A190978 a(n) = 8*a(n-1) - 6*a(n-2), with a(0)=0, a(1)=1.

0, 1, 8, 58, 416, 2980, 21344, 152872, 1094912, 7842064, 56167040, 402283936, 2881269248, 20636450368, 147803987456, 1058613197440, 7582081654784, 54304974053632, 388947302500352, 2785748575681024, 19952304790446080, 142903946869482496, 1023517746213183488

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, May 24 2011

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Pamela Fleischmann, Jonas Höfer, Annika Huch, and Dirk Nowotka, alpha-beta-Factorization and the Binary Case of Simon's Congruence, arXiv:2306.14192 [math.CO], 2023.
• Index entries for linear recurrences with constant coefficients, signature (8,-6).

Crossrefs

Cf. A190958 (index to generalized Fibonacci sequences).

Programs

Magma

[n le 2 select n-1 else 8*Self(n-1) -6*Self(n-2): n in [1..41]]; // G. C. Greubel, Jun 17 2022

Mathematica

LinearRecurrence[{8,-6}, {0,1}, 50]
CoefficientList[Series[x/(1-8x+6x^2),{x,0,30}],x] (* Harvey P. Dale, Aug 03 2021 *)

SageMath

[sum( (-1)^k*binomial(n-k-1, k)*6^k*8^(n-2*k-1) for k in (0..((n-1)//2))) for n in (0..40)] # G. C. Greubel, Jun 17 2022

Formula

a(n) = ((4 + sqrt(10))^n - (4 - sqrt(10))^n)/(2*sqrt(10)). - Giorgio Balzarotti, May 28 2011

G.f.: x/(1 - 8*x + 6*x^2). - Philippe Deléham, Oct 12 2011

From G. C. Greubel, Jun 17 2022: (Start)

a(n) = 6^((n-1)/2)*ChebyshevU(n-1, 4/sqrt(6)).

E.g.f.: (1/sqrt(10))*exp(4*x)*sinh(sqrt(10)*x). (End)