A274309 Related to integral generalized binomial coefficients of multiplicative functions (see Comments for precise definition).

2, 1, 3, 3, 1, 1, 3, 5, 3, 1, 1, 3, 3, 1, 2, 3, 2, 1, 3, 3, 1, 1, 3, 5, 3, 1, 1, 3, 3, 1, 2, 3, 2, 1, 3, 3, 1, 1, 3, 5, 3, 1, 1, 3, 3, 1, 2, 3, 2, 1, 3, 3, 1, 1, 3, 5, 3, 1, 1, 3, 3, 1, 2, 3, 2, 1, 3, 3, 1, 1, 3, 5, 3, 1, 1, 3, 3, 1, 2, 3, 2, 1, 3, 3, 1

Offset

1,1

Comments

Let C(n,m)_f = ( Product_{i=1..n} f(i) ) / (( Product_{i=1..n-m} f(i) ) * ( Product_{i=1..m} f(i) )). [I would have expected the lower limits in these products to be 0 rather than 1. - N. J. A. Sloane, Jun 21 2016]

Set m=3, and for any multiplicative function f, construct a binary sequence S by writing 1 if C(n,3)_f is an integer, 0 if not.

The present sequence gives the lengths of the successive runs in B.

For example the initial terms 2,1,3,3,... mean that B begins 1,1,0,1,1,1,0,0,0,...

Periodic with period length 16.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Imanuel Chen and Michael Z. Spivey, Integral Generalized Binomial Coefficients of Multiplicative Functions, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.

Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1).

Formula

G.f.: x*(2 - x + 4*x^2 - x^3 + 2*x^4 - x^5 + 4*x^6 + x^7 + 2*x^8 - x^9 + 2*x^10 + x^11 + 2*x^12 - x^13 + 3*x^14) / ((1 - x)*(1 + x^2)*(1 + x^4)*(1 + x^8)). - Colin Barker, Jun 22 2016

Prog

(PARI) Vec(x*(2 -x +4*x^2 -x^3 +2*x^4 -x^5 +4*x^6 +x^7 +2*x^8 -x^9 +2*x^10 +x^11 +2*x^12 -x^13 +3*x^14) / ((1 -x)*(1 +x^2)*(1 +x^4)*(1 +x^8)) + O(x^100)) \\ Colin Barker, Jun 22 2016

Crossrefs

Sequence in context: A212634 A162883 A081446 * A158440 A119803 A195916

Adjacent sequences: A274306 A274307 A274308 * A274310 A274311 A274312

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 21 2016

Status

approved