A100145 -id:A100145 - cp's OEIS frontend

A100177 Structured meta-prism numbers, the n-th number from a structured n-gonal prism number sequence.

1, 4, 18, 64, 175, 396, 784, 1408, 2349, 3700, 5566, 8064, 11323, 15484, 20700, 27136, 34969, 44388, 55594, 68800, 84231, 102124, 122728, 146304, 173125, 203476, 237654, 275968, 318739, 366300, 418996, 477184, 541233, 611524, 688450, 772416, 863839, 963148, 1070784, 1187200, 1312861, 1448244

Offset: 1

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

			There are no 1- or 2-gonal prisms, so 1 and (2n) 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. A002411, A000578, A050509, A006597, A100176, A100177 - structured prisms; A006484 for other meta structured numbers; and A100145 for more on structured numbers.

Cf. A060354, A000124.

[(1/6)*(3*n^4-9*n^3+12*n^2): n in [1..50] ]; // Vincenzo Librandi, Aug 02 2011

Table[(3n^4-9n^3+12n^2)/6,{n,50}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,4,18,64,175},50] (* Harvey P. Dale, Nov 07 2017 *)

PARI
```
a(n)=(1/6)*(3*n^4-9*n^3+12*n^2);
```

a(n) = (1/6)*(3*n^4 - 9*n^3 + 12*n^2).
G.f.: x*(1 - x + 8*x^2 + 4*x^3)/(1-x)^5. - Colin Barker, Jun 08 2012
a(n) = A060354(n) * n = A000124(n-2) * n^2. - Bruce J. Nicholson, Jul 11 2018

A100185 Structured meta-anti-prism numbers, the n-th number from a structured n-gonal anti-prism number sequence.

Original entry on oeis.org

1, 4, 19, 68, 185, 416, 819, 1464, 2433, 3820, 5731, 8284, 11609, 15848, 21155, 27696, 35649, 45204, 56563, 69940, 85561, 103664, 124499, 148328, 175425, 206076, 240579, 279244, 322393, 370360, 423491, 482144

Offset: 1

James A. Record (james.record(AT)gmail.com)

			There are no 1- or 2-gonal anti-prisms, so 1 and (2n) 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. A005900, A000447, A096000, A100178, A100157, A100185 - structured anti-prisms; A006484 for other structured meta numbers; and A100145 for more on structured numbers.

[(1/6)*(3*n^4-8*n^3+9*n^2+2*n): n in [1..40]]; // Vincenzo Librandi, Aug 03 2011

a(n) = (1/6)*(3*n^4 - 8*n^3 + 9*n^2 + 2*n).

G.f.: x*(1 - x + 9*x^2 + 3*x^3)/(1-x)^5. [Colin Barker, Jun 08 2012]

A051673 Cubic star numbers: a(n) = n^3 + 4*Sum_{i=0..n-1} i^2.

Original entry on oeis.org

0, 1, 12, 47, 120, 245, 436, 707, 1072, 1545, 2140, 2871, 3752, 4797, 6020, 7435, 9056, 10897, 12972, 15295, 17880, 20741, 23892, 27347, 31120, 35225, 39676, 44487, 49672, 55245, 61220, 67611, 74432, 81697, 89420, 97615, 106296, 115477, 125172

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

Also as a(n) = (1/6)*(14*n^3 - 12*n^2 + 4*n), n>0: structured cubeoctahedral numbers (vertex structure 7); and structured pentagonal anti-diamond numbers (vertex structure 7) (cf. A004466 = alternate vertex) (cf. A100188 = structured anti-diamonds). Cf. A100145 for more on structured polyhedral numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Starting with offset 1 = binomial transform of [1, 11, 24, 14, 0, 0, 0, ...]. - Gary W. Adamson, Aug 05 2009
This is prime for a(3) = 47. The subsequence of semiprimes begins: 707, 7435, 10897, 20741, 115477, 341797, 825091, 897097, no more through a(100). - Jonathan Vos Post, May 27 2010

			a(51) = 51*(51*(7*51-6)+2)/3 = 304351 = 17 * 17903 is semiprime. - _Jonathan Vos Post_, May 27 2010

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

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

Cf. A002378, A004466, A005915, A051662, A100145, A100188.

Cf. A214659, A214661.

