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A134662 Number of odd coefficients in (1 + x + x^4)^n. 4

%I

%S 1,3,3,9,3,7,9,17,3,9,7,21,9,17,17,33,3,9,9,27,7,17,21,43,9,27,17,51,

%T 17,35,33,67,3,9,9,27,9,21,27,51,7,21,17,51,21,41,43,83,9,27,27,81,17,

%U 43,51,113,17,51,35,105,33,67,67,137,3,9,9,27,9,21,27,51,9,27,21,63,27,51

%N Number of odd coefficients in (1 + x + x^4)^n.

%H S. R. Finch, P. Sebah and Z.-Q. Bai, <a href="http://arXiv.org/abs/0802.2654">Odd Entries in Pascal's Trinomial Triangle</a> (arXiv:0802.2654)

%e From _Omar E. Pol_, Mar 01 2015: (Start)

%e Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:

%e 1;

%e 3;

%e 3,9;

%e 3,7,9,17;

%e 3,9,7,21,9,17,17,33;

%e 3,9,9,27,7,17,21,43,9,27,17,51,17,35,33,67;

%e 3,9,9,27,9,21,27,51,7,21,17,51,21,41,43,83,9,27,27,81,17,43,51,113,17,51,35,105,33,67,67,137;

%e Thanks to _Michel Marcus_ we can see the first few terms of the next four rows as shown below:

%e 3,9,9,27,9,21,27,51,9,27,21,63,27,51,51,99,7,21,...

%e 3,9,9,27,9,21,27,51,9,27,21,63,27,51,51,99,9,27,27,...

%e 3,9,9,27,9,21,27,51,9,27,21,63,27,51,51,99,9,27,27,81,...

%e 3,9,9,27,9,21,27,51,9,27,21,63,27,51,51,99,9,27,27,81,21,...

%e ...

%e Apparently in each row the first quarter of the terms (and no more) are equal to 3 times the beginning of the sequence itself (comment corrected after Sloane's comment in A247649, Mar 03 2015).

%e (End)

%t Table[PolynomialMod[(1+x+x^4)^n,2]/.x->1,{n,0,80}]

%t Table[Count[CoefficientList[Expand[(1+x+x^4)^n],x],_?OddQ],{n,0,80}] (* _Harvey P. Dale_, Apr 15 2012 *)

%o (PARI) a(n) = {my(pol = (xx^4 + xx + 1)*Mod(1,2)); subst(lift(pol^n), xx, 1);} \\ _Michel Marcus_, Mar 01 2015

%o (PARI) tabf(nn, k=16) = {nbpt = 0; for (n=0, nn, if (n==0, nbt = 1, nbt = 2^(n-1)); for (m=nbpt, nbpt+nbt-1, if (m-nbpt >k, k++; break); print1(nbopd(m), ",");); print(); nbpt += nbt;);} \\ _Michel Marcus_, Mar 03 2015

%Y Cf. A071053.

%K nonn

%O 0,2

%A _Steven Finch_, Jan 25 2008

%E First Mathematica program corrected by _Harvey P. Dale_, Apr 15 2012

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Last modified August 26 00:35 EDT 2019. Contains 326324 sequences. (Running on oeis4.)