A071053 -id:A071053 - cp's OEIS frontend

A134660 Number of odd coefficients in (1 + x + x^2 + x^3)^n.

1, 4, 4, 4, 4, 16, 4, 8, 4, 16, 16, 4, 4, 16, 8, 16, 4, 16, 16, 16, 16, 64, 4, 8, 4, 16, 16, 8, 8, 32, 16, 32, 4, 16, 16, 16, 16, 64, 16, 32, 16, 64, 64, 4, 4, 16, 8, 16, 4, 16, 16, 16, 16, 64, 8, 16, 8, 32, 32, 16, 16, 64, 32, 64, 4, 16, 16, 16, 16, 64, 16, 32, 16, 64, 64, 16, 16, 64

Offset: 0

Steven Finch, Jan 25 2008

nonn

			From _Omar E. Pol_, Mar 01 2015: (Start)
Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
4;
4,4;
4,16,4,8;
4,16,16,4,4,16,8,16;
4,16,16,16,16,64,4,8,4,16,16,8,8,32,16,32;
4,16,16,16,16,64,16,32,16,64,64,4,4,16,8,16,4,16,16,16,16,64,8,16,8,32,32,16,16,64,32,64;
...
(End)

G. C. Greubel, Table of n, a(n) for n = 0..10000
S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.

Cf. A000120, A001316, A008287, A036555, A071053.

seq(igcd(4^n,binomial(4*n,n)),n=0..77); # Peter Luschny, Nov 08 2011

PolynomialMod[(1+x+x^2+x^3)^n, 2] /. x->1
A036555 = Total /@ IntegerDigits[3 Range[0, 100], 2]; Table[2^A036555[[n]], {n, 1, 20}] (* or *) Table[GCD[4^n, Binomial[4*n, n]], {n, 0, 50}] (* G. C. Greubel, Dec 31 2017 *)

a(n) = {my(pol= Pol([1,1,1,1], xx)*Mod(1,2)); subst(lift(pol^n), xx, 1);} \\ Michel Marcus, Mar 01 2015

a(n) = 2^hammingweight(3*n); \\ Joerg Arndt, Mar 10 2015

a(n) = 2^A036555(n).

a(n) = gcd(4^n, C(4*n, n)). - Peter Luschny, Nov 08 2011

A134661 Number of odd coefficients in (1 + x + x^3)^n.

Original entry on oeis.org

1, 3, 3, 7, 3, 9, 7, 13, 3, 9, 9, 19, 7, 21, 13, 27, 3, 9, 9, 21, 9, 27, 19, 35, 7, 21, 21, 41, 13, 39, 27, 55, 3, 9, 9, 21, 9, 27, 21, 39, 9, 27, 27, 55, 19, 57, 35, 73, 7, 21, 21, 49, 21, 63, 41, 75, 13, 39, 39, 79, 27, 81, 55, 109, 3, 9, 9, 21, 9, 27, 21, 39, 9, 27, 27, 57, 21, 63, 39

Offset: 0

Steven Finch, Jan 25 2008

nonn

			From _Omar E. Pol_, Mar 01 2015: (Start)
Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
3;
3,7;
3,9,7,13;
3,9,9,19,7,21,13,27;
3,9,9,21,9,27,19,35,7,21,21,41,13,39,27,55;
3,9,9,21,9,27,21,39,9,27,27,55,19,57,35,73,7,21,21,49,21,63,41,75,13,39,39,79,27,81,55,109;
3,9,9,21,9,27,21,39,9,27,27,57,21,63,39...
...
Note that in each row a fraction of the first terms are equal to 3 times the beginning of the sequence itself. For rows 0-6 the fractions are: 0, 1, 1/2, 1/2, 3/8, 3/8, 11/32. Apparently the fractions converge to a constant.
(End)

Joerg Arndt, Table of n, a(n) for n = 0..10000
S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.

Cf. A071053, A038717.

Mathematica
```
PolynomialMod[(1+x+x^3)^n, 2] /. x->1
```

a(n) = {my(pol= Pol([1,0,1,1], xx)*Mod(1,2)); subst(lift(pol^n), xx, 1);} \\ Michel Marcus, Mar 01 2015

A245565 a(n) = Product_{i in row n of A245562} Pell(i+1).

Original entry on oeis.org

1, 2, 2, 5, 2, 4, 5, 12, 2, 4, 4, 10, 5, 10, 12, 29, 2, 4, 4, 10, 4, 8, 10, 24, 5, 10, 10, 25, 12, 24, 29, 70, 2, 4, 4, 10, 4, 8, 10, 24, 4, 8, 8, 20, 10, 20, 24, 58, 5, 10, 10, 25, 10, 20, 25, 60, 12, 24, 24, 60, 29, 58, 70, 169, 2, 4, 4, 10, 4, 8, 10, 24, 4, 8, 8, 20, 10, 20, 24, 58, 4, 8, 8, 20, 8, 16

Offset: 0

N. J. A. Sloane, Aug 10 2014; revised Sep 05 2014

This is the Run Length Transform of S(n) = Pell(n+1) (cf. A000129).

