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%I #22 Sep 08 2022 08:45:10
%S 0,1,4,19,88,401,1804,8051,35760,158401,700564,3095731,13673224,
%T 60375953,266559388,1176763859,5194762080,22931453953,101225940772,
%U 446836798675,1972442421688,8706804701201,38433749994028
%N A Pell-related fourth-order recurrence.
%C Binomial transform of A084154.
%H G. C. Greubel, <a href="/A084155/b084155.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-18,8,7)
%F a(n) = (A083878(n) - A001333(n))/2.
%F a(n) = 8*a(n-1) - 18*a(n-2) + 8*a(n-3) + 7*a(n-4), a(0)=0, a(1)=1, a(2)=4, a(3)=19.
%F a(n) = ((3+sqrt(2))^n +(3-sqrt(2))^n -(1+sqrt(2))^n -(1-sqrt(2))^n)/4.
%F G.f.: x*(1-4*x+5*x^2)/((1-2*x-x^2)*(1-6*x+7*x^2)).
%F E.g.f.: exp(2*x)*sinh(x)*cosh(sqrt(2)*x).
%p seq(coeff(series(x*(1-4*x+5*x^2)/((1-2*x-x^2)*(1-6*x+7*x^2)),x,n+1), x, n), n = 0 .. 25); # _Muniru A Asiru_, Oct 18 2018
%t LinearRecurrence[{8,-18,8,7},{0,1,4,19},30] (* _Harvey P. Dale_, Aug 16 2015 *)
%o (PARI) m=40; v=concat([0,1,4,19], vector(m-4)); for(n=5, m, v[n] = 8*v[n-1] -18*v[n-2] +8*v[n-3] +7*v[n-4]); v \\ _G. C. Greubel_, Oct 17 2018
%o (Magma) I:=[0,1,4,19]; [n le 4 select I[n] else 8*Self(n-1) -18*Self(n-2) +8*Self(n-3) +7*Self(n-4): n in [1..40]]; // _G. C. Greubel_, Oct 17 2018
%o (GAP) a:=[0,1,4,19];; for n in [5..25] do a[n]:=8*a[n-1]-18*a[n-2]+8*a[n-3]+7*a[n-4]; od; a; # _Muniru A Asiru_, Oct 18 2018
%Y Cf. A001333, A083878.
%K easy,nonn
%O 0,3
%A _Paul Barry_, May 16 2003
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