A156712 Star numbers (A003154) that are also triangular numbers (A000217).

1, 7, 91, 1261, 17557, 244531, 3405871, 47437657, 660721321, 9202660831, 128176530307, 1785268763461, 24865586158141, 346332937450507, 4823795538148951, 67186804596634801, 935791468814738257, 13033893758809700791, 181538721154521072811

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

Cf. A000217, A001570, A001844, A001922, A003154.

Cf. A003215, A005448, A007655, A016754.

Sequence in context: A248226 A165230 A346939 * A004368 A243946 A130978

Adjacent sequences: A156709 A156710 A156711 * A156713 A156714 A156715

Keyword

easy,nonn

Author

Aaron Meyerowitz, Feb 14 2009

Extensions

a(11) onwards from Robert G. Wilson v, Oct 28 2010

Status

approved