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

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

0,3

Comments

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

Old name was "A divide and conquer sequence".

Links

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.

Formula

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

Maple

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

Mathematica

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

Prog

(PARI)
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
(PARI) a(n) = fromdigits([!b |b<-binary(n+2)], 4); \\ Kevin Ryde, Jul 15 2024
(Python)
def A115637(n): return int(bin((~(n+2))^(-1<<(n+2).bit_length()))[2:], 4) # Chai Wah Wu, Jul 17 2024

Crossrefs

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

Sequence in context: A133842 A199453 A245699 * A124602 A320060 A132707

Adjacent sequences: A115634 A115635 A115636 * A115638 A115639 A115640

Keyword

easy,nonn,look

Author

Paul Barry, Jan 27 2006

Extensions

New name from Kevin Ryde, Jul 15 2024

Status

approved