A272100 Integers n that are the sum of three nonzero squares while n*(n+1) is not.

12, 19, 44, 51, 76, 83, 108, 115, 140, 147, 172, 179, 204, 211, 236, 243, 268, 275, 300, 307, 332, 339, 364, 371, 396, 403, 428, 435, 460, 467, 492, 499, 524, 531, 556, 563, 588, 595, 620, 627, 652, 659, 684, 691, 716, 723, 748, 755, 780, 787, 812, 819, 844, 851, 876, 883, 908, 915

Offset

1,1

Comments

Values of a^2 + b^2 + c^2 such that (a^2 + b^2 + c^2)^2 + a^2 + b^2 + c^2 is not of the form x^2 + y^2 + z^2 where a, b, c, x, y, z are nonzero integers.

First differences of this sequence are 7, 25, 7, 25, 7, 25, 7, 25, 7, 25, ...

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Formula

From Colin Barker, Apr 30 2016: (Start)

a(n) = (32*n-17-9*(-1)^n)/2.

a(n) = 16*n-13 for n even.

a(n) = 16*n-4 for n odd.

a(n) = a(n-1)+a(n-2)-a(n-3) for n>3.

G.f.: x*(12+7*x+13*x^2) / ((1-x)^2*(1+x)).

(End)

Example

12 is a term because 12 = 2^2 + 2^2 + 2^2 = A000408(5) and 12*13 = A002378(12) = 156 is not in A000408.

Mathematica

Select[Range[10^3], Length[PowersRepresentations[#, 3, 2] /. {0, __} -> Nothing] > 0 && Length[PowersRepresentations[# (# + 1), 3, 2] /. {0, __} -> Nothing] == 0 &] (* Michael De Vlieger, Apr 20 2016, Version 10.2 *)
LinearRecurrence[{1, 1, -1}, {12, 19, 44}, 60] (* Harvey P. Dale, Mar 13 2017 *)

Prog

(PARI) isA000408(n) = my(a, b) ; a=1 ; while(a^2+1<n, b=1 ; while(b<=a && a^2+b^2<n, if(issquare(n-a^2-b^2), return(1) ) ; b++ ; ) ; a++ ; ) ; return(0);
lista(nn) = for(n=1, nn, if(isA000408(n) && !isA000408(n*(n+1)), print1(n, ", ")));
(PARI) Vec(x*(12+7*x+13*x^2)/((1-x)^2*(1+x)) + O(x^50)) \\ Colin Barker, Apr 30 2016

Crossrefs

Cf. A000408, A002378.

Sequence in context: A246343 A136770 A155574 * A119382 A166671 A357211

Adjacent sequences: A272097 A272098 A272099 * A272101 A272102 A272103

Keyword

nonn,easy

Author

Altug Alkan, Apr 20 2016

Status

approved