OFFSET
1,2
COMMENTS
From Colin Barker, Jan 06 2015: (Start)
Also indices of centered square numbers (A001844) which are also centered triangular numbers (A005448).
Also indices of centered octagonal numbers (A016754) which are also centered hexagonal numbers (A003215).
Also positive integers y in the solutions to 3*x^2-4*y^2-3*x+4*y = 0, the corresponding values of x being A001922.
(End)
LINKS
Colin Barker, Table of n, a(n) for n = 1..875
Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.
Wikipedia, Star Numbers
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).
FORMULA
a(n+3) = 15*a(n+2) - 15*a(n+1) + a(n).
If x^2 - 3*y^2 = 1 with x even then a(y) = (y+2)/4 evidently related to A001570 by: add 1 and halve.
G.f.: x*(1 - 8*x + x^2)/((1-x)*(1 - 14*x + x^2)). - Alexander R. Povolotsky, Feb 15 2009
a(n) = (4 + (2 + sqrt(3))*(7 - 4*sqrt(3))^n + (2 - sqrt(3))*(7 + 4*sqrt(3))^n)/8. - Colin Barker, Mar 05 2016
a(n) = (1/2)*( 1 + ChebyshevU(n, 7) - 13*ChebyshevU(n-1, 7) ). - G. C. Greubel, Oct 07 2022
MAPLE
f:= gfun[rectoproc]({a(n+3)=15*a(n+2)-15*a(n+1)+a(n), a(1)=1, a(2)=7, a(3)=91}, a(n), 'remember'):
seq(f(n), n=1..30); # Robert Israel, Jan 01 2015
MATHEMATICA
f[n_] := (Simplify[(2 + Sqrt@3)^(2 n - 1) + (2 - Sqrt@3)^(2 n - 1)] + 4)/8; Array[f, 17] (* Robert G. Wilson v, Oct 28 2010 *)
PROG
(PARI) Vec(-x*(x^2-8*x+1)/((x-1)*(x^2-14*x+1)) + O(x^100)) \\ Colin Barker, Jan 01 2015
(Magma) [(Evaluate(ChebyshevSecond(n+1), 7) - 13*Evaluate(ChebyshevU(n), 7) + 1)/2: n in [1..30]]; // G. C. Greubel, Oct 07 2022
(SageMath)
def A156712(n): return (1 + chebyshev_U(n, 7) - 13*chebyshev_U(n-1, 7))/2
[A156712(n) for n in range(1, 31)] # G. C. Greubel, Oct 07 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Aaron Meyerowitz, Feb 14 2009
EXTENSIONS
a(11) onwards from Robert G. Wilson v, Oct 28 2010
STATUS
approved