A084155 A Pell-related fourth-order recurrence.

0, 1, 4, 19, 88, 401, 1804, 8051, 35760, 158401, 700564, 3095731, 13673224, 60375953, 266559388, 1176763859, 5194762080, 22931453953, 101225940772, 446836798675, 1972442421688, 8706804701201, 38433749994028

Offset: 0

Author

Paul Barry, May 16 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A084154.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-18,8,7)

Crossrefs

Cf. A001333, A083878.

Programs

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

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

Maple

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

Mathematica

LinearRecurrence[{8,-18,8,7},{0,1,4,19},30] (* Harvey P. Dale, Aug 16 2015 *)

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

Formula

a(n) = (A083878(n) - A001333(n))/2.

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.

a(n) = ((3+sqrt(2))^n +(3-sqrt(2))^n -(1+sqrt(2))^n -(1-sqrt(2))^n)/4.

G.f.: x*(1-4*x+5*x^2)/((1-2*x-x^2)*(1-6*x+7*x^2)).

E.g.f.: exp(2*x)*sinh(x)*cosh(sqrt(2)*x).