The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).

Alois P. Heinz, Table of n, a(n) for n = 0..8191

Cf. A245562, A000129, A001045, A071053, A245564.

A000129 := proc(n) option remember; if n <=1 then n; else 2*A000129(n-1)+A000129(n-2); fi; end;
ans:=[];
for n from 0 to 100 do lis:=[]; t1:=convert(n,base,2); L1:=nops(t1);
out1:=1; c:=0;
for i from 1 to L1 do
   if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1;
   elif out1 = 0 and t1[i] = 1 then c:=c+1;
   elif out1 = 1 and t1[i] = 0 then c:=c;
   elif out1 = 0 and t1[i] = 0 then lis:=[c,op(lis)]; out1:=1; c:=0;
   fi;
   if i = L1 and c>0 then lis:=[c,op(lis)]; fi;
                   od:
a:=mul(A000129(i+1), i in lis);
ans:=[op(ans),a];
od:
ans;

A247282 Run Length Transform of A001317.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 1, 3, 3, 3, 5, 15, 1, 1, 1, 3, 1, 1, 3, 5, 3, 3, 3, 9, 5, 5, 15, 17, 1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 1, 3, 3, 3, 5, 15, 3, 3, 3, 9, 3, 3, 9, 15, 5, 5, 5, 15, 15, 15, 17, 51, 1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 1, 3, 3, 3, 5, 15, 1, 1, 1, 3, 1, 1, 3, 5, 3, 3, 3, 9, 5, 5, 15, 17, 3, 3, 3, 9, 3, 3, 9, 15, 3, 3, 3, 9, 9, 9, 15, 45

Offset: 0

Antti Karttunen, Sep 22 2014

nonn

The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).

This sequence is obtained by applying Run Length Transform to the right-shifted version of the sequence A001317: 1, 3, 5, 15, 17, 51, 85, 255, 257, ...

			115 is '1110011' in binary. The run lengths of 1-runs are 2 and 3, thus a(115) = A001317(2-1) * A001317(3-1) = 3*5 = 15.
From _Omar E. Pol_, Feb 15 2015: (Start)
Written as an irregular triangle in which row lengths are the terms of A011782:
1;
1;
1,3;
1,1,3,5;
1,1,1,3,3,3,5,15;
1,1,1,3,1,1,3,5,3,3,3,9,5,5,15,17;
1,1,1,3,1,1,3,5,1,1,1,3,3,3,5,15,3,3,3,9,3,3,9,15,5,5,5,15,15,15,17,51;
...
Right border gives 1 together with A001317.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A003714 (gives the positions of ones).
Cf. A001317, A051179.
A001316 is obtained when the same transformation is applied to A000079, the powers of two.
Run Length Transforms of other sequences: A071053, A227349, A246588, A246595, A246596, A246660, A246661, A246674, A246685.

a1317[n_] := FromDigits[ Table[ Mod[Binomial[n-1, k], 2], {k, 0, n-1}], 2];
Table[ Times @@ (a1317[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 100}] (* Jean-François Alcover, Jul 11 2017 *)

# uses RLT function from A278159
def A247282(n): return RLT(n,lambda m: int(''.join(str(int(not(~(m-1)&k))) for k in range(m)),2)) # Chai Wah Wu, Feb 04 2022

For all n >= 0, a(A051179(n)) = A246674(A051179(n)) = A051179(n).

A253065 Number of odd terms in f^n, where f = 1+x+x^2+x^2*y+x^2/y.

Original entry on oeis.org

1, 5, 5, 17, 5, 25, 17, 65, 5, 25, 25, 85, 17, 85, 65, 229, 5, 25, 25, 85, 25, 125, 85, 325, 17, 85, 85, 289, 65, 325, 229, 813, 5, 25, 25, 85, 25, 125, 85, 325, 25, 125, 125, 425, 85, 425, 325, 1145, 17, 85, 85, 289, 85, 425, 289, 1105, 65, 325, 325, 1105, 229, 1145, 813, 2945, 5, 25, 25, 85

Offset: 0

N. J. A. Sloane, Jan 26 2015

nonn

This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation.

This is the odd-rule cellular automaton defined by OddRule 171 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).

			Here is the neighborhood f:
[0, 0, X]
[X, X, X]
[0, 0, X]
which contains a(1) = 5 ON cells.

Alois P. Heinz, Table of n, a(n) for n = 0..8191
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata

Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035, A253064, A253066.

Cf. A253067.

