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A007588 Stella octangula numbers: n*(2*n^2 - 1).
(Formerly M4932)
26
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

REFERENCES

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 51.

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]

T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).

Eric Weisstein's World of Mathematics, Stella Octangula Number

Wikipedia, Stella octangula number

Index to sequences with 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

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

CROSSREFS

Backwards differences give star numbers A003154: A003154(n)=A007588(n)-A007588(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.

Sequence in context: A244804 A009961 A059997 * A129025 A113907 A125740

Adjacent sequences:  A007585 A007586 A007587 * A007589 A007590 A007591

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

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Last modified October 22 04:03 EDT 2014. Contains 248388 sequences.