OFFSET
1,1
COMMENTS
Members of A176541, for which there are only a finite number of solutions.
Integer k is in this sequence if k=2*m^2 and the equation (2*x-m*y)*(2*x+m*y)=A077415(k)/2 has integer solutions with y>=k. - Max Alekseyev, May 10 2010
It seems that a(n) = 2*A001651(n+2)^2. - Colin Barker, Sep 25 2015
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..10000
Sela Fried, Proof of the conjectures stated in A176542, 2025.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
FORMULA
Conjectures from Colin Barker, Sep 24 2015: (Start)
a(n) = (9*n^2+24*n+16)/2 for n even.
a(n) = (9*n^2+30*n+25)/2 for n odd.
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5.
G.f.: -2*x*(4*x^4-3*x^3-8*x^2+9*x+16) / ((x-1)^3*(x+1)^2). (End)
These conjectures are true (see Fried link). - Sela Fried, Jan 02 2026
EXAMPLE
32 is in this sequence because there is only one set of 32 consecutive triangular numbers that sum to a square (namely, A000217(26) thru A000217(57), which sum to 29584 = 172^2).
3 is NOT in this sequence, because there are infinitely many sets of 3 consecutive triangular numbers that sum to a square (cf. A165517).
4 is NOT in this sequence, because there are infinitely many sets of 4 consecutive triangular numbers that sum to a square (cf. A202391).
5 is NOT in this sequence, because there are NO sets of 5 consecutive triangular numbers that sum to a square.
11 is NOT in this sequence, since there are infinitely many sets of 11 consecutive triangular numbers that sum to a square (cf. A116476).
MATHEMATICA
LinearRecurrence[{1, 2, -2, -1, 1}, {32, 50, 98, 128, 200}, 50] (* Paolo Xausa, Mar 30 2026 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Andrew Weimholt, Apr 20 2010
EXTENSIONS
Terms a(6) onward from Max Alekseyev, May 10 2010
More terms from Paolo Xausa, Mar 30 2026
STATUS
approved