A115633 -id:A115633 - cp's OEIS frontend

A115636 A divide-and-conquer number triangle.

1, 1, -1, 4, 0, 1, 4, 0, 1, -1, 4, -4, 0, 0, 1, 4, -4, 0, 0, 1, -1, 16, 0, 4, 0, 0, 0, 1, 16, 0, 4, 0, 0, 0, 1, -1, 16, 0, 4, -4, 0, 0, 0, 0, 1, 16, 0, 4, -4, 0, 0, 0, 0, 1, -1, 16, -16, 0, 0, 4, 0, 0, 0, 0, 0, 1, 16, -16, 0, 0, 4, 0, 0, 0, 0, 0, 1, -1, 16, -16, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, 1, 16, -16, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, 1, -1

Offset: 0

Paul Barry, Jan 27 2006

			Triangle begins
   1;
   1,  -1;
   4,   0,  1;
   4,   0,  1, -1;
   4,  -4,  0,  0,  1;
   4,  -4,  0,  0,  1, -1;
  16,   0,  4,  0,  0,  0,  1;
  16,   0,  4,  0,  0,  0,  1, -1;
  16,   0,  4, -4,  0,  0,  0,  0,  1;
  16,   0,  4, -4,  0,  0,  0,  0,  1, -1;
  16, -16,  0,  0,  4,  0,  0,  0,  0,  0,  1;
  16, -16,  0,  0,  4,  0,  0,  0,  0,  0,  1, -1;
  16, -16,  0,  0,  4, -4,  0,  0,  0,  0,  0,  0,  1;
  16, -16,  0,  0,  4, -4,  0,  0,  0,  0,  0,  0,  1, -1;
  64,   0, 16,  0,  0,  0,  4,  0,  0,  0,  0,  0,  0,  0,  1;
  64,   0, 16,  0,  0,  0,  4,  0,  0,  0,  0,  0,  0,  0,  1, -1;
  64,   0, 16,  0,  0,  0,  4, -4,  0,  0,  0,  0,  0,  0,  0,  0,  1;

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

Cf. A115633 (inverse), A115637 (row sums), A115639 (first column).

A115633[n_, k_]:= If[k==n, (-1)^n, If[k==n-1, Mod[n,2], If[n==2*k+2, -4, 0]]];
T[n_, k_]:= T[n, k]= (-1)^k*If[k==n, 1, -Sum[T[n, j]*A115633[j, k], {j,k+1,n}] ];
Table[T[n, k], {n,0,18}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 24 2021 *)

@CachedFunction
def A115633(n, k):
    if (k==n): return (-1)^n
    elif (k==n-1): return n%2
    elif (n==2*k+2): return -4
    else: return 0
def A115636(n,k):
    if (k==0): return 4^(floor(log(n+2, 2)) -1)
    elif (k==n): return (-1)^n
    elif (k==n-1): return (n%2)
    else: return (-1)^(k+1)*sum( A115636(n, j)*A115633(j, k) for j in (k+1..n) )
flatten([[A115636(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Nov 24 2021

T(n, 0) = A115639(n).
Sum_{k=0..n} T(n, k) = A115637(n).
T(n, k) = (-1)^k*( 1 if k = n otherwise (-1)*Sum_{j=k+1..n} T(n, j)*A115633(j, k) ). - G. C. Greubel, Nov 24 2021

A115637 In the binary expansion of n+2, transform 0->1 and 1->0 then interpret as base 4.

Original entry on oeis.org

1, 0, 5, 4, 1, 0, 21, 20, 17, 16, 5, 4, 1, 0, 85, 84, 81, 80, 69, 68, 65, 64, 21, 20, 17, 16, 5, 4, 1, 0, 341, 340, 337, 336, 325, 324, 321, 320, 277, 276, 273, 272, 261, 260, 257, 256, 85, 84, 81, 80, 69, 68, 65, 64, 21, 20, 17, 16, 5, 4, 1, 0, 1365, 1364, 1361, 1360, 1349

Offset: 0

Paul Barry, Jan 27 2006

Row sums of number triangle A115636. Partial sums of A115638.

Old name was "A divide and conquer sequence".

Alois P. Heinz, Table of n, a(n) for n = 0..16384 (first 1025 terms from Antti Karttunen)
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003, see equation 2.4 with a(n) = a_{n+2} for case alpha=4, c=1, d=0.

