A117898 -id:A117898 - cp's OEIS frontend

A117901 Inverse of number triangle A117898.

1, -1, 1, 0, -2, 1, 0, 3, -2, 1, 0, 0, 0, -1, 1, 0, -6, 3, 0, -2, 1, 0, 12, -6, 0, 3, -2, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, -24, 12, 0, -6, 3, 0, -2, 1, 0, 48, -24, 0, 12, -6, 0, 3, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, -96, 48, 0, -24, 12, 0, -6, 3, 0, -2, 1, 0, 192, -96, 0, 48, -24, 0, 12, -6, 0, 3, -2, 1

Offset: 0

Paul Barry, Apr 01 2006

Row sums are A117902. Diagonal sums are A117903.

			Triangle begins
1,
-1, 1,
0, -2, 1,
0, 3, -2, 1,
0, 0, 0, -1, 1,
0, -6, 3, 0, -2, 1,
0, 12, -6, 0, 3, -2, 1,
0, 0, 0, 0, 0, 0, -1, 1,
0, -24, 12, 0, -6, 3, 0, -2, 1,
0, 48, -24, 0, 12, -6, 0, 3, -2, 1

G.f.:(2x^6*y^2+x^5*y(2-y)+4x^4(1-y)-x^3(2y^2-3y+4)+x^2*y(y-2)+x(y-1)+1)/((1-4x^3)(1-x^3*y^3))

A117904 Number triangle [k<=n]*0^abs(L(C(n,2)/3) - L(C(k,2)/3)) where L(j/p) is the Legendre symbol of j and p.

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1

Offset: 0

Paul Barry, Apr 01 2006

Row sums are A009947(n+2).
Diagonal sums are A117905.
Inverse is A117906.
Equals A117898 mod 2.

			Triangle begins
  1;
  1, 1;
  0, 0, 1;
  1, 1, 0, 1;
  1, 1, 0, 1, 1;
  0, 0, 1, 0, 0, 1;
  1, 1, 0, 1, 1, 0, 1;
  1, 1, 0, 1, 1, 0, 1, 1;
  0, 0, 1, 0, 0, 1, 0, 0, 1;
  1, 1, 0, 1, 1, 0, 1, 1, 0, 1;
  1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A009947, A117898, A117905, A117906.

T[n_, k_]:= If[Abs[JacobiSymbol[Binomial[n, 2], 3] - JacobiSymbol[Binomial[k, 2], 3]]==0, 1, 0];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 20 2021 *)

def A117904(n,k): return 1 if abs(jacobi_symbol(binomial(n,2), 3) - jacobi_symbol(binomial(k,2), 3))==0 else 0
flatten([[A117904(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Oct 20 2021

G.f.: (1 +x*(1+y) +x^2*y^2 +x^3*y)/((1-x^3)*(1-x^3*y^3)).

T(n, k) = [k<=n] * 2^abs(L(C(n,2)/3) - L(C(k,2)/3)) mod 2.

A117899 Expansion of (1 + 2x + 5x^2 + 3x^3 + 2x^4)/(1-x^3)^2.

Original entry on oeis.org

1, 2, 5, 5, 6, 10, 9, 10, 15, 13, 14, 20, 17, 18, 25, 21, 22, 30, 25, 26, 35, 29, 30, 40, 33, 34, 45, 37, 38, 50, 41, 42, 55, 45, 46, 60, 49, 50, 65, 53, 54, 70, 57, 58, 75, 61, 62, 80, 65, 66, 85, 69, 70, 90, 73, 74, 95, 77, 78, 100, 81, 82, 105, 85, 86, 110, 89, 90, 115, 93

Offset: 0

Paul Barry, Apr 01 2006

Row sums of A117898.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A117898.

I:=[1,2,5,5,6,10]; [n le 6 select I[n] else 2*Self(n-3) - Self(n-6): n in [1..91]]; // G. C. Greubel, Oct 01 2021

CoefficientList[Series[(1+2x+5x^2+3x^3+2x^4)/(1-x^3)^2,{x,0,90}],x] (* or *) LinearRecurrence[{0,0,2,0,0,-1},{1,2,5,5,6,10},90] (* Harvey P. Dale, Dec 18 2013 *)

def A117899_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+2*x+5*x^2+3*x^3+2*x^4)/(1-x^3)^2 ).list()
A117899_list(90) # G. C. Greubel, Oct 01 2021

a(n) = 2*a(n-3) - a(n-6).

a(n) = Sum_{k=0..n} 2^abs(L(C(n,2)/3) - L(C(k,2)/3)), L(j/p) the Legendre symbol of j and p.

A117900 Expansion of (1 + 2x + 4x^2 + 4x^3 + 2x^4)/((1+x)*(1-x^3)^2).

Original entry on oeis.org

1, 1, 3, 3, 3, 5, 6, 4, 8, 8, 6, 10, 11, 7, 13, 13, 9, 15, 16, 10, 18, 18, 12, 20, 21, 13, 23, 23, 15, 25, 26, 16, 28, 28, 18, 30, 31, 19, 33, 33, 21, 35, 36, 22, 38, 38, 24, 40, 41, 25, 43, 43, 27, 45, 46, 28, 48, 48, 30, 50, 51, 31, 53, 53, 33, 55, 56, 34, 58, 58, 36

Offset: 0

Paul Barry, Apr 01 2006

Diagonal sums of A117898.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,0,2,2,0,-1,-1).

Cf. A117898.

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+2*x+4*x^2+4*x^3+2*x^4)/((1+x)*(1-x^3)^2) )); // G. C. Greubel, Oct 01 2021

CoefficientList[Series[(1+2x+4x^2+4x^3+2x^4)/((1-x^3)(1+x-x^3-x^4)),{x,0,80}],x] (* or *) LinearRecurrence[{-1,0,2,2,0,-1,-1},{1,1,3,3,3,5,6},80] (* Harvey P. Dale, Mar 06 2018 *)

def A117899_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+2*x+4*x^2+4*x^3+2*x^4)/((1+x)*(1-x^3)^2) ).list()
A117899_list(80) # G. C. Greubel, Oct 01 2021

a(n) = -a(n-1) + 2*a(n-3) + 2*a(n-4) - a(n-6) - a(n-7).

a(n) = Sum_{k=0..floor(n/2)} 2^abs(L(C(n-k,2)/3) - L(C(k,2)/3)), L(j/p) the Legendre symbol of j and p.

cp's OEIS Frontend

A117901 Inverse of number triangle A117898.

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A117904 Number triangle [k<=n]*0^abs(L(C(n,2)/3) - L(C(k,2)/3)) where L(j/p) is the Legendre symbol of j and p.

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A117899 Expansion of (1 + 2*x + 5*x^2 + 3*x^3 + 2*x^4)/(1-x^3)^2.

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Magma

Mathematica

Sage

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A117900 Expansion of (1 + 2*x + 4*x^2 + 4*x^3 + 2*x^4)/((1+x)*(1-x^3)^2).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A117899 Expansion of (1 + 2x + 5x^2 + 3x^3 + 2x^4)/(1-x^3)^2.

A117900 Expansion of (1 + 2x + 4x^2 + 4x^3 + 2x^4)/((1+x)*(1-x^3)^2).