A051673 -id:A051673 - cp's OEIS frontend

A005902 Centered icosahedral (or cuboctahedral) numbers, also crystal ball sequence for f.c.c. lattice.

1, 13, 55, 147, 309, 561, 923, 1415, 2057, 2869, 3871, 5083, 6525, 8217, 10179, 12431, 14993, 17885, 21127, 24739, 28741, 33153, 37995, 43287, 49049, 55301, 62063, 69355, 77197, 85609, 94611, 104223, 114465, 125357, 136919, 149171, 162133, 175825, 190267, 205479

Offset: 0

N. J. A. Sloane

Called "magic numbers" in some chemical contexts.
Partial sums of A005901(n). - Lekraj Beedassy, Oct 30 2003
Equals binomial transform of [1, 12, 30, 20, 0, 0, 0, ...]. - Gary W. Adamson, Aug 01 2008
Crystal ball sequence for A_3 lattice. - Michael Somos, Jun 03 2012

			a(4) = 147 = (1, 3, 3, 1) dot (1, 12, 30, 20) = (1 + 36 + 90 + 20). - _Gary W. Adamson_, Aug 01 2008
G.f. = 1 + 13*x + 55*x^2 + 147*x^3 + 309*x^4 + 561*x^5 + 923*x^6 + 1415*x^7 + ...

H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
S. Bjornholm, Clusters, condensed matter in embryonic form, Contemp. Phys. 31 1990 pp. 309-324.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Nicolas Gastineau, Olivier Togni, Coloring of the d-th power of the face-centered cubic grid, arXiv:1806.08136 [cs.DM], 2018.
D. R. Herrick, Home Page (displays these numbers as sizes of clusters in chemistry)
Xiaogang Liang, Ilyar Hamid, and Haiming Duan, Dynamic stabilities of icosahedral-like clusters and their ability to form quasicrystals,>, AIP Advances 6, 065017 (2016).
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
K. Urner, Cuboctahedral Sphere Packing
Index entries for crystal ball sequences
Index entries for sequences related to f.c.c. lattice
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.
The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.
Cf. A100171, A100174, A051673.

[(2*n+1)*(5*n^2+5*n+3)/3: n in [0..30]]; // G. C. Greubel, Dec 01 2017

A005902 := n -> (2*n+1)*(5*n^2+5*n+3)/3;
A005902:=(z+1)*(z**2+8*z+1)/(z-1)**4; # Simon Plouffe in his 1992 dissertation

f[n_] := (2n + 1)(5n^2 + 5n + 3)/3; Array[f, 36, 0] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{4,-6,4,-1},{1,13,55,147},50] (* Harvey P. Dale, Oct 08 2015 *)
CoefficientList[Series[(x^3 + 9*x^2 + 9*x + 1)/(x - 1)^4, {x, 0, 50}], x] (* Indranil Ghosh, Apr 08 2017 *)

{a(n) = (2*n + 1) * (5*n^2 + 5*n + 3) / 3}; /* Michael Somos, Jun 03 2012 */

x='x+O('x^50); Vec((x^3 + 9*x^2 + 9*x + 1)/(x - 1)^4) \\ Indranil Ghosh, Apr 08 2017

def a(n): return (2*n+1)*(5*n**2+5*n+3)//3
print([a(n) for n in range(40)]) # Michael S. Branicky, Jan 13 2021

a(n) = (2*n+1)*(5*n^2+5*n+3)/3.
For n > 0, n*a(n) = (Sum_{i=0..n-1} a(i)) + 2*A005891(n)*A000217(n). - Bruno Berselli, Feb 02 2011
a(-1 - n) = -a(n). - Michael Somos, Jun 03 2012
From Indranil Ghosh, Apr 08 2017: (Start)
G.f.: (x^3 + 9x^2 + 9x + 1)/(x - 1)^4.
E.g.f.: (1/3)*exp(x)*(10x^3 + 45x^2 + 36x + 3).
(End)
a(n) = A100171(n+1) - A008778(n-1) = A100174(n+1) - A000290(n) = A005917(n+1) - A006331(n) = A051673(n+1) + A000578(n). - Bruce J. Nicholson, Jul 05 2018

A100145 Structured great rhombicosidodecahedral numbers.

Original entry on oeis.org

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

A004466 a(n) = n(5n^2 - 2)/3.

Original entry on oeis.org

0, 1, 12, 43, 104, 205, 356, 567, 848, 1209, 1660, 2211, 2872, 3653, 4564, 5615, 6816, 8177, 9708, 11419, 13320, 15421, 17732, 20263, 23024, 26025, 29276, 32787, 36568, 40629, 44980, 49631, 54592

Offset: 0

Albert D. Rich (Albert_Rich(AT)msn.com)

3-dimensional analog of centered polygonal numbers.
Also as a(n)=(1/6)*(10*n^3-4*n), n>0: structured pentagonal anti-diamond numbers (vertex structure 11) (Cf. A051673 = alternate vertex A100188 = structured anti-diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004
a(n+1)-10*a(n) = (n+1)*(5*(n+1)^2-2)/3 - (10n(n+1)(n+2)/6) = n. The unit digits are 0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,... . - Eric Desbiaux, Aug 18 2008

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A062786 (first differences), A264853 (partial sums).

