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URL: https://oeis.org/A115637

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A115637
In the binary expansion of n+2, transform 0->1 and 1->0 then interpret as base 4.
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
KEYWORD
easy,nonn,look
AUTHOR
Paul Barry, Jan 27 2006
EXTENSIONS
New name from Kevin Ryde, Jul 15 2024
STATUS
approved