A083333 a(n) = 10*a(n-2) - 16*a(n-4) for n>=4, with a(0)=a(1)=1, a(2)=6, a(3)=10.

1, 1, 6, 10, 44, 84, 344, 680, 2736, 5456, 21856, 43680, 174784, 349504, 1398144, 2796160, 11184896, 22369536, 89478656, 178956800, 715828224, 1431655424, 5726623744, 11453245440, 45812985856, 91625967616, 366503878656

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Apr 24 2003

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,10,0,-16).

Crossrefs

Cf. A016131, A082412 (bisections).

Cf. A001045, A016116, A083332.

Programs

Magma

I:=[1,1,6,10]; [n le 4 select I[n] else 10*Self(n-2) -16*Self(n-4): n in [1..41]]; // G. C. Greubel, Dec 27 2024

Mathematica

CoefficientList[Series[(1+x-4x^2)/(1-10x^2+16x^4), {x, 0, 30}], x]
LinearRecurrence[{0,10,0,-16},{1,1,6,10},30] (* Harvey P. Dale, Aug 04 2024 *)

SageMath

def A083333(n): return 2^((n-1)/2)*( (n%2)*(2^(n+1) -1) + ((n+1)%2)*sqrt(2)*(2^(n+1) +1))/3
print([A083333(n) for n in range(41)]) # G. C. Greubel, Dec 27 2024

Formula

G.f.: (1+x-4*x^2)/(1-10*x^2+16*x^4).

Limit_{n -> oo} A083332(n)/a(n) = 3.

a(n) = A001045(n+1)*A016116(n). - R. J. Mathar, Jul 08 2009

From G. C. Greubel, Dec 27 2024: (Start)

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

E.g.f.: (1/3)*(2*cosh(2*sqrt(2)*x) + cosh(sqrt(2)*x)) + (1/(3*sqrt(2)))*(2*sinh(2*sqrt(2)*x) - sinh(sqrt(2)*x)). (End)