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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. 3

%I

%S 1,3,9,9,3,15,33,18,36,24,75,96,51,96,159,165,255,168,27,60,333,255,

%T 66,18,441,291,735,258,390,696,480,696,1062,423,927,681,867,1731,300,

%U 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

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

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

%H Paul D. Hanna, <a href="/A083352/b083352.txt">Table of n, a(n) for n = 0..300</a>

%e 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 + ...

%e Let the B(x) denote the g.f. of A083353, then

%e 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 + ...

%e the coefficients of which are squares (see A083354).

%o (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}

%Y Cf. A083353, A083354.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Apr 26 2003

%E Extended by _Paul D. Hanna_, Nov 19 2017

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Last modified December 1 11:29 EST 2021. Contains 349429 sequences. (Running on oeis4.)