A367546 a(n) = Sum_{k=0..n} n^k * |(n - k | k)|, where (a | b) denotes the Kronecker symbol.

0, 2, 2, 12, 68, 780, 7782, 137256, 2130440, 47895390, 1010001010, 28531167060, 743044451340, 25239592216020, 797785000011174, 31147773583410240, 1157442765409226768, 51702516367896047760, 2185932446972586391986, 109912203092239643840220, 5255987282125313280008020

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Wikipedia, Kronecker symbol

Formula

a(n) = Sum_{k=0..n} [gcd(k, n) = 1] * n^k, where [] is the Iverson bracket.

Maple

KS := (n, k) -> NumberTheory:-KroneckerSymbol(n, k):
A367546 := n -> local k; add(n^k * abs(KS(n - k , k)), k = 0..n):
seq(A367546(n), n = 0..20);

Mathematica

A367546[n_]:=If[n==0, 0, Sum[n^k*Boole[CoprimeQ[n, k]], {k, 0, n}]]; Array[A367546, 25, 0] (* Paolo Xausa, Nov 24 2023 *)

Prog

(SageMath)
def a(n):
    return sum(abs(kronecker_symbol(n - k, k)) * n^k for k in range(n + 1))
# Alternative:  (For Python include 'import math' for math.gcd.)
def a(n):
    cop = [int(gcd(k, n) == 1) for k in (0..n)]
    return sum(p * n^k for k, p in enumerate(cop))
print([a(n) for n in range(21)])
(PARI) a(n) = sum(k=0, n, n^k*abs(kronecker(n-k, k))); \\ Michel Marcus, Nov 23 2023
(Python)
from math import gcd
def A367546(n): return sum(n**k for k in range(n+1) if gcd(n, k)==1) # Chai Wah Wu, Nov 24 2023

Crossrefs

Cf. A217831, A000010, A023896, A055034, A349136, A367544, A367545.

Sequence in context: A131444 A291541 A180072 * A013315 A032321 A013311

Adjacent sequences: A367543 A367544 A367545 * A367547 A367548 A367549

Keyword

nonn

Author

Peter Luschny, Nov 22 2023

Status

approved