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

[n*(5*n^2-2)/3: n in [0..50]]; // Vincenzo Librandi, May 15 2011

A004466:=n->n*(5*n^2 - 2)/3; seq(A004466(n), n=0..50); # Wesley Ivan Hurt, Mar 10 2014

Table[n(5n^2-2)/3,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

a(n)=n*(5*n^2-2)/3 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: x*(1+8*x+x^2)/(1-x)^4. - Colin Barker, Jan 08 2012

E.g.f.: (x/3)*(3 + 15*x + 5*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A214661 Odd numbers obtained by transposing the left half of A176271 into rows of a triangle: T(n,k) = A176271(n - 1 + k, k), 1 <= k <= n.

Original entry on oeis.org

1, 3, 9, 7, 15, 25, 13, 23, 35, 49, 21, 33, 47, 63, 81, 31, 45, 61, 79, 99, 121, 43, 59, 77, 97, 119, 143, 169, 57, 75, 95, 117, 141, 167, 195, 225, 73, 93, 115, 139, 165, 193, 223, 255, 289, 91, 113, 137, 163, 191, 221, 253, 287, 323, 361, 111, 135, 161, 189, 219, 251, 285, 321, 359, 399, 441

Offset: 1

Reinhard Zumkeller, Jul 25 2012

			.     Take the first n elements of the n-th diagonal (northwest to
.     southeast) of the triangle on the left side
.     and write this as n-th row on the triangle of the right side.
. 1:                1                    1
. 2:              3   _                  3  9
. 3:            7   9  __                7 15 25
. 4:         13  15  __  __             13 23 35 49
. 5:       21  23  25  __  __           21 33 47 63 ..
. 6:     31  33  35  __  __  __         31 45 61 .. .. ..
. 7:   43  45  47  49  __  __  __       43 59 .. .. .. .. ..
. 8: 57  59  61  63  __  __  __  __     57 .. .. .. .. .. .. .. .

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

Cf. A000290, A002061, A016754, A025581, A094727, A176271, A214604.

Cf. A051673 (row sums), A214675 (main diagonal).

import Data.List (transpose)
a214661 n k = a214661_tabl !! (n-1) !! (k-1)
a214661_row n = a214661_tabl !! (n-1)
a214661_tabl = zipWith take [1..] $ transpose $ map reverse a176271_tabl

[(n+k)^2-3*n-k+1: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 10 2024

Table[(n+k)^2-3*n-k+1, {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 10 2024 *)

flatten([[(n+k)^2-3*n-k+1 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 10 2024

T(n, k) = (n+k)^2 - 3*n - k + 1.
T(n,k) = A176271(n+k-1, k).
T(n, k) = A214604(n,k) - 2*A025581(n,k).
T(n, k) = 2*A000290(A094727(n,k)) - A214604(n,k).
T(2*n-1, n) = A214675() (main diagonal).
T(n,1) = A002061(n).
T(n,n) = A016754(n-1).
Sum_{k=1..n} T(n, k) = A051673(n) (row sums).

A100189 Equatorial structured meta-anti-diamond numbers, the n-th number from an equatorial structured n-gonal anti-diamond number sequence.

Original entry on oeis.org

1, 6, 27, 92, 245, 546, 1071, 1912, 3177, 4990, 7491, 10836, 15197, 20762, 27735, 36336, 46801, 59382, 74347, 91980, 112581, 136466, 163967, 195432, 231225, 271726, 317331, 368452, 425517, 488970, 559271, 636896

Offset: 1

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

			There are no 1- or 2-gonal anti-diamonds, so 1 and (2n+2) are used as the first and second terms since all the sequences begin as such.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A000578, A096000, A051673, A005915, A100186, A100187 - "equatorial" structured anti-diamonds; A100188 - "polar" structured meta-anti-diamond numbers; A006484 for other structured meta numbers; and A100145 for more on structured numbers.

[(1/6)*(4*n^4-12*n^3+20*n^2-6*n): n in [1..40]]; // Vincenzo Librandi, Aug 18 2011

Table[(4n^4-12n^3+20n^2-6n)/6,{n,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,6,27,92,245},40] (* Harvey P. Dale, Jul 05 2011 *)

a(n) = (1/6)*(4*n^4-12*n^3+20*n^2-6*n).
a(1)=1, a(2)=6, a(3)=27, a(4)=92, a(5)=245, a(n)=5*a(n-1)-10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5). - Harvey P. Dale, Jul 05 2011
G.f.: x*(1+x)*(1+7*x^2)/(1-x)^5. - Colin Barker, Jan 19 2012

A214659 a(n) = n(7n^2 - 3*n - 1)/3.

