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A084155 A Pell-related fourth-order recurrence. 3
0, 1, 4, 19, 88, 401, 1804, 8051, 35760, 158401, 700564, 3095731, 13673224, 60375953, 266559388, 1176763859, 5194762080, 22931453953, 101225940772, 446836798675, 1972442421688, 8706804701201, 38433749994028 (list; graph; refs; listen; history; text; internal format)
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: A290667 A291416 A192526 * A015530 A256959 A181880

Adjacent sequences:  A084152 A084153 A084154 * A084156 A084157 A084158

KEYWORD

easy,nonn

AUTHOR

Paul Barry, May 16 2003

STATUS

approved

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Last modified May 7 19:23 EDT 2021. Contains 343652 sequences. (Running on oeis4.)