What rule governs the generation of the following series 1, 4, 7, 9, 11, 41, 115

Question

want to know how that serie is generated

Answer 1

Assume f(n)=An⁶+Bn⁵+Cn⁴+Dn³+En²+Fx+G for integer n≥0.

The table above shows how the coefficients A-F are calculated, and G=1, the first term.

Row 21 (R21) shows that 720A=-64, from which A=-64/720=-4/45.

R19 shows that 1800A+120B=25, from which B=(25-1800A)/120=37/24.

R16 shows that C=(2-1560A-240B)/24=-86/9, and so on.

f(n)=-4n⁶/45+37n⁵/24-86n⁴/9+213n³/8-1501n²/45+107n/6+1.

This function gives all the integer terms exactly in the given series, where n starts from 0. For example the last term is f(6)=115.

I suspect that other solutions may be available.