OFFSET
1,1
COMMENTS
This is equivalent to the Pell equation A(n)^2 - 10*B(n)^2 = -9 with a(n) = (A(n)+7)/2, b(n) = (B(n)+1)/2, and the 3 fundamental solutions (1,1), (9,3), (41,13), and the solution (19,6) for the unit form.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,38,-38,0,-1,1).
FORMULA
G.f.: x*(4 +4*x +16*x^2 -133*x^3 -16*x^4 -4*x^5 +3*x^6)/((1-x)*(1 -38*x^3 +x^6)).
a(n+9) = 39*a(n+6) - 39*a(n+3) + a(n).
Let r = sqrt(10) then:
a(3*n+1) = (14 + (1+r)*(19+6*r)^n + (1-r)*(19-6*r)^n)/4.
a(3*n+2) = (14 + 3*(3+r)*(19+6*r)^n + 3*(3-r)*(19-6*r)^n)/4.
a(3*n+3) = (14 + (41+13*r)*(19+6*r)^n + (41-13*r)*(19-6*r)^n)/4.
a(n) = a(n-1) + 38*a(n-3) - 38*a(n-4) - a(n-6) + a(n-7). - G. C. Greubel, Mar 20 2019
EXAMPLE
For n=3: a(3)=24 and b(3)=7 since binomial(24,5) = binomial(24,3)*binomial(7,2) = 42504.
MATHEMATICA
Rest[CoefficientList[Series[x*(4+4*x+16*x^2-133*x^3-16*x^4-4*x^5 +3*x^6 )/((1-x)*(1-38*x^3+x^6)), {x, 0, 30}], x]] (* G. C. Greubel, Mar 20 2019 *)
LinearRecurrence[{1, 0, 38, -38, 0, -1, 1}, {4, 8, 24, 43, 179, 783, 1504}, 30] (* Harvey P. Dale, May 04 2024 *)
PROG
(PARI) my(x='x+O('x^30)); Vec(x*(4+4*x+16*x^2-133*x^3-16*x^4-4*x^5+3*x^6) /((1-x)*(1-38*x^3+x^6))) \\ G. C. Greubel, Mar 20 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x*(4+4*x+ 16*x^2-133*x^3-16*x^4-4*x^5+3*x^6)/((1-x)*(1-38*x^3+x^6)) )); // G. C. Greubel, Mar 20 2019
(Sage) a=(x*(4+4*x+16*x^2-133*x^3-16*x^4-4*x^5+3*x^6)/((1-x)*(1-38*x^3 +x^6))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Mar 20 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul Weisenhorn, Aug 05 2010
STATUS
approved