C:=f->subs({x=1, y=1}, f);
# Find number of ON cells in CA for generations 0 thru M defined by rule
# that cell is ON iff number of ON cells in nbd at time n-1 was odd
# where nbd is defined by a polynomial or Laurent series f(x, y).
OddCA:=proc(f, M) global C; local n, a, i, f2, p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1;
for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i], i=1..nops(a))]);
end;
f:=1+x+x^2+x^2*y+x^2/y;
OddCA(f, 130);

(* f = A253067 *) f[0]=1; f[1]=5; f[2]=17; f[3]=65; f[4]=229; f[5]=813; f[n_] := f[n] = 8 f[n-5] + 6 f[n-4] + 13 f[n-3] + 5 f[n-2] + f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[s = Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 67}] (* Jean-François Alcover, Jul 12 2017 *)

This is the Run Length Transform of A253067.

A253066 Number of odd terms in f^n, where f = 1/x+1+x+1/y+y/x+x*y.

Original entry on oeis.org

1, 6, 6, 28, 6, 36, 28, 112, 6, 36, 36, 168, 28, 168, 112, 456, 6, 36, 36, 168, 36, 216, 168, 672, 28, 168, 168, 784, 112, 672, 456, 1816, 6, 36, 36, 168, 36, 216, 168, 672, 36, 216, 216, 1008, 168, 1008, 672, 2736, 28, 168, 168, 784, 168, 1008, 784, 3136, 112, 672, 672, 3136, 456, 2736, 1816, 7288

Offset: 0

N. J. A. Sloane, Jan 29 2015

nonn

This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation.

This is the odd-rule cellular automaton defined by OddRule 275 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).

			Here is the neighborhood f:
[X, 0, X]
[X, X, X]
[0, X, 0]
which contains a(1) = 6 ON cells.

Alois P. Heinz, Table of n, a(n) for n = 0..8191
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata

Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035, A253064, A253065.

Cf. A253068.

C:=f->subs({x=1, y=1}, f);
# Find number of ON cells in CA for generations 0 thru M defined by rule
# that cell is ON iff number of ON cells in nbd at time n-1 was odd
# where nbd is defined by a polynomial or Laurent series f(x, y).
OddCA:=proc(f, M) global C; local n, a, i, f2, p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1;
for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i], i=1..nops(a))]);
end;
f:=1/x+1+x+1/y+y/x+x*y;
OddCA(f, 130);

(* f = A253068 *) f[0] = 1; f[n_] := ((-2)^n + 4^(n+2)-8)/9; Table[Times @@ (f[Length[#]]&) /@ Select[s = Split[IntegerDigits[n, 2]], #[[1]] == 1 &], {n, 0, 63}] (* Jean-François Alcover, Jul 12 2017 *)

This is the Run Length Transform of A253068.

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

Original entry on oeis.org

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, 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, 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

Offset: 0

Steven Finch, Jan 25 2008

nonn

			From _Omar E. Pol_, Mar 01 2015: (Start)
Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
3;
3,9;
3,7,9,17;
3,9,7,21,9,17,17,33;
3,9,9,27,7,17,21,43,9,27,17,51,17,35,33,67;
3,9,9,27,9,21,27,51,7,21,17,51,21,41,43,83,9,27,27,81,17,43,51,113,17,51,35,105,33,67,67,137;
Thanks to _Michel Marcus_ we can see the first few terms of the next four rows as shown below:
3,9,9,27,9,21,27,51,9,27,21,63,27,51,51,99,7,21,...
3,9,9,27,9,21,27,51,9,27,21,63,27,51,51,99,9,27,27,...
3,9,9,27,9,21,27,51,9,27,21,63,27,51,51,99,9,27,27,81,...
3,9,9,27,9,21,27,51,9,27,21,63,27,51,51,99,9,27,27,81,21,...
...
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).
(End)

S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle (arXiv:0802.2654)

Cf. A071053.

Table[PolynomialMod[(1+x+x^4)^n,2]/.x->1,{n,0,80}]
Table[Count[CoefficientList[Expand[(1+x+x^4)^n],x],?OddQ],{n,0,80}] (* _Harvey P. Dale, Apr 15 2012 *)

a(n) = {my(pol = (xx^4 + xx + 1)*Mod(1,2)); subst(lift(pol^n), xx, 1);} \\ Michel Marcus, Mar 01 2015

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

First Mathematica program corrected by Harvey P. Dale, Apr 15 2012

A246037 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+y).

Original entry on oeis.org

1, 6, 6, 20, 6, 36, 20, 88, 6, 36, 36, 120, 20, 120, 88, 336, 6, 36, 36, 120, 36, 216, 120, 528, 20, 120, 120, 400, 88, 528, 336, 1376, 6, 36, 36, 120, 36, 216, 120, 528, 36, 216, 216, 720, 120, 720, 528, 2016, 20, 120, 120, 400, 120, 720, 400, 1760, 88, 528, 528, 1760, 336, 2016, 1376, 5440

Offset: 0

N. J. A. Sloane, Aug 21 2014

nonn

This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation.
This is the odd-rule cellular automaton defined by OddRule 077 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
Run Length Transform of A246036.
The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).

