A181443 Solutions a(n) to (r(n)-5)*(r(n)-6) = 21 *a(n)*(a(n)-1).

1, 2, 5, 22, 34, 161, 494, 2365, 3685, 17654, 54281, 260074, 405262, 1941725, 5970362, 28605721, 44575081, 213572042, 656685485, 3146369182, 4902853594, 23490982841, 72229432934, 346072004245, 539269320205, 2583794540414, 7944580937201, 38064774097714, 59314722368902, 284193908462645, 873831673659122, 4186779078744241, 6524080191258961, 31258746136350482, 96113539521566165, 460507633887768742, 717589506316116754

Offset

0,2

Comments

The associated r(n) are in A180509, which gives a combinatorial interpretation of the pairs (r(n),a(n)).

Links

Table of n, a(n) for n=0..36.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,110,-110,0,0,-1,1).

Formula

G.f. ( -1-x-3*x^2-17*x^3+98*x^4-17*x^5-3*x^6-x^7-x^8 ) / ( (x-1)*(x^8-110*x^4+1) ). - R. J. Mathar, Feb 05 2011

Explicit formulas: r=sqrt(21), s=55+12*r, t=55-12*r:

a(4*n)=(42+(21+r)*s^n+(21-r)*t^n)/84.

a(4*n+1)=(42+(63+13*r)*s^n+(63-13*r)*t^n)/84.

a(4*n+2)=(42+(189+41*r)*s^n+(189-41*r)*t^n)/84.

a(4*n+3)=(42+(903+197*r)*s^n+(903-197*r)*t^n)/84.

a(n) = 111*a(n-4) - 111*a(n-8) + a(n-12).

a(n) = +a(n-1) +110*a(n-4) -110*a(n-5) -a(n-8) +a(n-9). - R. J. Mathar, Jan 05 2011

Example

For n=2: a(2)=5; b(2)=26; binomial(26,7)=657800; binomial(26,5)*binomial(5,2)=657800.

Maple

n:=0: for s from 1 to 100 do r:=(sqrt(84*s^2-84*s+1)+11)/2: if (floor(r)=r) then a[n]:=s: b[n]:=r: n:=n+1: end if: end do:

Mathematica

LinearRecurrence[{1, 0, 0, 110, -110, 0, 0, -1, 1}, {1, 2, 5, 22, 34, 161, 494, 2365, 3685}, 40] (* Harvey P. Dale, Aug 03 2024 *)

Crossrefs

Sequence in context: A024600 A131510 A357020 * A041165 A041006 A346557

Adjacent sequences: A181440 A181441 A181442 * A181444 A181445 A181446

Keyword

nonn

Author

Paul Weisenhorn, Jan 29 2011

Status

approved