A111607 Fourth column of A109626.

1, 2, 3, 3, 5, 3, 7, 2, 9, 10, 11, 9, 13, 7, 15, 4, 17, 18, 19, 15, 21, 11, 23, 6, 25, 26, 27, 21, 29, 15, 31, 8, 33, 34, 35, 27, 37, 19, 39, 10, 41, 42, 43, 33, 45, 23, 47, 12, 49, 50, 51, 39, 53, 27, 55, 14, 57, 58, 59, 45, 61, 31, 63, 16, 65, 66, 67, 51, 69, 35, 71, 18, 73, 74, 75

Offset: 1

Author

Paul D. Hanna and Robert G. Wilson v, Aug 01 2005

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A109626.

Programs

Magma

R:=PowerSeriesRing(Integers(), 102);
p:= func< x | x*(1+2*x+3*x^2+3*x^3 +5*x^4 +3*x^5 +7*x^6 +2*x^7 +7*x^8 +6*x^9 +5*x^10 +3*x^11 +3*x^12 +x^13 +x^14)/(1-x^8)^2 >;
Coefficients(R!( p(x) )); // G. C. Greubel, Jan 29 2025

Mathematica

(* First program *)
f[n_] := f[n] = Block[{a}, a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j, 0, 128}]]; g[n_, m_] := f[n][[m]]; Table[g[n, 4 + 1], {n, 75}]
(* Second program *)
CoefficientList[Series[(1+2*x+3*x^2+3*x^3+5*x^4+3*x^5+7*x^6+2*x^7+7*x^8 +6*x^9+5*x^10+3*x^11+3*x^12+x^13+x^14)/(1-x^8)^2, {x,0,100}], x] (* G. C. Greubel, Jan 29 2025 *)

SageMath

def p(x): return x*(1+2*x+3*x^2+3*x^3 +5*x^4 +3*x^5 +7*x^6 +2*x^7 +7*x^8 +6*x^9 +5*x^10 +3*x^11 +3*x^12 +x^13 +x^14)
def A111607_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( p(x)/(1-x^8)^2 ).list()
a=A111607_list(101); a[1:] # G. C. Greubel, Jan 29 2025

Formula

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