A005891 -id:A005891 - cp's OEIS frontend

A164015 5 times centered pentagonal numbers: 5(5n^2 + 5*n + 2)/2.

5, 30, 80, 155, 255, 380, 530, 705, 905, 1130, 1380, 1655, 1955, 2280, 2630, 3005, 3405, 3830, 4280, 4755, 5255, 5780, 6330, 6905, 7505, 8130, 8780, 9455, 10155, 10880, 11630, 12405, 13205, 14030, 14880, 15755, 16655, 17580, 18530

Offset: 0

Omar E. Pol, Nov 07 2009

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005891, A152734, A164013, A108099, A164016.

Table[5(5n^2+5n+2)/2,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{5,30,80},40] (* Harvey P. Dale, Oct 08 2011 *)

a(n)=25*n*(n+1)/2+5 \\ Charles R Greathouse IV, Jul 17 2011

a(n) = 5*A005891(n).
a(n) = a(n-1) + 25*n (with a(0)=5). - Vincenzo Librandi, Nov 30 2010
a(0)=5, a(1)=30, a(2)=80, a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Oct 08 2011
G.f.: (5*(x*(x+3)+1))/(1-x)^3. - Harvey P. Dale, Oct 08 2011
E.g.f.: (5/2)*(2 + 10*x + 5*x^2)*exp(x). - G. C. Greubel, Sep 06 2017

A362863 Centered hecatonicosachoral numbers.

Original entry on oeis.org

1, 1441, 11521, 44641, 122401, 273601, 534241, 947521, 1563841, 2440801, 3643201, 5243041, 7319521, 9959041, 13255201, 17308801, 22227841, 28127521, 35130241, 43365601, 52970401, 64088641, 76871521, 91477441, 108072001, 126828001, 147925441, 171551521, 197900641

Offset: 1

Léo Cymrot Cymbalista, May 06 2023

A hecatonicosachoral number is a centered figurate number that represents a hecatonicosachoron, which is a four-dimensional regular polytope composed of 120 cells.

One of the 6 centered regular polichoral (centered pentachoral, centered hexadecachoral, centered octachoral, centered icositetrachoral, centered hexacosichoral and centered hecatonicosachoral) numbers.

Eric Weisstein's World of Mathematics, Figurate Number.
Wikipedia, 120-cell.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A005891 (2D), A005904 (3D), A006322, A151989.

Table[300*n^4 - 600*n^3 + 420*n^2 - 120*n + 1, {n, 1, 100}]

a(n) = 300*n^4 - 600*n^3 + 420*n^2 - 120*n + 1.
a(n) = 1440*A006322(n-1) + 1 for n > 1.
a(n) = 288*(A151989(n-1)-1)/25 + 1.
G.f.: x*(1 + 1436*x + 4326*x^2 + 1436*x^3 + x^4)/(1 - x)^5. - Stefano Spezia, May 12 2023

A364610 Centered pentagonal numbers which are products of three distinct primes.

Original entry on oeis.org

1266, 1626, 2806, 3706, 4731, 6126, 7426, 7701, 9766, 10726, 13506, 15801, 18706, 19581, 25251, 26266, 26781, 31641, 35106, 36906, 40006, 50766, 52926, 56626, 57381, 62806, 69306, 71826, 74391, 76126, 85101, 90726, 93606, 95551, 96531, 99501, 106606, 108681, 109726, 117181, 121551, 123766

Offset: 1

Massimo Kofler, Sep 07 2023

nonn

			A005891(22) = 1266 = (5*22^2 + 5*22 + 2)/2 = 2 * 3 * 211.
A005891(25) = 1626 = (5*25^2 + 5*25 + 2)/2 = 2 * 3 * 271.
A005891(33) = 2806 = (5*33^2 + 5*33 + 2)/2 = 2 * 23 * 61.

Intersection of A005891 and A007304.

Select[Table[5*n*(n + 1)/2 + 1, {n, 0, 225}], FactorInteger[#][[;; , 2]] == {1, 1, 1} &] (* Amiram Eldar, Sep 07 2023 *)

A071910 a(n) = t(n)t(n+1)t(n+2), where t() are the triangular numbers.

