A144138 -id:A144138

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A007588

Stella octangula numbers: a(n) = n*(2*n^2 - 1).
(Formerly M4932)

+10
38

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

REFERENCES

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).

LINKS

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).

FORMULA

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

MAPLE

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

MATHEMATICA

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 *)

PROG

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

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

(Python)

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

CROSSREFS

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.

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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.

+10
14

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Wikipedia, Chebyshev polynomials.

Index entries for sequences related to Chebyshev polynomials.

FORMULA

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

EXAMPLE

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, ...

PROG

(PARI) T(n, k) = polchebyshev(n, 2, k);

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

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

CROSSREFS

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).

KEYWORD

sign,tabl

AUTHOR

Seiichi Manyama, Jan 06 2019

STATUS

approved

A242850

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

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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

FORMULA

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).

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A000466, A144138, A144139, A242851.

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, May 29 2014

EXTENSIONS

Edited by Bruno Berselli, May 29 2014

STATUS

approved

A242851

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

+10
7

-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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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

FORMULA

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).

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A000466, A144138, A144139, A242850.

KEYWORD

sign,easy

AUTHOR

Vincenzo Librandi, May 29 2014

EXTENSIONS

Edited by Bruno Berselli, May 29 2014

STATUS

approved

A242852

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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).

FORMULA

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).

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A144138, A144139, A242850, A242851.

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, May 30 2014

STATUS

approved

A242853

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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).

FORMULA

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).

MATHEMATICA

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}]

PROG

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

CROSSREFS

Cf. A144138, A144139, A242850, A242851.

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, May 30 2014

STATUS

approved

A242854

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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).

FORMULA

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).

MAPLE

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

MATHEMATICA

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}]

PROG

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

CROSSREFS

Cf. A144138, A144139, A242850, A242851.

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, May 30 2014

STATUS

approved

A243130

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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).

FORMULA

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).

MATHEMATICA

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 *)

PROG

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

CROSSREFS

Cf. A144138, A144139, A242850, A242851.

KEYWORD

sign,easy

AUTHOR

Vincenzo Librandi, May 30 2014

STATUS

approved

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