221, 1369

Offset: 1

Gary W. Adamson, Aug 02 2008

Row sums = A100147: (1, 30, 135, 364, 765,...).

			First few rows of the triangle =
1;
5, 25;
9, 45, 81;
13, 65, 117, 169;
17, 85, 153, 221, 289;
21, 105, 189, 273, 357, 441;
...
T(5,3) = 153 = (4n-3)*(4k-3) = 17*9

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

Cf. A016813, A100147.

Flatten[Table[(4 n - 3) (4 k - 3), {n, 10}, {k, n}]] (* Harvey P. Dale, Feb 28 2015 *)

Triangle read by rows, T(n,k) = (4n-3)*(4k-3); 1<=k<=n, where (4k-3) = A016813: (1, 5, 9, 13, 17,...). Let X = an infinite lower triangular matrix with (1, 5, 9, 13,...) in the main diagonal and the rest zeros. The triangle = X * A000012 * X, where A000012 = an infinite lower triangular matrix with all 1's: (1; 1,1; 1,1,1;...)

cp's OEIS Frontend

A100145 Structured great rhombicosidodecahedral numbers.

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A079588 a(n) = (n+1)*(2*n+1)*(4*n+1).

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A155103 Triangle read by rows: Matrix inverse of A155102.

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A100146 Structured great rhombicubeoctahedral numbers.

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A100148 Structured small rhombicosidodecahedral numbers.

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A143254 Triangle read by rows, T(n,k) = (4n-3)*(4k-3); 1<=k<=n.

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A079588 a(n) = (n+1)(2n+1)(4n+1).