[n*(n*(7*n-6)+2)/3: n in [0..50]]; // Vincenzo Librandi, May 12 2011

A051673:=n->n*(n*(7*n-6)+2)/3; seq(A051673(n), n=0..40); # Wesley Ivan Hurt, Feb 02 2014

Table[n^3+4Sum[i^2,{i,0,n-1}],{n,0,40}] (* or *) LinearRecurrence[ {4,-6,4,-1},{0,1,12,47},40] (* Harvey P. Dale, Jul 22 2011 *)

a(n)=n*(n*(7*n-6)+2)/3 \\ Charles R Greathouse IV, Oct 07 2015

[n*(7*n^2-6*n+2)/3 for n in range(51)] # G. C. Greubel, Mar 10 2024

a(n) = n*(n*(7*n-6) + 2)/3.
G.f.: x*(1+8*x+5*x^2)/(1-x)^4. - Bruno Berselli, May 12 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=12, a(3)=47. - Harvey P. Dale, Jul 22 2011
From Reinhard Zumkeller, Jul 25 2012: (Start)
a(n) = A214659(n) - A002378(n).
a(n) = Sum_{k=1..n} A214661(n, k), for n > 0 (row sums). (End)
E.g.f.: (x/3)*(3 + 15*x + 7*x^2)*exp(x). - G. C. Greubel, Mar 10 2024

Corrected by T. D. Noe, Nov 01 2006, Nov 08 2006

A096000 Cupolar numbers: a(n) = (n+1)(5n^2 + 7*n + 3)/3.

Original entry on oeis.org

1, 10, 37, 92, 185, 326, 525, 792, 1137, 1570, 2101, 2740, 3497, 4382, 5405, 6576, 7905, 9402, 11077, 12940, 15001, 17270, 19757, 22472, 25425, 28626, 32085, 35812, 39817, 44110, 48701, 53600, 58817, 64362, 70245, 76476, 83065, 90022, 97357, 105080, 113201, 121730

Offset: 0

N. J. A. Sloane, in memory of Harold Scott MacDonald Coxeter [Feb 09 1907 - Mar 31 2003], May 08 2004

Number of equal balls that will fill a triangular cupola, formed by splitting a cuboctahedron along one of its four "equilateral" hexagons.

Also as a(n) = (1/6)*(10*n^3 - 6*n^2 + 10*n), n>0: structured pentagonal anti-prism numbers (Cf. A100185 = structured anti-prisms); and structured tetragonal anti-diamond numbers (vertex structure 7) (Cf. A000447 = alternate vertex; A100188 = structured anti-diamonds). Cf. A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004

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.

T. D. Noe, Table of n, a(n) for n = 0..1000
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000447, A005902, A100145, A100185, A100188.

[(n+1)*(5*n^2+7*n+3)/3 : n in [0..50]]; // Wesley Ivan Hurt, May 23 2015

A096000:=n->(n+1)*(5*n^2+7*n+3)/3; seq(A096000(n), n=0..50); # Wesley Ivan Hurt, Mar 11 2014

Table[(n + 1)(5n^2 + 7n + 3)/3, {n, 0, 50}] (* Wesley Ivan Hurt, Mar 11 2014 *)
CoefficientList[Series[(1 + 6 x + 3 x^2)/(1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 23 2015 *)

a(n) = (1/3)*(n+1)*(5*n^2+7*n+3) \\ Michel Marcus, Jul 11 2013

a(n) = (1/2)*(Q(n) + 3*n^2 + 3*n + 1), where Q(n) are the cuboctahedral numbers, A005902.
G.f.: (1+6*x+3*x^2)/(1-x)^4. - Paul Barry, Oct 28 2006
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3. - Wesley Ivan Hurt, May 23 2015
E.g.f.: exp(x)*(3 + 27*x + 27*x^2 + 5*x^3)/3. - Elmo R. Oliveira, Aug 11 2025

A100147 Structured icosidodecahedral numbers.

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, 424650

Offset: 1

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

Equals row sums of triangle A143254 & binomial transform of [1, 29, 76, 48, 0, 0, 0, ...]. - Gary W. Adamson, Aug 02 2008

Apart from offset, same as A079588.

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. A100146, A100148 for adjacent structured Archimedean solids; and A100145 for more on structured polyhedral numbers.

Cf. also A079588.

[(1+(n-1))*(1+2*(n-1))*(1+4*(n-1)): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011

Table[(48*n^3 - 60*n^2 + 18*n)/6, {n,1,50}] (* G. C. Greubel, Oct 18 2018 *)

a(n)=n*(8*n^2-10*n+3) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (1/6)*(48*n^3 - 60*n^2 + 18*n).
a(n) = A079588(n-1) = n*(2*n-1)*(4*n-3). - R. J. Mathar, Sep 02 2008
From Jaume Oliver Lafont, Sep 08 2009: (Start)
a(n) = (1+(n-1))*(1+2*(n-1))*(1+4*(n-1)).
G.f.: x*(1 + 26*x + 21*x^2)/(1-x)^4. (End)
E.g.f.: x*(1 + 14*x + 8*x^2)*exp(x). - G. C. Greubel, Oct 18 2018
From Amiram Eldar, Sep 20 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*sqrt(2)*log(sqrt(2)+1)/3 + log(2)/3 - (3 - 2*sqrt(2))*Pi/6. (End)

A100175 Structured triakis tetrahedral numbers (vertex structure 4).

