OFFSET
0,2
COMMENTS
AESZ #182 is entry 182 in the Almkvist-van Enckevort-van Straten-Zudilin tabulation of Calabi-Yau differential equations and case 2.64 in the Almkvist-van Straten classification of degree-two Calabi-Yau operators. The operator is theta^4 - z*(43*theta^4 + 86*theta^3 + 77*theta^2 + 34*theta + 6) + 12*z^2*(theta+1)^2*(6*theta+5)*(6*theta+7), with theta = z*d/dz.
Discriminant 1 - 43*z + 432*z^2 = (1 - 16*z)*(1 - 27*z); conifold positions at z = 1/27 and z = 1/16; topological invariants H^3 = 132, c_2*H = 132, c_3 = -96, |H| = 33.
Conjectured Apery-limit: let b(n) satisfy the same recurrence with b(0) = 0, b(1) = 1. Then lim_{n->infinity} b(n)/a(n) = (3/11)*zeta(3). Equivalently, this conjecture is equivalent to the polynomial continued fraction 11/(3*zeta(3)) = 6 - 420/(246 - 109824/(1758 - 2825604/(6606 - ...))), with quartic partial denominators P_n = 43*n^4 + 86*n^3 + 77*n^2 + 34*n + 6 and degree-8 partial numerators C_n = 12*(6*n-1)*(6*n+1)*n^6. Numerically checked to over 45 decimal digits.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..700
G. Almkvist, C. van Enckevort, D. van Straten, and W. Zudilin, Tables of Calabi-Yau equations, arXiv:math/0507430 [math.AG], 2010, entry #182.
G. Almkvist and D. van Straten, Calabi-Yau operators of degree two, J. Algebraic Combin. 58 (2023), 1203-1259, case 2.64.
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
P. Metelitsyn and D. van Straten, Calabi-Yau differential operator database, entry AESZ #182.
FORMULA
n^4*a(n) = (43*n^4 - 86*n^3 + 77*n^2 - 34*n + 6)*a(n-1) - 12*(6*n-7)*(6*n-5)*(n-1)^2*a(n-2), with a(0) = 1, a(1) = 6.
Conjecturally, a(n) ~ (9/(2*Pi^2)) * 27^n / n^2.
MATHEMATICA
a[0] = 1; a[1] = 6;
a[n_] := a[n] = ((43*n^4 - 86*n^3 + 77*n^2 - 34*n + 6)*a[n-1] - 12*(6*n-7)*(6*n-5)*(n-1)^2*a[n-2])/n^4;
Table[a[n], {n, 0, 19}]
PROG
(PARI) a(n) = if(n<2, [1, 6][n+1], ((43*n^4-86*n^3+77*n^2-34*n+6)*a(n-1) - 12*(6*n-7)*(6*n-5)*(n-1)^2*a(n-2))/n^4);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Alex Shvets, May 21 2026
STATUS
approved