cp's OEIS Frontend

Empirical: a(n) = (7/2)*n^4 + 21*n^3 - (7/2)*n^2 - 7*n.

Conjectures from Colin Barker, Mar 07 2018: (Start)

G.f.: 14*x*(1 + 9*x - 3*x^2 - x^3) / (1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.

(End)

Empirical formula (and thus Barker's conjectures) proved by Robert Israel, Mar 07 2018: see link.

Empirical: a(n) = 7*n^5 + 70*n^4 + (35/3)*n^3 - (105/2)*n^2 - (7/6)*n.

Conjectures from Colin Barker, Mar 07 2018: (Start)

G.f.: 35*x*(1 + x)*(1 + 28*x - 17*x^2) / (1 - x)^6.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.

(End)

Empirical: a(n) = 9*a(n-1) -11*a(n-2) -108*a(n-3) +272*a(n-4) +480*a(n-5) -1754*a(n-6) -942*a(n-7) +5482*a(n-8) +552*a(n-9) -9304*a(n-10) +888*a(n-11) +8539*a(n-12) -1755*a(n-13) -3871*a(n-14) +1092*a(n-15) +648*a(n-16) -216*a(n-17) for n>18

Empirical: a(n) = 13*a(n-1) -15*a(n-2) -453*a(n-3) +1521*a(n-4) +6613*a(n-5) -32909*a(n-6) -51675*a(n-7) +380097*a(n-8) +219811*a(n-9) -2799167*a(n-10) -347187*a(n-11) +14122641*a(n-12) -1318649*a(n-13) -50390909*a(n-14) +9520521*a(n-15) +128386635*a(n-16) -28476569*a(n-17) -232248311*a(n-18) +53023599*a(n-19) +292800435*a(n-20) -67701287*a(n-21) -248742653*a(n-22) +60344055*a(n-23) +134128372*a(n-24) -35691432*a(n-25) -40789736*a(n-26) +12200640*a(n-27) +5184000*a(n-28) -1728000*a(n-29) for n>30

Empirical: a(n) = 9*n^3 + (9/2)*n^2 - (9/2)*n.

Formula confirmed by Robert Israel, Mar 07 2018: see link.

From Colin Barker, Jul 12 2018: (Start)

G.f.: 9*x*(1 + 5*x) / (1 - x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.

(End)

Empirical: a(n) = 22*n^4 + (88/3)*n^3 - 22*n^2 - (22/3)*n.

Conjectures from Colin Barker, Jul 12 2018: (Start)

G.f.: 22*x*(1 + 17*x + 7*x^2 - x^3) / (1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.

(End)

Empirical: a(n) = 11*a(n-1) -13*a(n-2) -243*a(n-3) +709*a(n-4) +2169*a(n-5) -9198*a(n-6) -9930*a(n-7) +61958*a(n-8) +23330*a(n-9) -254062*a(n-10) -16614*a(n-11) +669763*a(n-12) -53961*a(n-13) -1146177*a(n-14) +176529*a(n-15) +1247849*a(n-16) -252499*a(n-17) -820948*a(n-18) +202824*a(n-19) +291592*a(n-20) -85440*a(n-21) -41472*a(n-22) +13824*a(n-23) for n>24

Empirical: a(n) = 15*a(n-1) -17*a(n-2) -754*a(n-3) +2878*a(n-4) +16186*a(n-5) -93370*a(n-6) -189146*a(n-7) +1639426*a(n-8) +1204022*a(n-9) -18805282*a(n-10) -2292182*a(n-11) +152576908*a(n-12) -28963300*a(n-13) -912064552*a(n-14) +308639746*a(n-15) +4101425146*a(n-16) -1601886466*a(n-17) -13987156774*a(n-18) +5410333402*a(n-19) +36132409294*a(n-20) -12837363742*a(n-21) -70095351814*a(n-22) +22168488862*a(n-23) +100539442517*a(n-24) -28492814059*a(n-25) -104090698583*a(n-26) +27535964120*a(n-27) +74934038256*a(n-28) -19682095616*a(n-29) -35224075008*a(n-30) +9752617152*a(n-31) +9586454976*a(n-32) -2904906240*a(n-33) -1119744000*a(n-34) +373248000*a(n-35) for n>36