Original entry on oeis.org

0, 18, 180, 900, 3150, 8820, 21168, 45360, 89100, 163350, 283140, 468468, 745290, 1146600, 1713600, 2496960, 3558168, 4970970, 6822900, 9216900, 12273030, 16130268, 20948400, 26910000, 34222500, 43120350, 53867268, 66758580, 82123650, 100328400, 121777920

Offset: 0

N. J. A. Sloane, Jun 13 2002

nonn

a(n) is also the number of three-dimensional cage assemblies such that the assembly is not a cube. See also A052149 for the two-dimensional version and to A059827 for the non-exclusive version. - Alejandro Rodriguez, Oct 20 2020

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A006542, (first differences of a(n) /18) A006414, (second differences of a(n) /18) A006322, (third differences of a(n) /18) A004068, (fourth differences of a(n) /18) A005891, (fifth differences of a(n) /18) A008706.

Join[{0},Times@@@Partition[Accumulate[Range[40]],3,1]] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,18,180,900,3150,8820,21168},40] (* Harvey P. Dale, Aug 08 2025 *)

t(n) = n*(n+1)/2;
a(n) = t(n)*t(n+1)*t(n+2); \\ Michel Marcus, Oct 21 2015

a(n) = 18*A006542(n+3). - Vladeta Jovovic, Jun 14 2002
G.f.: 18*x*(1+3*x+x^2)/(1-x)^7. - Vladeta Jovovic, Jun 14 2002
a(n) = ((n+1)*(n+2))^3/8 - Sum_{i=1..n+1} i^3. - Jon Perry, Feb 13 2004
a(n) = C(2+n, n)*C(3+n, 1+n)*C(4+n, 2+n). - Zerinvary Lajos, Jul 29 2005
a(n) = A059827(n+1) - A000537(n+1). - Michel Marcus, Oct 21 2015

A121722 Triangle T(n,k) = 1 + kn(n+1)/2, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 4, 7, 1, 7, 13, 19, 1, 11, 21, 31, 41, 1, 16, 31, 46, 61, 76, 1, 22, 43, 64, 85, 106, 127, 1, 29, 57, 85, 113, 141, 169, 197, 1, 37, 73, 109, 145, 181, 217, 253, 289, 1, 46, 91, 136, 181, 226, 271, 316, 361, 406, 1, 56, 111, 166, 221, 276, 331, 386, 441, 496, 551

Offset: 0

Roger L. Bagula, Sep 08 2006

A triangular form based on the Hex number recursion: a(n) = 2*a(n-1) - a(n-1) + 6: A003215 form as generalized to Integer m.

A solution for the general type for m held constant: a(n) = 2*a(n-1) - a(n-2) + m, with first two values as {1, 1+m}.

			Triangle begins as:
  1;
  1,  2;
  1,  4,  7;
  1,  7, 13, 19;
  1, 11, 21, 31, 41;
  1, 16, 31, 46, 61, 76;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A001844, A002061, A003215, A005448, A005891.

Flat(List([0..10], n-> List([0..n], k-> 1 + k*Binomial(n+1,2) ))); # G. C. Greubel, Nov 21 2019

[1+k*Binomial(n+1,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 21 2019

seq(seq( 1 + k*binomial(n+1,2), k=0..n), n=0..10); # G. C. Greubel, Nov 21 2019

f[n_Integer] = Module[{a}, a[n]/.RSolve[{a[n]==2*a[n-1]-a[n-2]+m, a[0] ==1, a[1]==1+m}, a[n], n][[1]]//FullSimplify] (* formula of triangle *)
Table[Table[1+k*n*(1+n)/2, {k,0,n}], {n,0,10}]//Flatten

T(n, k) = 1 + k*binomial(n+1,2); \\ G. C. Greubel, Nov 21 2019

[[1+k*binomial(n+1,2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 21 2019

T(n, k) = 1 + k*binomial(n+1,2).

