A063522 -id:A063522 - cp's OEIS frontend

A004188 a(n) = n(3n^2 - 1)/2.

0, 1, 11, 39, 94, 185, 321, 511, 764, 1089, 1495, 1991, 2586, 3289, 4109, 5055, 6136, 7361, 8739, 10279, 11990, 13881, 15961, 18239, 20724, 23425, 26351, 29511, 32914, 36569, 40485, 44671, 49136, 53889, 58939, 64295, 69966, 75961

Offset: 0

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

3-dimensional analog of centered polygonal numbers.

(1), (4+7), (10+13+16), (19+22+25+28), ... - Jon Perry, Sep 10 2004

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).

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

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.
Cf. A002412, A016061, A051682.
Cf. A236770 (partial sums).

[n*(3*n^2-1)/2: n in [0..50]]; //Vincenzo Librandi, May 15 2011

seq(binomial(2*n+1,3)+binomial(n+1,3), n=0..37); # Zerinvary Lajos, Jan 21 2007

Table[n(3n^2-1)/2,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4,-6,4,-1},{0,1,11,39},40] (* Harvey P. Dale, Jul 19 2019 *)

PARI
```
vector(40, n, n*(3*n^2-1)/2)
```

Partial sums of n-1 3-spaced triangular numbers, e.g., a(4) = t(1) + t(4) + t(7) = 1 + 10 + 28 = 39. - Jon Perry, Jul 23 2003
a(n) = C(2*n+1,3) + C(n+1,3), n >= 0. - Zerinvary Lajos, Jan 21 2007
a(n) = A000447(n) + A000292(n). - Zerinvary Lajos, Jan 21 2007
G.f.: x*(1+7*x+x^2) / (x-1)^4. - R. J. Mathar, Oct 08 2011
From Miquel Cerda, Dec 25 2016: (Start)
a(n) = A000578(n) + A135503(n).
a(n) = A007588(n) - A135503(n). (End)
E.g.f.: (x/2)*(2 + 9*x + 3*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

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

A063521 a(n) = n(7n^2-4)/3.

Original entry on oeis.org

0, 1, 16, 59, 144, 285, 496, 791, 1184, 1689, 2320, 3091, 4016, 5109, 6384, 7855, 9536, 11441, 13584, 15979, 18640, 21581, 24816, 28359, 32224, 36425, 40976, 45891, 51184, 56869, 62960, 69471, 76416, 83809, 91664, 99995, 108816, 118141, 127984, 138359, 149280, 160761

Offset: 0

N. J. A. Sloane, Aug 02 2001

Also as a(n)=(1/6)*(14*n^3-8*n), n>0: structured heptagonal anti-diamond numbers (vertex structure 15) (Cf. A100186 = alternate vertex; A100188 = structured anti-diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Harry J. Smith, Table of n, a(n) for n = 0..1000
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).

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.

A063521:=n->n*(7*n^2-4)/3; seq(A063521(k), k=0..100); # Wesley Ivan Hurt, Oct 24 2013

lst={};Do[AppendTo[lst, n*(7*n^2-4)/3], {n, 1, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 02 2008 *)
CoefficientList[Series[x*(1+12*x+x^2)/(1-x)^4, {x, 0, 50}], x] (* G. C. Greubel, Sep 01 2017 *)

a(n) = { n*(7*n^2 - 4)/3 } \\ Harry J. Smith, Aug 25 2009

G.f.: x*(1+12*x+x^2)/(1-x)^4. - Colin Barker, Jan 10 2012

E.g.f.: (x/3)*(3 + 21*x + 7*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A004126 a(n) = n(7n^2 - 1)/6.

Original entry on oeis.org

0, 1, 9, 31, 74, 145, 251, 399, 596, 849, 1165, 1551, 2014, 2561, 3199, 3935, 4776, 5729, 6801, 7999, 9330, 10801, 12419, 14191, 16124, 18225, 20501, 22959, 25606, 28449, 31495, 34751, 38224, 41921, 45849, 50015, 54426, 59089, 64011

Offset: 0

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

3-dimensional analog of centered polygonal numbers.
Sum of n triangular numbers starting from T(n), where T = A000217. E.g., a(4) = T(4) + T(5) + T(6) + T(7) = 10 + 15 + 21 + 28 = 74. - Amarnath Murthy, Jul 16 2004
Also as a(n) = (1/6)*(7*n^3-n), n>0: structured heptagonal diamond numbers (vertex structure 8). Cf. A100179 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Partial sums of A069099, centered heptagonal numbers (A000566). - Jonathan Vos Post, Mar 16 2006
Binomial transform of (0, 1, 7, 7, 0, 0, 0, ...) and third partial sum of (0, 1, 6, 7, 7, 7, ...). - Gary W. Adamson, Oct 05 2015

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).

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.
Cf. A000217, A000566, A016993, A069099.
Cf. A000447, A000292.

