A115636 -id:A115636 - cp's OEIS frontend

A115639 First column of divide-and-conquer triangle A115636.

1, 1, 4, 4, 4, 4, 16, 16, 16, 16, 16, 16, 16, 16, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256

Offset: 0

Paul Barry, Jan 27 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000523, A115636.

[4^(Ilog2(n+2) -1) : n in [0..80] ]; // G. C. Greubel, Nov 23 2021

4^(Floor[Log[2, Range[0, 80] +2]] -1) (* G. C. Greubel, Nov 23 2021 *)

[4^(floor(log(n+2, 2)) -1) for n in (0..80)] # G. C. Greubel, Nov 23 2021

G.f.: (1/(1-x))*( 1/4 + (3/4)*Sum_{k>=1} 4^(k-1)*x^(2^k-2) ).

a(n) = 4^(floor(log_2(n+2)) - 1). - G. C. Greubel, Nov 23 2021

A115633 A divide and conquer-related triangle: see formula for T(n,k), n >= k >= 0.

Original entry on oeis.org

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

Offset: 0

Paul Barry, Jan 27 2006

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

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

Cf. A115634 (row sums), A115635 (diagonal sums), A115636 (inverse).

T[n_, k_]:= If[k==n, (-1)^n, If[k==n-1, (1-(-1)^n)/2, If[n==2*k+2, -4, 0]]];
Table[T[n, k], {n,0,18}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 23 2021 *)

A115633(n,k)=if(n==k, (-1)^n, bittest(n,0), k==n-1, k+1==n\2, -4) \\ M. F. Hasler, Nov 24 2021

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
flatten([[A115633(n,k) for k in (0..n)] for n in (0..18)]) # G. C. Greubel, Nov 23 2021

T(n, k) = (-1)^n if n = k; else -4 if n = 2k+2; else (n mod 2) if n = k+1; else 0.
G.f.: (1+x-x*y)/(1-x^2*y^2) - 4*x^2/(1-x^2*y).
(1, -x) + (x, x)/2 + (x, -x)/2 - 4(x^2, x^2) expressed in the notation of stretched Riordan arrays.
Column k has g.f.: (-x)^k + (x*(-x)^k + x^(k+1))/2 - 4*x^(2*k+2).
Sum_{k=0..n} T(n, k) = A115634(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A115635(n).

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

cp's OEIS Frontend

A115639 First column of divide-and-conquer triangle A115636.

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A115633 A divide and conquer-related triangle: see formula for T(n,k), n >= k >= 0.

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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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