OFFSET
1,3
COMMENTS
This was formerly conjectured to be "values of maximal number of regions obtained by joining each triple of n points on a 3-dimensional sphere (S_2) by a plane", a generalization of A000127. However, members of Sugaku_day pointed out that there could be 48 (or even 54) regions for n=5 and not 41. Those counterexamples were visualized, so the conjecture was incorrect. - Taichi Aoki, Nov 13 2025
The problem of partitioning a sphere by binomial(n,3) planes determined by n points is discussed in a thread on Math StackExchange. It appears quite plausible that the value a(5)=41 currently listed (e.g., in the OEIS entry for A144841) may indeed be too low. - Seiichi Manyama, Nov 13 2025
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..10000
Sugaku_day, There could exist 48 regions for n=5, Nov 12 2025.
Taichi Aoki, Example of 48 regions with n=5 points on the sphere
Sugaku_day, Our member found 54 regions for n=5, Nov 20 2025.
Taichi Aoki, Example of 54 regions with n=5 points on the sphere
Math StackExchange, Maximum number of regions of a sphere partitioned by binomial(n,3) planes from n points, Nov 04 2016
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
FORMULA
G.f.: x*(1-9*x+37*x^2-84*x^3+111*x^4-76*x^5+9*x^6+18*x^7+274*x^8-x^9)/(1-x)^10. - Colin Barker, Feb 15 2012
MATHEMATICA
A144841[n_] := 280*#[9] + 10*#[6] + 6*#[4] + #[3] + 1 & [Binomial[n, #] &];
Array[A144841, 35] (* Paolo Xausa, Nov 15 2025 *)
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {1, 1, 2, 11, 41, 121, 316, 757, 1961, 6281}, 40] (* Harvey P. Dale, Dec 25 2025 *)
PROG
(Magma) function a(n) return 280*Binomial(n, 9)+10*Binomial(n, 6)+6*Binomial(n, 4)+Binomial(n, 3)+1; end function;
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Sep, Jorik & Noud Thijssen (sep.thijssen(AT)gmail.com), Sep 22 2008
EXTENSIONS
More terms and new name using formula given from Joerg Arndt, Aug 21 2025
Edited by N. J. A. Sloane, Nov 15 2025
STATUS
approved