Original entry on oeis.org

1, 8, 30, 76, 155, 276, 448, 680, 981, 1360, 1826, 2388, 3055, 3836, 4740, 5776, 6953, 8280, 9766, 11420, 13251, 15268, 17480, 19896, 22525, 25376, 28458, 31780, 35351, 39180, 43276, 47648, 52305, 57256, 62510, 68076, 73963, 80180, 86736, 93640, 100901, 108528

Offset: 1

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

Equals binomial transform of [1, 7, 15, 9, 0, 0, 0, ...] where (1, 7, 15, 9) = row 3 of triangle A038763. - Gary W. Adamson, Jul 19 2008

Equals convolution square of 1, 4, 7, 10, 13, 16, 19, ..., A016777. - Gary W. Adamson, Jul 28 2009

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

Cf. A000578 (alternate vertex), A100145 for more on structured numbers.

Cf. A038763.

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

CoefficientList[Series[x (2x+1)^2/((x-1)^4),{x,0,50}],x] (* or *) LinearRecurrence[{4,-6,4,-1},{0,1,8,30},50] (* Harvey P. Dale, Mar 18 2023 *)

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

G.f.: x*(2*x+1)^2 / ( (x-1)^4 ).

A006597 a(n) = n^2(5n-3)/2.

Original entry on oeis.org

0, 1, 14, 54, 136, 275, 486, 784, 1184, 1701, 2350, 3146, 4104, 5239, 6566, 8100, 9856, 11849, 14094, 16606, 19400, 22491, 25894, 29624, 33696, 38125, 42926, 48114, 53704, 59711, 66150, 73036, 80384, 88209, 96526, 105350, 114696, 124579, 135014, 146016, 157600

Offset: 0

N. J. A. Sloane

Structured heptagonal prism numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Apart from 0, partial sums of A220083. - Bruno Berselli, Dec 11 2012

W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 29.

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

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. A006592, A220083.

[n^2*(5*n-3)/2: n in [0..40]]; // Vincenzo Librandi, Jul 20 2011

A006597:=n->n^2*(5*n-3)/2; seq(A006597(n), n=0..40); # Wesley Ivan Hurt, Mar 11 2014

Table[n^2*(5*n-3)/2, {n, 0, 40}] (* Wesley Ivan Hurt, Mar 11 2014 *)

a(n)=n^2*(5*n-3)/2; \\ Joerg Arndt, Jul 20 2011

a(n) = (1/6)*(15*n^3 - 9*n^2). - James A. Record (james.record(AT)gmail.com), Nov 07 2004
G.f.: x*(1+10*x+4*x^2)/(1-x)^4. - Colin Barker, Jun 08 2012
a(n) = Sum_{i=0..n-1} n*(5*i+1) for n>0. - Bruno Berselli, Sep 08 2015
Sum_{n>=1} 1/a(n) = 1.1080093773051638036... = (sqrt(5*(5 - 2*sqrt(5)))*Pi - Pi^2 - 5*sqrt(5)*arccoth(sqrt(5)) + (25*log(5))/2)/9. - Vaclav Kotesovec, Oct 04 2016
From Elmo R. Oliveira, Aug 06 2025: (Start)
E.g.f.: exp(x)*x*(2 + 12*x + 5*x^2)/2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = A006592(n)/4. (End)

Name corrected by Arkadiusz Wesolowski, Jul 20 2011

A050509 House numbers (version 2): a(n) = (n+1)^3 + (n+1)*Sum_{i=0..n} i.

Original entry on oeis.org

1, 10, 36, 88, 175, 306, 490, 736, 1053, 1450, 1936, 2520, 3211, 4018, 4950, 6016, 7225, 8586, 10108, 11800, 13671, 15730, 17986, 20448, 23125, 26026, 29160, 32536, 36163, 40050, 44206, 48640, 53361, 58378, 63700, 69336, 75295, 81586, 88218, 95200, 102541

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 28 1999

Also as a(n) = (1/6)*(9*n^3 - 3*n^2), n>0: structured pentagonal prism numbers (Cf. A100177 - structured prisms; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Number of inequivalent tetrahedral edge colorings using at most n+1 colors so that no color appears only once. - David Nacin, Feb 22 2017

			        *     *
a(2) = * * + * * = 10.
       * *   * *

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000217, A000578, A051662, A100145, A100177.

Cf. similar sequences, with the formula (k*n - k + 2)*n^2/2, listed in A262000.