			Here is the neighborhood:
[X, X, X]
[0, 0, 0]
[X, X, X]
which contains a(1) = 6 ON cells.

Alois P. Heinz, Table of n, a(n) for n = 0..8192
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata

Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035.

Cf. A246036.

C:=f->subs({x=1, y=1}, f);
# Find number of ON cells in CA for generations 0 thru M defined by rule
# that cell is ON iff number of ON cells in nbd at time n-1 was odd
# where nbd is defined by a polynomial or Laurent series f(x, y).
OddCA:=proc(f, M) global C; local n, a, i, f2, p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1;
for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i], i=1..nops(a))]);
end;
f:=(1/x+1+x)*(1/y+y);
OddCA(f, 70);

(* f = A246036 *) f[0] = 1; f[n_] := (4^(n+1)-(-2)^n)/3; Table[Times @@ (f[Length[#]]&) /@ Select[s = Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* Jean-François Alcover, Jul 12 2017 *)

A255485 Number of odd terms in expansion of (1 + x + x^2 + x^4)^n.

Original entry on oeis.org

1, 4, 4, 8, 4, 12, 8, 14, 4, 16, 12, 24, 8, 24, 14, 30, 4, 16, 16, 32, 12, 36, 24, 44, 8, 32, 24, 48, 14, 46, 30, 60, 4, 16, 16, 32, 16, 48, 32, 56, 12, 48, 36, 72, 24, 76, 44, 92, 8, 32, 32, 64, 24, 72, 48, 88, 14, 56, 46, 92, 30, 96, 60, 118, 4, 16, 16, 32, 16, 48, 32, 56, 16, 64, 48, 96

Offset: 0

N. J. A. Sloane, Feb 28 2015

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A001316, A071053, A134660, A134661, A134662, A255486.

r1:=proc(f) local g,n; g:=n->nops(expand(f^n) mod 2); [seq(g(n),n=0..90)]; end;
r1(1+x+x^2+x^4);
# Alternative:
P:= 1:
for n from 0 to 100 do
  A[n]:= nops(P);
  P:= expand(P*(1+x+x^2+x^4)) mod 2;
od:
seq(A[i],i=0..100); # Robert Israel, Jan 07 2018

a[n_] := Count[(List @@ Expand[(1+x+x^2+x^4)^n]) /. x -> 1, _?OddQ]; a[0] = 1;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 06 2023 *)

a(n) = {my(pol=(1+x+x^2+x^4)*Mod(1,2)); subst(lift(pol^n), x, 1);} \\ Michel Marcus, Mar 01 2015

From Robert Israel, Jan 07 2018: (Start)
a(2*n) = a(n).
a(8*n+1) = 4*a(n). (End)

A255486 Number of odd terms in expansion of (1+x+x^3+x^4)^n.

Original entry on oeis.org

1, 4, 4, 10, 4, 12, 10, 18, 4, 16, 12, 28, 10, 28, 18, 38, 4, 16, 16, 40, 12, 40, 28, 52, 10, 40, 28, 64, 18, 52, 38, 74, 4, 16, 16, 40, 16, 48, 40, 72, 12, 48, 40, 96, 28, 88, 52, 108, 10, 40, 40, 100, 28, 96, 64, 120, 18, 72, 52, 120, 38

Offset: 0

N. J. A. Sloane, Mar 01 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..8191

Cf. A001316, A071053, A134660, A134661, A134662, A255485, A247649.

r1:=proc(f) local g,n; g:=n->nops(expand(f^n) mod 2); [seq(g(n),n=0..90)]; end;
r1(1+x+x^2+x^3);

a[n_] := Count[(List @@ Expand[(1+x+x^3+x^4)^n]) /. x -> 1, _?OddQ]; a[0] = 1;
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Apr 04 2017 *)

a(n) = {my(pol=(1+x+x^3+x^4)*Mod(1,2)); subst(lift(pol^n), x, 1);} \\ Michel Marcus, Mar 01 2015

cp's OEIS Frontend

A134660 Number of odd coefficients in (1 + x + x^2 + x^3)^n.

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A134661 Number of odd coefficients in (1 + x + x^3)^n.

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A245565 a(n) = Product_{i in row n of A245562} Pell(i+1).

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A247282 Run Length Transform of A001317.

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A253065 Number of odd terms in f^n, where f = 1+x+x^2+x^2*y+x^2/y.

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A253066 Number of odd terms in f^n, where f = 1/x+1+x+1/y+y/x+x*y.

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

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