Original entry on oeis.org

0, 1, 14, 53, 132, 265, 466, 749, 1128, 1617, 2230, 2981, 3884, 4953, 6202, 7645, 9296, 11169, 13278, 15637, 18260, 21161, 24354, 27853, 31672, 35825, 40326, 45189, 50428, 56057, 62090, 68541, 75424, 82753, 90542, 98805, 107556, 116809, 126578, 136877

Offset: 0

Reinhard Zumkeller, Jul 25 2012

a(n) = the sum of the n X n matrices of A204008. For example, for n = 3, the sum of the 9 elements of the 3 X 3 submatrix of A204008 is 1 + 4 + 7 + 4 + 5 + 8 + 7 + 8 + 9 = 53. - J. M. Bergot, Jul 15 2013

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

Cf. A002378, A051673, A204008, A214604, A214675.

a214659 n = ((7 * n - 3) * n - 1) * n `div` 3

[(7*n^3-3*n^2-n)/3 : n in [0..50]]; // Wesley Ivan Hurt, Apr 11 2015

A214659:=n->(7*n^3-3*n^2-n)/3: seq(A214659(n), n=0..50); # Wesley Ivan Hurt, Apr 11 2015

Table[(7 n^3 -3 n^2 -n)/3, {n,0,50}] (* Wesley Ivan Hurt, Apr 11 2015 *)
LinearRecurrence[{4,-6,4,-1}, {0,1,14,53}, 51] (* G. C. Greubel, Mar 09 2024 *)

[(7*n^3-3*n^2-n)/3 for n in range(51)] # G. C. Greubel, Mar 09 2024

a(n) = Sum_{k=0..n} A214604(n, k) for n > 0 (row sums).
a(n) = A002378(n) + A051673(n).
From Wesley Ivan Hurt, Apr 11 2015: (Start)
a(n) = (7*n^3 - 3*n^2 - n)/3.
G.f.: x*(1+10*x+3*x^2)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
E.g.f.: (x/3)*(3 + 18*x + 7*x^2)*exp(x). - G. C. Greubel, Mar 09 2024

A006060 Triangular star numbers.

Original entry on oeis.org

1, 253, 49141, 9533161, 1849384153, 358770992581, 69599723176621, 13501987525271953, 2619315980179582321, 508133798167313698381, 98575337528478677903653, 19123107346726696199610361

Offset: 1

N. J. A. Sloane

nonn

M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 20.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

B. Berselli, Table of n, a(n) for n = 1..400. [From _Bruno Berselli_, Jul 07 2010]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Star Number
Index entries for linear recurrences with constant coefficients, signature (195, -195, 1).

Cf. A000567, A003154, A006061, A051673.

A006060:=-(1+58*z+z**2)/(z-1)/(z**2-194*z+1); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation
a:= n-> (Matrix([[253,1,1]]). Matrix([[195,1,0], [ -195,0,1], [1,0,0]])^n)[1,3]: seq(a(n), n=1..20); # Alois P. Heinz, Aug 14 2008

a006060 = {}; Do[
If[Length[a006060] < 2, AppendTo[a006060, 1],
  AppendTo[a006060, 194*a006060[[-1]] + 60 - a006060[[-2]]]],  {n,
  20}]; TableForm[Transpose[List[Range[Length[a006060]], a006060]]] (* Michael De Vlieger *)
LinearRecurrence[{195,-195,1},{1,253,49141},20] (* Harvey P. Dale, Jan 12 2017 *)

G.f.: (1 + 58x + x^2)/((x-1)(1 - 194x + x^2)). - Ralf Stephan, Apr 23 2004
From Bruno Berselli, Jul 07 2010: (Start)
a(n) = 194*a(n-1) - a(n-2) + 60 (n>2).
a(n) = (3*((7 + 4*sqrt(3))^(2*n-1) + (7 - 4*sqrt(3))^(2*n-1)) - 10)/32 (n>0).
(End)

Extended by Eric W. Weisstein, Mar 01 2002

cp's OEIS Frontend

A005902 Centered icosahedral (or cuboctahedral) numbers, also crystal ball sequence for f.c.c. lattice.

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A100145 Structured great rhombicosidodecahedral numbers.

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A004466 a(n) = n*(5*n^2 - 2)/3.

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A214661 Odd numbers obtained by transposing the left half of A176271 into rows of a triangle: T(n,k) = A176271(n - 1 + k, k), 1 <= k <= n.

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A100189 Equatorial structured meta-anti-diamond numbers, the n-th number from an equatorial structured n-gonal anti-diamond number sequence.

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A214659 a(n) = n*(7*n^2 - 3*n - 1)/3.

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A004466 a(n) = n(5n^2 - 2)/3.

A214659 a(n) = n(7n^2 - 3*n - 1)/3.