A105045 a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=12; for n > 4, a(n) = 8*a(n-2) - a(n-4) - 3.

0, 1, 2, 3, 12, 20, 91, 154, 713, 1209, 5610, 9515, 44164, 74908, 347699, 589746, 2737425, 4643057, 21551698, 36554707, 169676156, 287794596, 1335857547, 2265802058, 10517184217, 17838621865, 82801616186, 140443172859, 651895745268, 1105706761004, 5132364345955

Offset: 0

Author

Gerald McGarvey, Apr 03 2005

Keywords

• nonn
• easy

Comments

It appears that this sequence gives all nonnegative m such that 60*m^2 - 60*m + 1 is a square and that for n > 3, a(n+1) = A103200(n) + 1.

From Paul Weisenhorn, Jun 30 2010: (Start)

Place b(n) red and a(n) blue balls in an urn, then draw 6 balls without replacement.

This gives binomial(b(n), 6) = binomial(b(n), 4) * binomial(a(n), 2), where b(n) = A179123(n). (End)

Examples

			For n=3, a(3)=3; b(3)=14; binomial(14,6)=3003; binomial(14,4)*binomial(3,2) = 1001*3 = 3003. - _Paul Weisenhorn_, Jun 30 2010
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..2000
• Index entries for linear recurrences with constant coefficients, signature (1,8,-8,-1,1).

Crossrefs

Cf. A001090, A103200, A179123.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+x-7*x^2+x^3+x^4)/((1-x)*(1-8*x^2+x^4)) )); // G. C. Greubel, Mar 14 2023

Maple

n:=1: for m from 1 to 2000 do w:=sqrt(1+60*m*(m-1)):
if (w=floor(w)) then a(n)=m: b(n)=(9+w)/2: inc(n): end if: end do # Paul Weisenhorn, Jun 30 2010

Mathematica

Join[{0},RecurrenceTable[{a[1]==1,a[2]==2,a[3]==3,a[4]==12,a[n] == 8a[n-2]-a[n-4]-3},a,{n,30}]] (* or *) LinearRecurrence[{1,8,-8,-1, 1}, {0,1,2,3,12,20}, 30] (* Harvey P. Dale, Nov 10 2011 *)

SageMath

@CachedFunction
def a(n): # a = A105045
    if (n<6): return (0,1,2,3,12,20)[n]
    else: return a(n-1) +8*a(n-2) -8*a(n-3) -a(n-4) +a(n-5)
[a(n) for n in range(41)] # G. C. Greubel, Mar 14 2023

Formula

a(n) = 8*a(n-2) - a(n-4) - 3, for n > 4.

From Paul Weisenhorn, Jun 30 2010: (Start)

Let r=sqrt(15), then

a(n) = ((15+r)*(4+r)^((n-1)/2) + (15-r)*(4-r)^((n-1)/2) + 30)/60 when n is odd, and

a(n) = ((45+11*r)*(4+r)^((n-2)/2) + (45-11*r)*(4-r)^((n-2)/2) + 30)/60 when n is even. (End)

a(n) = a(n-1) + 8*a(n-2) - 8*a(n-3) - a(n-4) + a(n-5), a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=12, a(5)=20. - Harvey P. Dale, Nov 10 2011

G.f.: x*(1 +x -7*x^2 +x^3 +x^4)/((1-x)*(1-8*x^2+x^4)). - Colin Barker, Jan 01 2013

Extensions

More terms from Colin Barker, Jan 01 2013