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

1,2

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).

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)

Prog

(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

Crossrefs

Cf. A016777 (another such sequence), A210380.

Sequence in context: A241987 A189464 A130333 * A080579 A325274 A303264

Adjacent sequences: A304865 A304866 A304867 * A304869 A304870 A304871

Keyword

nonn,easy

Author

Michel Marcus, May 20 2018

Status

approved