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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.
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
1,1
COMMENTS
Row numbers of Pascal's triangle where only every 5th binomial coefficient in the row is divisible by 5.
Numbers of the form (i*5^j) - 2 (i = 2,3,4,5, j >= 1).
The binomial coefficients which are multiples of 5 are easily seen in the display of Pascal's triangle given in the link below by N. J. A. Sloane.
REFERENCES
Thomas M. Green, Prime Patterns in Pascal's Triangle, paper in review process, 2013.
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
EXAMPLE
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).
MATHEMATICA
LinearRecurrence[{1, 0, 0, 5, -5}, {8, 13, 18, 23, 48}, 40] (* Harvey P. Dale, May 15 2016 *)
CROSSREFS
Cf. A007318 (Pascal's triangle), A181287.
Sequence in context: A319879 A095097 A187227 * A006619 A023487 A045636
KEYWORD
nonn,easy
AUTHOR
Thomas M. Green, Apr 02 2013
EXTENSIONS
More terms from Colin Barker, Apr 02 2013
STATUS
approved