%I #42 Jan 19 2025 09:28:31

%S 1,2,5,4,21,8,85,16,341,32,1365,64,5461,128,21845,256,87381,512,

%T 349525,1024,1398101,2048,5592405,4096,22369621,8192,89478485,16384,

%U 357913941,32768,1431655765,65536,5726623061,131072,22906492245,262144,91625968981,524288,366503875925

%N Column 1 of the array in A107735.

%D S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.

%H Chai Wah Wu, <a href="/A107732/b107732.txt">Table of n, a(n) for n = 3..3324</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,7,0,-14,0,8).

%F a(2*k+2) = 2^k = A000079(k), a(2*k+1) = (4^k-1)/3 = A002450(k) = A001045(2*k).

%F a(n) = 7*a(n-2) - 14*a(n-4) + 8*a(n-6) for n > 8. - _Chai Wah Wu_, Jun 19 2016

%F G.f.: x^3*(1 + 2*x - 2*x^2 - 10*x^3 + 8*x^5)/(1 - 7*x^2 + 14*x^4 - 8*x^6). - _Chai Wah Wu_, Jun 19 2016

%F a(n) = (3*(1 + (-1)^n)*2^(n/2) - (1 - (-1)^n)*(2 - 2^n))/12. - _Colin Barker_, Mar 26 2019

%F a(n) = (2^n - 2)/6 if n is odd else 2^(n/2 - 1). - _Peter Luschny_, Mar 26 2019

%t Table[(3 (1 + (-1)^n) 2^(n/2) - (1 - (-1)^n) (2 - 2^n))/12, {n, 3, 50}] (* _Bruno Berselli_, Mar 26 2019 *)

%o (PARI) Vec(x^3*(1 + 2*x - 2*x^2 - 10*x^3 + 8*x^5) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 - 2*x^2)) + O(x^40)) \\ _Colin Barker_, Mar 26 2019

%o (SageMath)

%o def a(n): return (2^n-2)//6 if is_odd(n) else 2^(n//2-1)

%o print([a(n) for n in (3..41)]) # _Peter Luschny_, Mar 26 2019

%Y Cf. A000079, A001045, A002450, A107735.

%K nonn,easy

%O 3,2

%A _N. J. A. Sloane_, Jun 10 2005

%E More terms from _Chai Wah Wu_, Jun 19 2016