A241201 a(n) is the least r such that there are n+2 consecutive increasing terms in the r-th row of Pascal's triangle (binomial(r,*)) which satisfy a polynomial of degree n.

7, 14, 62, 31, 339, 1022

Offset: 1

Author

T. D. Noe, Apr 21 2014

Keywords

• nonn
• more

Comments

Old definition: "Numbers k such that n+2 consecutive terms of binomial(n,k) satisfy a polynomial relation of degree n for some k in the range 0 <= k <= n/2.".

Is this sequence finite?

Examples

			a(1) = 7 because the 3 terms 7, 21, 35 are linear.
		

Crossrefs

Cf. A008865 (binomial(n,k) has 3 consecutive terms in a linear relation).

Cf. A062730 (3 terms in arithmetic progression in Pascal's triangle).

Cf. A241199, A241200 (similar, but quadratic).

Cf. A241202 (position of the first of terms).

Programs

Mathematica

t = Table[k = 1; While[b = Binomial[k, Range[0, k/2]]; d = Differences[b, n + 1]; ! MemberQ[d, 0], k++]; {k, Position[d, 0, 1, 1][[1, 1]] - 1}, {n, 6}]; Transpose[t][[1]]

Extensions

Definition clarified by Don Reble, Dec 14 2020