cp's OEIS Frontend

Empirical: a(n) = 3*a(n-1) for n>2

From Manuel Kauers and Christoph Koutschan, Mar 02 2023: (Start)

a(n) = (2^(n-2) + 2*Sum_{k=0..floor((n+1)/2)} Sum_{l=k+1..floor((n+1)/2)} binomial(n-1, n-k-l) - binomial(n-1, n+k-l+1)) * (2^(n-2) + 2*Sum_{k=0..floor(n/2)} Sum_{l=k+1..floor(n/2)} binomial(n-1, n-k-l-1) - binomial(n-1, n+k-l+1)) for n>1.

Recurrence: (n-2)*(n+3)^2*(n+4)*(2*n^6 - 3*n^5 - 22*n^4 - 17*n^3 - 16*n^2 - 61*n - 33)*a(n+5) - 4*(n+3)*(2*n^9 + 15*n^8 - 24*n^7 - 278*n^6 - 279*n^5 + 622*n^4 + 1327*n^3 + 1792*n^2 + 2619*n + 1314)*a(n+4) - 16*(n+2)*(4*n^9 + 8*n^8 - 91*n^7 - 251*n^6 + 183*n^5 + 509*n^4 - 1161*n^3 - 1955*n^2 - 399*n - 207)*a(n+3) + 64*(4*n^10 + 32*n^9 + 17*n^8 - 455*n^7 - 1362*n^6 - 754*n^5 + 2250*n^4 + 4669*n^3 + 5364*n^2 + 4509*n + 1566)*a(n+2) + 256*(n+1)*(2*n^9 + n^8 - 52*n^7 - 121*n^6 + 79*n^5 + 255*n^4 - 476*n^3 - 1533*n^2 - 1665*n - 810)*a(n+1) - 1024*(n-1)*n*(n+1)^2*(2*n^6 + 9*n^5 - 7*n^4 - 95*n^3 - 199*n^2 - 235*n - 150)*a(n) = 0. (End)

Empirical: a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).

Conjectures from Colin Barker, Apr 29 2018: (Start)

G.f.: x*(8 + 9*x - 2*x^2 + x^3) / ((1 - x)^3*(1 + x)).

a(n) = (2 + 8*n + 8*n^2) / 2 for n even.

a(n) = (8*n + 8*n^2) / 2 for n odd.

(End)

Empirical: a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>5.

Conjectures from Colin Barker, Apr 29 2018: (Start)

G.f.: x*(32 + 105*x + 94*x^2 + 41*x^3 - 16*x^4) / ((1 - x)^3*(1 + x)).

a(n) = 9 - 48*n + 64*n^2 for n even.

a(n) = -48*n + 64*n^2 for n>1 and odd.

(End)

Empirical: a(n) = 256*n^2 - 384*n + 144 for n>2.

Conjectures from Colin Barker, Apr 30 2018: (Start)

G.f.: x*(64 + 249*x + 165*x^2 + 75*x^3 - 41*x^4) / (1 - x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>5.

(End)

Empirical: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4) for n>7.

Conjectures from Colin Barker, Apr 30 2018: (Start)

G.f.: 4*x*(32 + 225*x + 394*x^2 + 329*x^3 + 72*x^4 + 8*x^5 - 36*x^6) / ((1 - x)^3*(1 + x)).

a(n) = 2*(578 - 1088*n + 512*n^2) for n>3 and even.

a(n) = 2*(528 - 1088*n + 512*n^2) for n>3 and odd.

(End)

Empirical: a(n) = 4096*n^2 - 11264*n + 7744 for n>4.

Empirical g.f.: x*(256 + 2257*x + 3357*x^2 + 2388*x^3 + 108*x^4 + 163*x^5 - 337*x^6) / (1 - x)^3. - Colin Barker, Apr 30 2018

Empirical: a(n) = 3*a(n-1) + 5*a(n-2) - 15*a(n-3) + 3*a(n-4) + 5*a(n-5) - a(n-6) for n>7.

Empirical g.f.: x*(16 - 23*x - 74*x^2 + 128*x^3 - 5*x^4 - 39*x^5 + 7*x^6) / ((1 - 5*x + 6*x^2 - x^3)*(1 + 2*x - x^2 - x^3)). - Colin Barker, Apr 30 2018

Empirical: a(n) = 4*a(n-1) -8*a(n-3) +4*a(n-4) for n>5.

Empirical g.f.: 4*x*(8 - 23*x - 6*x^2 + 44*x^3 - 20*x^4) / ((1 - 4*x + 2*x^2)*(1 - 2*x^2)). - Colin Barker, Apr 30 2018

Empirical: a(n) = 4*a(n-1) +9*a(n-2) -45*a(n-3) +108*a(n-5) -42*a(n-6) -57*a(n-7) +18*a(n-8) +7*a(n-9) -a(n-10) for n>11.

Empirical g.f.: x*(64 - 207*x - 603*x^2 + 2339*x^3 + 405*x^4 - 5535*x^5 + 1831*x^6 + 2835*x^7 - 840*x^8 - 340*x^9 + 48*x^10) / ((1 - x)*(1 - 6*x + 9*x^2 - x^3)*(1 + 3*x - x^3)*(1 - 3*x^2 - x^3)). - Colin Barker, Apr 30 2018