A397085 - OEIS

A397085

a(n) = (-1)^(n-floor(sqrt(n))^2-1) if n is not a perfect square and 0 otherwise.

0, 0, 1, -1, 0, 1, -1, 1, -1, 0, 1, -1, 1, -1, 1, -1, 0, 1, -1, 1, -1, 1, -1, 1, -1, 0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1

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OFFSET

0,1

COMMENTS

For n >= 1, if a(n) = -1 or a(n) = 0, then A161886(n) is odd; if a(n) = 1, then A161886(n) is even. Equivalently, a(n) is -1 or 0 precisely when n belongs to A133280, and a(n) = 1 precisely when n belongs to A195437.

REFERENCES

Ian Connell, Elementary Problem E1382, Amer. Math. Monthly, 66 (1959), no. 8, 724.

Ian Connell and Andrew Korsak, Solution to Elementary Problem E1382, Amer. Math. Monthly, 67 (1960), no. 4, 380.

D. E. Iannucci and D. Mills-Taylor, On Generalizing the Connell Sequence, Journal of Integer Sequences, Vol. 2 (1999), Article 99.1.7.

R. Redheffer, "Eine explizit loesbare Optimierungsaufgabe" (in German), in Numerische Methoden bei Optimierungsaufgaben, Band 3 (Tagung, Math. Forschungsinst., Oberwolfach, 1976), pp. 213-216, Internat. Ser. Numer. Math., Vol. 36, Birkhaeuser, Basel, 1977.

G. E. Stevens, A Connell-Like Sequence, Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.4.

LINKS

Paolo Xausa, Table of n, a(n) for n = 0..10000

Anthony Hernandez, The Redheffer Matrix Parity Problem, arXiv:2607.08962 [math.CO], 2026.

FORMULA

G.f.: Sum_{m>=1} x^(m^2+1) * (1 - x^(2*m)) / (1+x).

D.g.f. for |a(n)|: zeta(s) - zeta(2s).

EXAMPLE

For small values of n we have:

a(0) = 0 because 0 = 0^2 + 0.

a(1) = 0 because 1 = 1^2 + 0.

a(2) = 1 because 2 = 1^2 + 1 and (-1)^(1-1) = 1.