OFFSET
0,3
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)
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).
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
EXAMPLE
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
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 *)
PROG
(Magma) R<x>:=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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gerald McGarvey, Apr 03 2005
EXTENSIONS
More terms from Colin Barker, Jan 01 2013
STATUS
approved