Cf. A000695, A035327, A115633, A115636, A115638 (first differences), A374625.

b:= n-> 1-(n mod 2)+`if`(n<2, 0, b(iquo(n, 2))*4):
a:= n-> b(n+2):
seq(a(n), n=0..66);  # Alois P. Heinz, Jul 16 2024

A115637[n_] := FromDigits[1 - IntegerDigits[n + 2, 2], 4];
Array[A115637, 100, 0] (* Paolo Xausa, Jul 16 2024 *)

up_to = 1024;
A115633array(n, k) = (((-1)^n)*if(n==k,1, if((k+k+2)==n, -4, if((k+1)==n, -(1+(-1)^k)/2, 0))));
A115637list(up_to) = { my(mA115633=matrix(up_to,up_to,n,k,A115633array(n-1,k-1)), mA115636 = matsolve(mA115633,matid(up_to)), v = vector(up_to)); for(n=1,up_to,v[n] = vecsum(mA115636[n,])); (v); };
v115637 = A115637list(up_to+1);
A115637(n) = v115637[1+n]; \\ Antti Karttunen, Nov 02 2018

a(n) = fromdigits([!b |b<-binary(n+2)], 4); \\ Kevin Ryde, Jul 15 2024

def A115637(n): return int(bin((~(n+2))^(-1<<(n+2).bit_length()))[2:],4) # Chai Wah Wu, Jul 17 2024

G.f.: (1/(1-x))*Sum_{k>=0} 4^k*x^(2^(k+1)-2)/(1+x^(2^k)); the g.f. G(x) satisfies G(x) - 4(1+x)*x^2*G(x^2) = 1/(1-x^2).

a(n) = A000695(A035327(n+2)). - Kevin Ryde, Jul 15 2024

New name from Kevin Ryde, Jul 15 2024

A115634 Expansion of (1-4*x^2)/(1-x^2).

Original entry on oeis.org

1, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0

Offset: 0

Paul Barry, Jan 27 2006

Row sums of number triangle A115633.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010674, A115633.

[4*0^n -3*(1+(-1)^n)/2: n in [0..100]]; // G. C. Greubel, Nov 23 2021

Join[{1}, -3*Mod[Range[100] -1, 2]] (* G. C. Greubel, Nov 23 2021 *)
CoefficientList[Series[(1-4x^2)/(1-x^2),{x,0,100}],x] (* or *) LinearRecurrence[{0,1},{1,0,-3},100] (* or *) PadRight[{1},100,{-3,0}] (* Harvey P. Dale, Dec 06 2024 *)

[1]+[-3*((n-1)%2) for n in (1..100)] # G. C. Greubel, Nov 23 2021

a(n) = 4*0^n - 3*(1 + (-1)^n)/2.
a(n) = Sum_{k=0..n} A115633(n, k).
From G. C. Greubel, Nov 23 2021: (Start)
a(n) = 1 if n = 0, otherwise a(n) = -A010674(n-1).
E.g.f.: 4 - 3*cosh(x). (End)

A115635 Periodic {1,1,-5,0,1,-3,-1,0,-3,1,-1,-4}.

Original entry on oeis.org

1, 1, -5, 0, 1, -3, -1, 0, -3, 1, -1, -4, 1, 1, -5, 0, 1, -3, -1, 0, -3, 1, -1, -4, 1, 1, -5, 0, 1, -3, -1, 0, -3, 1, -1, -4, 1, 1, -5, 0, 1, -3, -1, 0, -3, 1, -1, -4, 1, 1, -5, 0, 1, -3, -1, 0, -3, 1, -1, -4, 1, 1, -5, 0, 1, -3, -1, 0, -3, 1, -1, -4

Offset: 0

Paul Barry, Jan 27 2006

Diagonal sums of number triangle A115633.

Index entries for linear recurrences with constant coefficients, signature (-1,-1,0,1,1,1).

G.f.: (1+2x-3x^2-4x^3-5x^4-4x^5)/((1-x^3)*(1+x+x^2+x^3)); a(n)=sum{k=0..floor(n/2), A115633(n-k, k)}.

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A115636 A divide-and-conquer number triangle.

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A115637 In the binary expansion of n+2, transform 0->1 and 1->0 then interpret as base 4.

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A115634 Expansion of (1-4*x^2)/(1-x^2).

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A115635 Periodic {1,1,-5,0,1,-3,-1,0,-3,1,-1,-4}.

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