A304868 Numbers x satisfying x == 1 (mod 4) or x == 14, 26, 30 (mod 32).

1, 5, 9, 13, 14, 17, 21, 25, 26, 29, 30, 33, 37, 41, 45, 46, 49, 53, 57, 58, 61, 62, 65, 69, 73, 77, 78, 81, 85, 89, 90, 93, 94, 97, 101, 105, 109, 110, 113, 117, 121, 122, 125, 126, 129, 133, 137, 141, 142, 145, 149, 153, 154, 157, 158, 161, 165, 169, 173, 174, 177

Offset: 1

Author

Michel Marcus, May 20 2018

Keywords

• nonn
• easy

Comments

The sum of two distinct terms of this sequence is never a square.

Sequence has density 11/32, the maximal density that can be attained with such a sequence.

References

• J. P. Massias, Sur les suites dont les sommes des termes 2 à 2 ne sont pas des carrés, Publications du département de mathématiques de Limoges, 1982.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• J. C. Lagarias, A. M. Odlyzko, J. B. Shearer, On the density of sequences of integers the sum of no two of which is a square. I. Arithmetic progressions, Journal of Combinatorial Theory. Series A, 33 (1982), pp. 167-185.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,1,-1).

Crossrefs

Cf. A016777 (another such sequence), A210380.

Programs

PARI

isok(n) = ((n%4)==1) || ((n%32)==14) || ((n%32)==26) || ((n%32)==30);

PARI

Vec(x*(1 + 4*x + 4*x^2 + 4*x^3 + x^4 + 3*x^5 + 4*x^6 + 4*x^7 + x^8 + 3*x^9 + x^10 + 2*x^11) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10)) + O(x^40)) \\ Colin Barker, May 20 2018

Formula

From Colin Barker, May 20 2018: (Start)

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

a(n) = a(n-1) + a(n-11) - a(n-12) for n>12.

(End)