A144138 -id:A144138 - cp's OEIS frontend

A007588 Stella octangula numbers: a(n) = n(2n^2 - 1).

0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, 2651, 3444, 4381, 5474, 6735, 8176, 9809, 11646, 13699, 15980, 18501, 21274, 24311, 27624, 31225, 35126, 39339, 43876, 48749, 53970, 59551, 65504, 71841, 78574, 85715, 93276, 101269, 109706, 118599, 127960

Offset: 0

N. J. A. Sloane

Also as a(n)=(1/6)*(12*n^3-6*n), n>0: structured hexagonal anti-diamond numbers (vertex structure 13) (Cf. A005915 = alternate vertex; A100188 = structured anti-diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004
The only known square stella octangula number for n>1 is a(169) = 169*(2*169^2 - 1) = 9653449 = 3107^2. - Alexander Adamchuk, Jun 02 2008
Ljunggren proved that 9653449 = (13*239)^2 is the only square stella octangula number for n>1. See A229384 and the Wikipedia link. - Jonathan Sondow, Sep 30 2013
4*A007588 = A144138(ChebyshevU[3,n]). - Vladimir Joseph Stephan Orlovsky, Jun 30 2011
If A016813 is regarded as a regular triangle (with leading terms listed in A001844), a(n) provides the row sums of this triangle: 1, 5+9=14, 13+17+21=51 and so on. - J. M. Bergot, Jul 05 2013
Shares its digital root, A267017, with n*(n^2 + 1)/2 ("sum of the next n natural numbers" see A006003). - Peter M. Chema, Aug 28 2016

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 51.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
W. Ljunggren, Zur Theorie der Gleichung x^2 + 1 = Dy^4, Avh. Norske Vid. Akad. Oslo. I. 1942 (5): 27.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alexander Adamchuk and Vincenzo Librandi, Table of n, a(n) for n = 0..10000 [Alexander Adamchuk computed terms 0 - 169, Jun 02, 2008; Vincenzo Librandi computed the first 10000 terms, Aug 18 2011]
A. Bremner, R. Høibakk and D. Lukkassen, Crossed ladders and Euler’s quartic, Annales Mathematicae et Informaticae, 36 (2009) pp. 29-41. See p. 33.
John Elias, Nesting Cubes of the Surface Points of a Hexagonal Prism
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Amelia Carolina Sparavigna, Generalized Sum of Stella Octangula Numbers, Politecnico di Torino (Italy, 2021).
Amelia Carolina Sparavigna, Cardano Formula and Some Figurate Numbers, Politecnico di Torino (Italy, 2021).
Eric Weisstein's World of Mathematics, Stella Octangula Number
Wikipedia, Stella octangula number
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Backwards differences give star numbers A003154: A003154(n)=a(n)-a(n-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. A001653 = Numbers n such that 2*n^2 - 1 is a square.
a(169) = (A229384(3)*A229384(4))^2.
Cf. A267017, A006003, A135503.
Cf. A002378, A005843, A061317, A062392.

[n*(2*n^2 - 1): n in [0..40]]; // Vincenzo Librandi, Aug 18 2011

A007588:=n->n*(2*n^2 - 1); seq(A007588(n), n=0..40); # Wesley Ivan Hurt, Mar 10 2014

Table[ n(2n^2-1), {n,0,169} ] (* Alexander Adamchuk, Jun 02 2008 *)
LinearRecurrence[{4,-6,4,-1},{0,1,14,51},50] (* Harvey P. Dale, Sep 16 2011 *)

PARI
```
a(n)=n*(2*n^2-1)
```

def A007588(n): return n*(2*n**2-1) # Chai Wah Wu, Feb 18 2022

G.f.: x*(1+10*x+x^2)/(1-x)^4.
a(n) = n*A056220(n).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3. - Harvey P. Dale, Sep 16 2011
From Ilya Gutkovskiy, Jul 02 2016: (Start)
E.g.f.: x*(1 + 6*x + 2*x^2)*exp(x).
Dirichlet g.f.: 2*zeta(s-3) - zeta(s-1). (End)
a(n) = A004188(n) + A135503(n). - Miquel Cerda, Dec 25 2016
a(n) = A061317(n) - A005843(n) = A062392(n) - A062392(n-1). - J.S. Seneschal, Jul 01 2025

In the formula given in the 1995 Encyclopedia of Integer Sequences, the second 2 should be an exponent.

