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, 2528508202404485318557

Offset: 1

Author

Aaron Meyerowitz, Feb 14 2009

Keywords

• easy
• nonn

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
• Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 30, 56.
• 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).

Crossrefs

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

Cf. A003215, A005448, A007655, A016754.

Programs

Magma

[(Evaluate(ChebyshevSecond(n+1),7) - 13*Evaluate(ChebyshevU(n), 7) + 1)/2: n in [1..30]]; // 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 *)

PARI

Vec(-x*(x^2-8*x+1)/((x-1)*(x^2-14*x+1)) + O(x^100)) \\ Colin Barker, Jan 01 2015

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

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

Extensions

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