Edited by G. C. Greubel, Nov 21 2019

A131751 Numbers that are both centered triangular and centered pentagonal.

Original entry on oeis.org

1, 31, 1891, 117181, 7263301, 450207451, 27905598631, 1729696907641, 107213302675081, 6645495068947351, 411913480972060651, 25531990325198812981, 1582571486681354344141, 98093900183918770523731, 6080239239916282418127151, 376876738974625591153359601

Offset: 1

Richard Choulet, Sep 20 2007

We solve 0.5*(3*p^2+3*p+2)=0.5*(5*r^2+5*r+2), i.e., 3*(2*p+1)^2=5*(2*r+1)^2-2.

The Diophantine equation 3*X^2=5*Y^2-2is such that : X is given by A057080 which satisfies the new formula a(n+1)=4*a(n)+(15*a(n)^2+10)^0.5, Y is given by A070997 which satisfies the new formula a(n+1)=4*a(n)+(15*a(n)^2-6)^0.5 while r is given by the sequence 0,3,27,216,1704,... which satisfies a(n+2)=8*a(n+1)-a(n)+3 and a(n+1)=4*a(n)+1.5+0.5*(60*a(n)^2+60*a(n)+9)^0.5, p is given by the sequence 0,4,35,279,2200,... which satisfies a(n+2)=8*a(n+1)-a(n)+3 and a(n+1)=4*a(n)+1.5+0.5*sqrt(60*a(n)^2+60*a(n)+25).

Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.
Index entries for linear recurrences with constant coefficients, signature (63,-63,1).

Cf. A050807 A070997.

A131751 := proc(n) coeftayl(x*(1-32*x+x^2)/(1-x)/(1-62*x+x^2),x=0,n) ; end: seq(A131751(n),n=1..20) ; # R. J. Mathar, Oct 24 2007

LinearRecurrence[{63,-63,1},{1,31,1891},20] (* Harvey P. Dale, Oct 01 2017 *)

a(n+2) = 62*a(n+1) - a(n) - 30, a(n+1) = 31*a(n) - 15 + sqrt(960*a(n)^2 - 960*a(n)+225).
G.f.: f(z) = a(1)*z+a(2)*z^2+... = z*(1-32*z+z^2)/((1-z)*(1-62*z+z^2)).
A005891 INTERSECT A005448. - R. J. Mathar, Oct 24 2007

Corrected and extended by R. J. Mathar, Oct 24 2007

A140702 Main diagonal of array A(k,n) = product of first n centered n-gonal numbers.

Original entry on oeis.org

40, 1625, 151776, 27316471, 8429601664, 4108830350625, 2977546171600000, 3062351613203813051, 4308809606735976861696, 8050856986181775515023417, 19490752185922086291273856000, 59888297825402713913058605859375, 229474927848540723655596345639141376

Offset: 3

Jonathan Vos Post, May 24 2008

For analog with regular (not centered) n-gonal numbers, see A133401.
Array A(k,n) = k-th polygorial(n,k) begins:
k  |  CenteredPolygorial(n,k)
---+-------------------------
3  | 1 4   40   760   23560    1083760      69360640      5895654400   A140701
4  | 1 5   65  1625   66625    4064125     345450625     39035920625
5  | 1 6   96  2976   151776   11534976   1222707456    172401751296
6  | 1 7  133  4921   300181   27316471   3469191817    586293417073
7  | 1 8  176  7568   537328   56956768   8429601664   1660631527808
8  | 1 9  225 11025   893025  108056025  18261468225   4108830350625
9  | 1 10 280 15400  1401400  190590400  36212176000   9161680528000

			a(3) = 3rd centered polygorial number polygorial(3,3) = A140701(3) = product of the first 3 centered triangular numbers = 1 * 4 * 10 = 40.
a(4) = 4th centered polygorial number centered polygorial(4,4) = product of the first 4 centered square numbers A001844 = 1 * 5 * 13 * 25 = 1625.
a(5) = 5th centered pentagorial number centered polygorial(5,5) = product of the first 5 centered pentagonal numbers A005891 = 1 * 5 * 12 * 22 * 35 = 151776.
a(6) = 6th centered hexagorial number centered polygorial(6,6) = product of the first 6 centered hexagonal numbers A003215 = 1 * 7 * 19 * 37 * 61 * 91 = 27316471.