A323182 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Chebyshev polynomial of the second kind U_{n}(x), evaluated at x=k.

Original entry on oeis.org

1, 1, 0, 1, 2, -1, 1, 4, 3, 0, 1, 6, 15, 4, 1, 1, 8, 35, 56, 5, 0, 1, 10, 63, 204, 209, 6, -1, 1, 12, 99, 496, 1189, 780, 7, 0, 1, 14, 143, 980, 3905, 6930, 2911, 8, 1, 1, 16, 195, 1704, 9701, 30744, 40391, 10864, 9, 0, 1, 18, 255, 2716, 20305, 96030, 242047, 235416, 40545, 10, -1

Offset: 0

Seiichi Manyama, Jan 06 2019

			Square array begins:
   1, 1,    1,     1,      1,      1,       1, ...
   0, 2,    4,     6,      8,     10,      12, ...
  -1, 3,   15,    35,     63,     99,     143, ...
   0, 4,   56,   204,    496,    980,    1704, ...
   1, 5,  209,  1189,   3905,   9701,   20305, ...
   0, 6,  780,  6930,  30744,  96030,  241956, ...
  -1, 7, 2911, 40391, 242047, 950599, 2883167, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Mirror of A228161.
Columns 0-19 give A056594, A000027(n+1), A001353(n+1), A001109(n+1), A001090(n+1), A004189(n+1), A004191, A007655(n+2), A077412, A049660(n+1), A075843(n+1), A077421, A077423, A097309, A097311, A097313, A029548, A029547, A144128(n+1), A078987.
Rows 0-10 give A000012, A005843, A000466, A144138, A144139, A242850, A242851, A242852, A242853, A242854, A243130.
Main diagonal gives A323118.
Cf. A179943, A322836 (Chebyshev polynomial of the first kind).

T(n,k)  = polchebyshev(n, 2, k);
matrix(7, 7, n, k, T(n-1,k-1)) \\ Michel Marcus, Jan 07 2019

T(n, k) = sum(j=0, n, (2*k-2)^j*binomial(n+1+j, 2*j+1)); \\ Seiichi Manyama, Mar 03 2021

T(0,k) = 1, T(1,k) = 2 * k and T(n,k) = 2 * k * T(n-1,k) - T(n-2,k) for n > 1.

T(n, k) = Sum_{j=0..n} (2*k-2)^j * binomial(n+1+j,2*j+1). - Seiichi Manyama, Mar 03 2021

A242850 32n^5 - 32n^3 + 6*n.

Original entry on oeis.org

0, 6, 780, 6930, 30744, 96030, 241956, 526890, 1032240, 1866294, 3168060, 5111106, 7907400, 11811150, 17122644, 24192090, 33423456, 45278310, 60279660, 79015794, 102144120, 130395006, 164575620, 205573770, 254361744, 312000150, 379641756, 458535330, 550029480, 655576494

Offset: 0

Vincenzo Librandi, May 29 2014

Chebyshev polynomial of the second kind U(5,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000466, A144138, A144139, A242851.

Magma
```
[32*n^5-32*n^3+6*n: n in [0..40]];
```

Table[32 n^5 - 32 n^3 + 6 n, {n, 0, 40}] (* or *) Table[ChebyshevU[5, n], {n, 0, 40}]

G.f.: x*(6 + 744*x + 2340*x^2 + 744*x^3 + 6*x^4)/(1 - x)^6.

a(n) = 2*n*(2*n-1)*(2*n+1)*(4*n^2-3).

