A172511 a(n) = a(n-1) * (11*a(n-1) - a(n-2)) / (a(n-1) + 4*a(n-2)), with a(0) = a(1) = 1.

1, 1, 2, 7, 35, 210, 1365, 9165, 62322, 425867, 2915551, 19974626, 136884937, 938162617, 6430103330, 44072167855, 302074043195, 2070443441970, 14191023001437, 97266699113157, 666675822475026, 4569463931720051

Offset: 0

Author

Michael Somos, Feb 05 2010

Keywords

• nonn
• easy

Examples

			G.f. = 1 + x + 2*x^2 + 7*x^3 + 35*x^4 + 210*x^5 + 1365*x^6 + 9165*x^7 + ...
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (11,-33,33,-11,1).

Crossrefs

Cf. A005247, A001519, A049685.

Programs

Mathematica

RecurrenceTable[{a[0]==a[1]==1,a[n]==a[n-1] (11a[n-1]-a[n-2])/(a[n-1]+ 4a[n-2])},a,{n,30}] (* or *) LinearRecurrence[{11,-33,33,-11,1},{1,1,2,7,35},30] (* Harvey P. Dale, Nov 18 2013 *)

PARI

{a(n) = (4 + fibonacci(4*n - 1)/3 + fibonacci(4*n - 3)/3 + 5 * fibonacci(2*n - 1)) / 10};

PARI

{a(n) = my(A); if( n<1, n = 1-n); if( n<3, n, A = vector(n, k, k); for(k=3, n, A[k] = A[k-1] * (11*A[k-1] - A[k-2]) / (A[k-1] + 4*A[k-2])); A[n])}; /* Michael Somos, Sep 22 2014 */

Formula

a(n) = (4 + A049685(n-1) + 5 * A001519(n)) / 10 = a(1 - n).

G.f.: (4 / (1 - x) + (1 - 6*x) / (1 - 7*x + x^2) + (5 - 10*x) / (1 - 3*x + x^2)) / 10.

a(0)=1, a(1)=1, a(2)=2, a(3)=7, a(4)=35, a(n)=11*a(n-1)-33*a(n-2)+ 33*a(n-3)- 11*a(n-4)+a(n-5). - Harvey P. Dale, Nov 18 2013

a(n) = a(1-n) for all n in Z. - Michael Somos, Sep 22 2014

0 = a(n)*(+a(n+1) + 4*a(n+2)) + a(n+1)*(-11*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Sep 22 2014

a(n) = b(n+1) * b(n) * b(n-1) * b(n-2) / 6 for all n in Z where b = A005247. - Michael Somos, Sep 22 2014