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A054318 a(n)-th star number (A003154) is a square. 5
1, 5, 45, 441, 4361, 43165, 427285, 4229681, 41869521, 414465525, 4102785725, 40613391721, 402031131481, 3979697923085, 39394948099365, 389969783070561, 3860302882606241, 38213059042991845, 378270287547312205 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A two-way infinite sequence which is palindromic.

Also indices of centered hexagonal numbers (A003215) which are also centered square numbers (A001844). - Colin Barker, Jan 02 2015

Also positive integers y in the solutions to 4*x^2 - 6*y^2 - 4*x + 6*y = 0. - Colin Barker, Jan 02 2015

REFERENCES

Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427; http://forumgeom.fau.edu/FG2016volume16/FG2016volume16.pdf#page=423

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for two-way infinite sequences

Index entries for linear recurrences with constant coefficients, signature (11,-11,1).

FORMULA

a(n) = 11*(a(n-1) - a(n-2)) + a(n-3).

a(n) = 1/2 + (3 - sqrt(6))/12*(5 + 2*sqrt(6))^n + (3 + sqrt(6))/12*(5 - 2*sqrt(6))^n.

G.f.: x(1-6x+x^2)/((1-x)(1-10x+x^2)). 12a(n)a(n-1)+4 = (a(n)+a(n-1)+2)^2. a(n) = a(1-n) = 10a(n-1)-a(n-2)-4 = 12a(n-1)^2/(a(n-1)+a(n-2))-a(n-1) =(a(n-1)+4)a(n-1)/a(n-2). - Michael Somos, Mar 18 2003

a(n+1) = 1 + 1/2*sum {k = 1..n} 8^k*binomial(n+k,2*k). a(n+1) = R(n,4), where R(n,x) is the n-th row polynomial of A211955. a(n+1) = 1/u*T(n,u)*T(n+1,u) with u = sqrt(3) and T(n,x) the Chebyshev polynomial of the first kind. Sum {k = 0..inf} 1/a(k) = sqrt(3/2). - Peter Bala, May 01 2012

A003154(a(n)) = A006061(n). - Zak Seidov, Oct 22 2012

a(n) = (4*a(n-1) + a(n-1)^2) / a(n-2), n >= 3. - Seiichi Manyama, Aug 11 2016

EXAMPLE

a(2) = 5 because the 5th Star number (A003154) 121=11^2 is the 2nd that is a square.

MATHEMATICA

CoefficientList[Series[x (1 - 6 x + x^2)/((1 - x) (1 - 10 x + x^2)), {x, 0, 19}], x] (* Michael De Vlieger, Aug 11 2016 *)

LinearRecurrence[{11, -11, 1}, {1, 5, 45}, 30] (* Harvey P. Dale, Nov 05 2016 *)

PROG

(PARI) a(n)=if(n<1, a(1-n), 1/2+subst(poltchebi(n)+poltchebi(n-1), x, 5)/12)

(PARI) Vec(x*(1-6*x+x^2)/((1-x)*(1-10*x+x^2)) + O(x^100)) \\ Colin Barker, Jan 02 2015

CROSSREFS

A031138 is 3*a(n)-2. Cf. A003154, A006061, A182432, A211955.

Quintisection of column k=2 of A233427.

Cf. A001844, A003215, A253475.

Sequence in context: A005979 A165225 A121272 * A093140 A137233 A001449

Adjacent sequences:  A054315 A054316 A054317 * A054319 A054320 A054321

KEYWORD

easy,nonn,changed

AUTHOR

Ignacio Larrosa CaƱestro, Feb 27 2000

EXTENSIONS

More terms from James A. Sellers, Mar 01 2000

STATUS

approved

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Last modified November 23 19:08 EST 2017. Contains 295128 sequences.