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

0,3

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)

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).

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A001333, A083878.

Sequence in context: A357572 A291416 A192526 * A015530 A256959 A181880

Adjacent sequences: A084152 A084153 A084154 * A084156 A084157 A084158

Keyword

easy,nonn

Author

Paul Barry, May 16 2003

Status

approved