[(3*n+2)*(n+1)^2/2: n in [0..40]]; // Vincenzo Librandi, Jul 19 2011

Table[((1+n)^2*(2+3n))/2,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,10,36,88},40] (* Harvey P. Dale, Jun 26 2011 *)

a(n)=(1/2)*(3*n+2)*(n+1)^2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A000578(n+1) + (n+1)*A000217(n).
a(n) = (1/2)*(3*n+2)*(n+1)^2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=10, a(2)=36, a(3)=88. - Harvey P. Dale, Jun 26 2011
G.f.: (1+6*x+2*x^2)/(1-x)^4. - Colin Barker, Jun 08 2012
a(n) = Sum_{i=0..n} (n+1)*(3*i+1). - Bruno Berselli, Sep 08 2015
Sum_{n>=0} 1/a(n) = 9*log(3) - sqrt(3)*Pi - Pi^2/3 = 1.15624437161388... . - Vaclav Kotesovec, Oct 04 2016
E.g.f.: exp(x)*(2 + 18*x + 17*x^2 + 3*x^3)/2. - Elmo R. Oliveira, Aug 06 2025

A100165 Structured rhombic triacontahedral numbers (vertex structure 7).

Original entry on oeis.org

1, 32, 147, 400, 845, 1536, 2527, 3872, 5625, 7840, 10571, 13872, 17797, 22400, 27735, 33856, 40817, 48672, 57475, 67280, 78141, 90112, 103247, 117600, 133225, 150176, 168507, 188272, 209525, 232320, 256711, 282752

Offset: 1

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

Also structured pentakis dodecahedral numbers (vertex structure 7) (cf. A100173 = alternate vertex).

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. A100164 = alternate vertex; A100145 for more on structured numbers.

Cf. A260260 (comment). [Bruno Berselli, Jul 22 2015]

[(1/6)*(54*n^3-72*n^2+24*n): n in [1..40]]; // Vincenzo Librandi, Jul 26 2011

Table[(3(n-1)+1)^2(3(n-1)+3)/3,{n,40}] (* Harvey P. Dale, Jan 02 2020 *)

a(n) = (1/6)*(54*n^3 - 72*n^2 + 24*n) = n*(3*n-2)^2.
From Jaume Oliver Lafont, Sep 08 2009: (Start)
a(n) = (3*(n-1) + 1)^2*(3*(n-1) + 3)/3.
G.f.: x*(1 + 28*x + 25*x^2)/(1-x)^4. (End)

A100158 Structured disdyakis triacontahedral numbers (vertex structure 11).

Original entry on oeis.org

1, 62, 293, 804, 1705, 3106, 5117, 7848, 11409, 15910, 21461, 28172, 36153, 45514, 56365, 68816, 82977, 98958, 116869, 136820, 158921, 183282, 210013, 239224, 271025, 305526, 342837, 383068, 426329, 472730, 522381, 575392

Offset: 1

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

Also structured deltoidal hexacontahedral numbers (vertex structure 11) (cf. A100166, A100159 = alternate vertices).

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. A100159, A100160 = alternate vertices; A100145 for more on structured polyhedral numbers.

[(1/6)*(110*n^3-150*n^2+46*n): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011

Table[(110*n^3 - 150*n^2 + 46*n)/6, {n,1,50}] (* or *) LinearRecurrence[{4,-6,4,-1}, {1, 62, 293, 804}, 50] (* G. C. Greubel, Oct 18 2018 *)

vector(50, n, (110*n^3 - 150*n^2 + 46*n)/6) \\ G. C. Greubel, Oct 18 2018

a(n) = (1/6)*(110*n^3 - 150*n^2 + 46*n).
G.f.: x*(1 + 58*x + 51*x^2)/(1-x)^4. - Colin Barker, Apr 16 2012
E.g.f.: x*(3 + 90*x + 55*x^2)*exp(x)/3. - G. C. Greubel, Oct 18 2018

cp's OEIS Frontend

A100177 Structured meta-prism numbers, the n-th number from a structured n-gonal prism number sequence.

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A100185 Structured meta-anti-prism numbers, the n-th number from a structured n-gonal anti-prism number sequence.

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A051673 Cubic star numbers: a(n) = n^3 + 4*Sum_{i=0..n-1} i^2.

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A096000 Cupolar numbers: a(n) = (n+1)*(5*n^2 + 7*n + 3)/3.

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A100147 Structured icosidodecahedral numbers.

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A100175 Structured triakis tetrahedral numbers (vertex structure 4).

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A006597 a(n) = n^2*(5*n-3)/2.

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A096000 Cupolar numbers: a(n) = (n+1)(5n^2 + 7*n + 3)/3.

A006597 a(n) = n^2(5n-3)/2.