Nathaniel Johnston, Table of n, a(n) for n = 3..100
Eric W. Weisstein, Centered Triangular Number.

Cf. A005448, A006003, A006472, A133401, A140701.

A140702 := proc(n) mul(n*k*(k-1)/2+1,k=1..n): end: seq(A140702(n),n=3..15); # Nathaniel Johnston, Oct 01 2011

Table[Product[n*k*(k-1)/2+1,{k,1,n}],{n,3,20}] (* Vaclav Kotesovec, Jul 11 2015 *)

a(n) ~ Pi * n^(3*n-1) / (exp(2*n) * 2^(n-2)). - Vaclav Kotesovec, Jul 11 2015

a(9) corrected and more terms from Nathaniel Johnston, Oct 01 2011

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

A249852 a(n) is the total number of pentagons on the left or the right of the vertical symmetry axis of a pentagon expansion (vertex to vertex) after n iterations.

Original entry on oeis.org

0, 2, 7, 14, 23, 35, 50, 67, 86, 108, 133, 160, 189, 221, 256, 293, 332, 374, 419, 466, 515, 567, 622, 679, 738, 800, 865, 932, 1001, 1073, 1148, 1225, 1304, 1386, 1471, 1558, 1647, 1739, 1834, 1931, 2030, 2132, 2237, 2344, 2453, 2565, 2680, 2797, 2916, 3038, 3163, 3290, 3419

Offset: 0

Kival Ngaokrajang, Mar 07 2015

nonn

a(n) is also total number of pentagrams on the left or the right of the vertical symmetry axis of a pentagram expansion (vertex to vertex) after n iterations.

The total pentagons (or pentagrams) after n iterations is A005891. See illustration in the links.

Kival Ngaokrajang, Illustration of initial terms

Cf. A005891, A247619.

{a=2; s=2; d=2; print1(0,", ",s,", "); for(n=2, 100, if(Mod(n,4)==3, d=2, if(Mod(n,4)==4, d=2, d=3)); a=a+d; s=s+a; print1(s, ", ");)}

Conjectures from Colin Barker, Mar 07 2015: (Start)
a(n) = 3*a(n-1)-4*a(n-2)+4*a(n-3)-3*a(n-4)+a(n-5).
G.f.: -x*(x^3+x^2+x+2) / ((x-1)^3*(x^2+1)). (End)
Conjectured e.g.f.: (exp(x)*(1 + 8*x + 5*x^2) - cos(x) - sin(x))/4. - Stefano Spezia, Feb 23 2025

A338491 Least number of centered pentagonal numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6

Offset: 1

Ilya Gutkovskiy, Oct 30 2020

nonn

Eric Weisstein's World of Mathematics, Centered Pentagonal Number
Index entries for sequences related to centered polygonal numbers

Cf. A005891, A100878, A101412, A234533, A338482, A338484, A338492, A338494.

cp's OEIS Frontend

A164015 5 times centered pentagonal numbers: 5*(5*n^2 + 5*n + 2)/2.

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A362863 Centered hecatonicosachoral numbers.

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A364610 Centered pentagonal numbers which are products of three distinct primes.

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A071910 a(n) = t(n)*t(n+1)*t(n+2), where t() are the triangular numbers.

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A121722 Triangle T(n,k) = 1 + k*n*(n+1)/2, read by rows.

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A131751 Numbers that are both centered triangular and centered pentagonal.

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A140702 Main diagonal of array A(k,n) = product of first n centered n-gonal numbers.

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A164015 5 times centered pentagonal numbers: 5(5n^2 + 5*n + 2)/2.

A071910 a(n) = t(n)t(n+1)t(n+2), where t() are the triangular numbers.

A121722 Triangle T(n,k) = 1 + kn(n+1)/2, read by rows.