Edited by Bruno Berselli, May 29 2014

A242851 64n^6 - 80n^4 + 24*n^2 - 1.

Original entry on oeis.org

-1, 7, 2911, 40391, 242047, 950599, 2883167, 7338631, 16451071, 33489287, 63202399, 112211527, 189447551, 306634951, 478821727, 724955399, 1068505087, 1538129671, 2168392031, 3000519367, 4083209599, 5473483847, 7237584991, 9451922311, 12204062207, 15593764999

Offset: 0

Vincenzo Librandi, May 29 2014

Chebyshev polynomial of the second kind U(6,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000466, A144138, A144139, A242850.

Magma
```
[64*n^6-80*n^4+24*n^2-1: n in [0..40]];
```

Table[64 n^6 - 80 n^4 + 24 n^2 - 1, {n, 0, 40}] (* or *) Table[ChebyshevU[6, n], {n, 0, 40}]

G.f.: (-1 + 14*x + 2841*x^2 + 20196*x^3 + 20161*x^4 + 2862*x^5 + 7*x^6)/(1 - x)^7.

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

Edited by Bruno Berselli, May 29 2014

A242852 128n^7-192n^5+80n^3-8n.

Original entry on oeis.org

0, 8, 10864, 235416, 1905632, 9409960, 34356048, 102213944, 262184896, 600940872, 1260879920, 2463542488, 4538833824, 7960697576, 13389885712, 21724469880, 34158739328, 52251130504, 78001833456, 113940720152, 163226239840, 229755926568, 318289163984, 434582852536

Offset: 0

Vincenzo Librandi, May 30 2014

Chebyshev polynomial of the second kind U(7,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A144138, A144139, A242850, A242851.

[128*n^7-192*n^5+80*n^3-8*n: n in [0..30]];

Table[ChebyshevU[7, n], {n, 0, 30}] (* or *) Table[128 n^7 - 192 n^5 + 80 n^3 - 8 n, {n, 0, 30}]

G.f.: (8*x + 10800*x^2 + 148728*x^3 + 326048*x^4 + 148728*x^5 + 10800*x^6 + 8*x^7)/(1-x)^8.

a(n) = 8*n*(2*n^2-1)*(8*n^4-8*n^2+1).

A242853 256n^8 - 448n^6 + 240n^4 - 40n^2 + 1.

Original entry on oeis.org

1, 9, 40545, 1372105, 15003009, 93149001, 409389409, 1423656585, 4178507265, 10783446409, 25154396001, 54085723209, 108742564225, 206671502025, 374437978209, 651009141001, 1092011153409, 1775000307465, 2805897612385, 4326746846409, 6524966384001, 9644275432009

Offset: 0

Vincenzo Librandi, May 30 2014

Chebyshev polynomial of the second kind U(8,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A144138, A144139, A242850, A242851.

[256*n^8-448*n^6+240*n^4-40*n^2+1: n in [0..30]];

Table[ChebyshevU[8, n], {n, 0, 30}] (* or *) Table[256 n^8 - 448 n^6 + 240 n^4 - 40 n^2 + 1, {n, 0, 30}]

G.f.: (1 + 40500*x^2 + 1007440*x^3 + 4113054*x^4 + 4112928*x^5 + 1007524*x^6 + 40464*x^7 + 9*x^8)/(1 - x)^9.

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

A242854 a(n) = 512n^9 - 1024n^7 + 672n^5 - 160n^3 + 10*n.

Original entry on oeis.org

0, 10, 151316, 7997214, 118118440, 922080050, 4878316860, 19828978246, 66593931344, 193501094490, 501827040100, 1187422368110, 2605282707576, 5365498355074, 10470873504140, 19508549760150, 34910198169760, 60297759323306, 100934312212404, 164302439443390

Offset: 0

Vincenzo Librandi, May 30 2014

Chebyshev polynomial of the second kind U(9,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A144138, A144139, A242850, A242851.

