A083352 Least positive integer coefficients of power series A(x) such that the coefficients of A(x)^2 + A(x) - 1 consist entirely of squares.

1, 3, 9, 9, 3, 15, 33, 18, 36, 24, 75, 96, 51, 96, 159, 165, 255, 168, 27, 60, 333, 255, 66, 18, 441, 291, 735, 258, 390, 696, 480, 696, 1062, 423, 927, 681, 867, 1731, 300, 1131, 969, 1959, 222, 1812, 1047, 2487, 3027, 1020, 537, 2859, 1788, 1698, 63, 2712, 2250, 2934, 282, 2409, 1533, 3243, 1188, 7002, 1779, 1038, 207, 4110, 6156, 7068, 7470, 825, 3501, 3612, 6984, 9945, 2733, 7383, 1581, 2730, 2616, 7476, 13959, 13224, 15078, 12837, 2979, 2805, 2943, 15426, 8871, 4536

Offset

0,2

Comments

After the first term, all terms seem to be multiples of 3.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Example

G.f.: A(x) = 1 + 3*x + 9*x^2 + 9*x^3 + 3*x^4 + 15*x^5 + 33*x^6 + 18*x^7 + 36*x^8 + 24*x^9 + 75*x^10 + 96*x^11 + 51*x^12 + 96*x^13 + 159*x^14 + 165*x^15 + 255*x^16 + 168*x^17 + 27*x^18 + 60*x^19 + 333*x^20 + 255*x^21 + 66*x^22 + 18*x^23 + 441*x^24 + 291*x^25 + 735*x^26 + 258*x^27 + 390*x^28 + 696*x^29 + 480*x^30 + ...
Let the B(x) denote the g.f. of A083353, then
B(x) = A(x)^2 + A(x) - 1 = 1 + 9*x + 36*x^2 + 81*x^3 + 144*x^4 + 225*x^5 + 324*x^6 + 576*x^7 + 1089*x^8 + 1296*x^9 + 1764*x^10 + 2916*x^11 + 4356*x^12 + 6084*x^13 + 7569*x^14 + 9801*x^15 + 16641*x^16 + 20736*x^17 + 25281*x^18 + 32400*x^19 + 39204*x^20 + 53361*x^21 + 69696*x^22 + 76176*x^23 + 90000*x^24 + 110889*x^25 + 149769*x^26 + 176400*x^27 + 184041*x^28 + 207936*x^29 + 281961*x^30 + ...
the coefficients of which are squares (see A083354).

Prog

(PARI) {for(i=1, 30, A=[1]; print1(A[1], ", "); for(i=1, 200, A0=concat(A, 0); for(n=1, 100*A[#A], A0[#A0]=n; B=Vec(Ser(A0)^2 + Ser(A0) - 1); if(issquare(B[#B]), print1(n, ", "); A0[#A0]=n; A=A0; break)))); A}

Crossrefs

Cf. A083353, A083354.

Sequence in context: A157349 A159811 A279134 * A243350 A091559 A268107

Adjacent sequences: A083349 A083350 A083351 * A083353 A083354 A083355

Keyword

nonn

Author

Paul D. Hanna, Apr 26 2003

Extensions

Extended by Paul D. Hanna, Nov 19 2017

Status

approved