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a(n) = (2^n - 1)*(4^n - 1).
1

%I #36 Nov 06 2024 04:14:32

%S 0,3,45,441,3825,31713,257985,2080641,16711425,133955073,1072692225,

%T 8585738241,68702695425,549688696833,4397778059265,35183298314241,

%U 281470681677825,2251782633684993,18014329789743105,144114913197424641,1152920405094170625

%N a(n) = (2^n - 1)*(4^n - 1).

%H Harry J. Smith, <a href="/A060242/b060242.txt">Table of n, a(n) for n = 0..100</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (15,-70,120,-64).

%F G.f.: 3*x*(1-8*x^2)/((1-x)*(1-2*x)*(1-4*x)*(1-8*x)). - _Alois P. Heinz_, Feb 19 2021

%F E.g.f.: exp(x) - exp(2*x) - exp(4*x) + exp(8*x). - _G. C. Greubel_, Nov 05 2024

%p f:= gfun:-rectoproc({a(n) - 15*a(n-1) + 70*a(n-2) - 120*a(n-3) + 64*a(n-4) = 0, a(0) = 0, a(1)=3, a(2)=45, a(3)=441}, a(n), remember): map(f, [$0..20]); # _Georg Fischer_, Feb 19 2021

%t Table[(2^n-1)(4^n-1),{n,0,20}] (* _Harvey P. Dale_, May 13 2019 *)

%o (PARI) a(n) = (2^n - 1)*(4^n - 1); \\ _Harry J. Smith_, Jul 04 2009

%o (Magma) [(2^n-1)^2*(2^n+1): n in [0..40]]; // _G. C. Greubel_, Nov 05 2024

%o (SageMath)

%o def A060242(n): return (2^n-1)*(4^n-1)

%o [A060242(n) for n in range(41)] # _G. C. Greubel_, Nov 05 2024

%Y Cf. A000225, A024036.

%K nonn,easy,changed

%O 0,2

%A _N. J. A. Sloane_, Mar 22 2001