A224251 Numbers, a(n) where binomial(a(n), 5n-1) == 0 (mod 5) and binomial(a(n), k) != 0 (mod 5) for k != 5n - 1.
8, 13, 18, 23, 48, 73, 98, 123, 248, 373, 498, 623, 1248, 1873, 2498, 3123, 6248, 9373, 12498, 15623, 31248, 46873, 62498, 78123, 156248, 234373, 312498, 390623, 781248, 1171873, 1562498, 1953123, 3906248, 5859373, 7812498, 9765623, 19531248, 29296873
Offset: 1
Examples
a(4) = 23. In the 23rd row of Pascal's triangle, the binomial coefficients C(23, 4), C(23, 9), C(23, 14) and C(23, 19) are divisible by 5 and none of the others are. C(23, 4) = 8855 = C(23, 19) and C(23, 9) = 817,190 = C(23, 14).
References
- Thomas M. Green, Prime Patterns in Pascal's Triangle, paper in review process, 2013.
Links
- Harvey P. Dale, Table of n, a(n) for n = 1..1000
- N. J. A. Sloane, First 141 rows of Pascal's triangle, formatted as a simple linear sequence n, a(n)
- Index entries for linear recurrences with constant coefficients, signature (1,0,0,5,-5).
Programs
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Mathematica
LinearRecurrence[{1,0,0,5,-5},{8,13,18,23,48},40] (* Harvey P. Dale, May 15 2016 *)
Formula
a(n) = 5*(n + 1)- 2, for n <= 4; a(5) = (5^2)*2 - 2; a(n)= a(n-1)+ 5*a(n-4)- 5*a(n-5) for n>=6.
G.f.: -x*(15*x^4-5*x^3-5*x^2-5*x-8) / ((x-1)*(5*x^4-1)). - Colin Barker, Apr 02 2013
Extensions
More terms from Colin Barker, Apr 02 2013
Comments