[n*(7*n^2-1)/6: n in [0..50]]; // Vincenzo Librandi, May 15 2011

seq(binomial(2*n+1,3)-binomial(n+1,3), n=0..38); # Zerinvary Lajos, Jan 21 2007

Table[n (7 n^2 - 1)/6, {n, 0, 80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

makelist(n*(7*n^2-1)/6,n,0,30); /* Martin Ettl, Jan 08 2013 */

vector(100, n, n--; n*(7*n^2 - 1)/6) \\ Altug Alkan, Oct 06 2015

a(n) = C(2*n+1,3)-C(n+1,3), n>=0. - Zerinvary Lajos, Jan 21 2007
a(n) = A000447(n) - A000292(n). - Zerinvary Lajos, Jan 21 2007
G.f.: x*(1+5*x+x^2)/(1-x)^4. - Colin Barker, Mar 02 2012
E.g.f.: (x/6)*(7*x^2 + 21*x + 6)*exp(x). - G. C. Greubel, Oct 05 2015
a(n) = Sum_{i = n..2*n-1} A000217(i). - Bruno Berselli, Sep 06 2017
a(n) = n^3 + Sum_{k=0..n-1} k*(k+1)/2. Alternately, a(n) = A000578(n) + A000292(n-1) for n>0. - Bruno Berselli, May 23 2018

A004467 a(n) = n(11n^2 - 5)/6.

Original entry on oeis.org

0, 1, 13, 47, 114, 225, 391, 623, 932, 1329, 1825, 2431, 3158, 4017, 5019, 6175, 7496, 8993, 10677, 12559, 14650, 16961, 19503, 22287, 25324, 28625, 32201, 36063, 40222, 44689, 49475, 54591, 60048

Offset: 0

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

3-dimensional analog of centered polygonal numbers, that is: centered hendecagonal pyramidal numbers (see Deza paper in References).

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).

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*(11*n^2-5)/6: n in [0..50]]; // Vincenzo Librandi, May 15 2011

Table[n(11n^2-5)/6,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4,-6,4,-1},{0,1,13,47},80] (* Harvey P. Dale, Sep 22 2013 *)

a(n)=n*(11*n^2-5)/6 \\ Charles R Greathouse IV, Sep 28 2011

G.f.: x*(1+9*x+x^2)/(1-x)^4. - Colin Barker, Jan 08 2012
a(0)=0, a(1)=1, a(2)=13, a(3)=47; for n>3, a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Sep 22 2013
E.g.f.: (x/6)*(6 + 33*x + 11*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A062025 a(n) = n(13n^2 - 7)/6.

Original entry on oeis.org

0, 1, 15, 55, 134, 265, 461, 735, 1100, 1569, 2155, 2871, 3730, 4745, 5929, 7295, 8856, 10625, 12615, 14839, 17310, 20041, 23045, 26335, 29924, 33825, 38051, 42615, 47530, 52809, 58465, 64511, 70960, 77825, 85119, 92855, 101046, 109705, 118845, 128479, 138620, 149281

Offset: 0

N. J. A. Sloane, Aug 02 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
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).

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.

Table[n(13n^2-7)/6,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

a(n) = n*(13*n^2 - 7)/6 \\ Harry J. Smith, Jul 29 2009

From G. C. Greubel, Sep 01 2017: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (x + 11*x^2 + x^3)/(1 - x)^4.
E.g.f.: (x/6)*(6 + 39*x + 13*x^2)*exp(x). (End)

A063523 a(n) = n(8n^2 - 5)/3.

Original entry on oeis.org

0, 1, 18, 67, 164, 325, 566, 903, 1352, 1929, 2650, 3531, 4588, 5837, 7294, 8975, 10896, 13073, 15522, 18259, 21300, 24661, 28358, 32407, 36824, 41625, 46826, 52443, 58492, 64989, 71950, 79391, 87328, 95777, 104754, 114275, 124356, 135013, 146262, 158119, 170600

Offset: 0

N. J. A. Sloane, Aug 02 2001

Also as a(n)=(1/6)*(16*n^3-10*n), n>0: structured octagonal anti-diamond numbers (vertex structure 17) (Cf. A100187 = alternate vertex; A100188 = structured anti-diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Harry J. Smith, Table of n, a(n) for n = 0..1000
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).

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.

Table[n(8n^2-5)/3,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4,-6,4,-1},{0,1,18,67},81] (* or *) CoefficientList[ Series[ (x+14 x^2+x^3)/(x-1)^4,{x,0,80}],x] (* Harvey P. Dale, Jul 11 2011 *)

a(n) = n*(8*n^2 - 5)/3 \\ Harry J. Smith, Aug 25 2009

a(0)=0, a(1)=1, a(2)=18, a(3)=67, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)- a(n-4). - Harvey P. Dale, Jul 11 2011
G.f.: (x+14*x^2+x^3)/(x-1)^4. - Harvey P. Dale, Jul 11 2011
E.g.f.: (x/3)*(3 + 24*x + 8*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

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A004188 a(n) = n*(3*n^2 - 1)/2.

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

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A063521 a(n) = n*(7*n^2-4)/3.

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A004126 a(n) = n*(7*n^2 - 1)/6.

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A004467 a(n) = n*(11*n^2 - 5)/6.

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A062025 a(n) = n*(13*n^2 - 7)/6.

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A063523 a(n) = n*(8*n^2 - 5)/3.

A004188 a(n) = n(3n^2 - 1)/2.

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

A063521 a(n) = n(7n^2-4)/3.

A004126 a(n) = n(7n^2 - 1)/6.

A004467 a(n) = n(11n^2 - 5)/6.

A062025 a(n) = n(13n^2 - 7)/6.

A063523 a(n) = n(8n^2 - 5)/3.