[512*n^9-1024*n^7+672*n^5-160*n^3+10*n: n in [0..20]];

A242854:=n->512*n^9 - 1024*n^7 + 672*n^5 - 160*n^3 + 10*n: seq(A242854(n), n=0..30); # Wesley Ivan Hurt, Feb 04 2017

Table[ChebyshevU[9, n], {n, 0, 20}] (* or *) Table[512 n^9 - 1024 n^7 + 672 n^5 - 160 n^3 + 10 n, {n, 0, 20}]

G.f.: x*(10 + 151216*x + 6484504*x^2 + 44954320*x^3 + 82614460*x^4 + 44954320*x^5 + 6484504*x^6 + 151216*x^7 + 10*x^8)/(1 - x)^10.

a(n) = 2*n*(4*n^2-2*n-1)*(4*n^2+2*n-1)*(16*n^4-20*n^2+5).

A243130 1024n^10 - 2304n^8 + 1792n^6 - 560n^4 + 60*n^2 - 1.

Original entry on oeis.org

-1, 11, 564719, 46611179, 929944511, 9127651499, 58130412911, 276182038859, 1061324394239, 3472236254411, 10011386405999, 26069206375211, 62418042417599, 139296285729899, 292810020137711, 584605483663499, 1116034330278911, 2048348816684939, 3630829342034159

Offset: 0

Vincenzo Librandi, May 30 2014

Chebyshev polynomial of the second kind U(10,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A144138, A144139, A242850, A242851.

[1024*n^10-2304*n^8+1792*n^6-560*n^4+60*n^2-1: n in [0..20]];

Table[ChebyshevU[10, n], {n, 0, 20}] (* or *) Table[1024 n^10 - 2304 n^8 + 1792 n^6 - 560 n^4 + 60 n^2 - 1, {n, 0, 20}]
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{-1,11,564719,46611179,929944511,9127651499,58130412911,276182038859,1061324394239,3472236254411,10011386405999},20] (* Harvey P. Dale, Dec 10 2023 *)

G.f.: (-1 + 22*x + 564543*x^2 + 40400040*x^3 + 448278942*x^4 + 1368702180*x^5 + 1368701718*x^6 + 448279272*x^7 + 40399875*x^8 + 564598*x^9 + 11*x^10)/(1 - x)^11.

a(n) = (32*n^5 - 16*n^4 - 32*n^3 +12*n^2 + 6*n - 1)*(32*n^5 + 16*n^4 - 32*n^3 -12*n^2 + 6*n + 1).

cp's OEIS Frontend

A007588 Stella octangula numbers: a(n) = n*(2*n^2 - 1).

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A323182 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Chebyshev polynomial of the second kind U_{n}(x), evaluated at x=k.

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A242850 32*n^5 - 32*n^3 + 6*n.

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A242851 64*n^6 - 80*n^4 + 24*n^2 - 1.

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A242852 128*n^7-192*n^5+80*n^3-8*n.

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A242853 256*n^8 - 448*n^6 + 240*n^4 - 40*n^2 + 1.

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Magma

Mathematica

Formula

A242854 a(n) = 512*n^9 - 1024*n^7 + 672*n^5 - 160*n^3 + 10*n.

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A007588 Stella octangula numbers: a(n) = n(2n^2 - 1).

A242850 32n^5 - 32n^3 + 6*n.

A242851 64n^6 - 80n^4 + 24*n^2 - 1.

A242852 128n^7-192n^5+80n^3-8n.

A242853 256n^8 - 448n^6 + 240n^4 - 40n^2 + 1.

A242854 a(n) = 512n^9 - 1024n^7 + 672n^5 - 160n^3 + 10*n.

A243130 1024n^10 - 2304n^8 + 1792n^6 - 560n^4 + 60*n^2 - 1.