<html><P>Seqfans, <BR> David probably already noticed this, but I make some quick observations<BR>by mere inspection (I am on my lunch break and lack time). <BR> The reciprocal of the g.f.s of p_k seems to reveal what is going on <BR>with these nice sequences:</P>
<P>1/GF(p_2) = GF([1,-1,0,0,1,0,0,0,0,-1,0,0,0,0])<BR>1/GF(p_3) = GF([1,-1,-1,0,0,1,0,1,0,0,0,0,-1,0])<BR>1/GF(p_4) = GF([1,-1,-1,-1,0,0,1,0,1,0,1,0,0,0])<BR>1/GF(p_5) = GF([1,-1,-1,-1,-1,0,0,1,0,1,0,1,0,1])<BR>1/GF(p_6) = GF([1,-1,-1,-1,-1,-1,0,0,1,0,1,0,1,0])<BR>1/GF(p_7) = GF([1,-1,-1,-1,-1,-1,-1,0,0,1,0,1,0,1])<BR>1/GF(p_8) = GF([1,-1,-1,-1,-1,-1,-1,-1,0,0,1,0,1,0])<BR>1/GF(p_9) = GF([1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,0,1])<BR>1/GF(p_10) = GF([1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,0])<BR><BR>It appears that the g.f. of p_2 is <BR> Sum_{n>=0} (-1)^n*x^(n^2)<BR> <BR>The g.f.s of the other p_k seem forthcoming.<BR> Paul<BR> <BR><BR>-- "David Wilson" <davidwwilson@comcast.net> wrote:<BR>[...]<BR><FONT face=Arial size=2><FONT size=2>Computing a few terms of p_k for small k gives</P>
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<P>p_1 = (1 0 0 0 0 0 0 0 0 0 0 0 0 0 ...)<BR>p_2 = (1 1 1 1 0 -1 -2 -3 -3 -1 2 6 10 11 ...)<BR>p_3 = (1 1 2 3 5 7 11 15 22 30 42 56 77 101 ...)<BR>p_4 = (1 1 2 4 7 13 23 42 75 135 242 434 779 1396 ...)<BR>p_5 = (1 1 2 4 8 15 29 55 106 202 387 739 1414 2702 ...)<BR>p_6 = (1 1 2 4 8 16 31 61 119 234 458 898 1759 3447 ...)<BR>p_7 = (1 1 2 4 8 16 32 63 125 247 490 970 1922 3806 ...)<BR>p_8 = (1 1 2 4 8 16 32 64 127 253 503 1002 1994 3970 ...)<BR>p_9 = (1 1 2 4 8 16 32 64 128 255 509 1015 2026 4042 ...)<BR>p_10 = (1 1 2 4 8 16 32 64 128 256 511 1021 2039 4074 ...)<BR>[...]</P></FONT></FONT></DIV></html>