A106400 -id:A106400 - cp's OEIS frontend

A132971 a(2n) = a(n), a(4n+1) = -a(n), a(4*n+3) = 0, with a(0) = 1.

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

Offset: 0

Michael Somos, Sep 17 2007, Sep 19 2007

sign

If binary(n) has adjacent 1 bits then a(n) = 0 else a(n) = (-1)^A000120(n).

Fibbinary numbers (A003714) gives the numbers n for which a(n) = A106400(n). - Antti Karttunen, May 30 2017

			G.f. = 1 - x - x^2 - x^4 + x^5 - x^8 + x^9 + x^10 - x^16 + x^17 + x^18 + ...

Antti Karttunen, Table of n, a(n) for n = 0..10922
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238, DOI: 10.1007/978-3-642-23283-1_15.
Index entries for sequences related to binary expansion of n

Cf. A000120, A000621, A003714, A005252, A005940, A008683, A024490, A027935, A106400.

Cf. A085357 (gives the absolute values: -1 -> 1), A286576 (when reduced modulo 3: -1 -> 2).

m = 100; A[_] = 1;
Do[A[x_] = A[x^2] - x A[x^4] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 16 2019 *)

{a(n) = if( n<1, n==0, if( n%2, if( n%4 > 1, 0, -a((n-1)/4) ), a(n/2) ) )};

{a(n) = my(A, m); if( n<0, 0, m = 1; A = 1 + O(x); while( m<=n, m *= 2; A = subst(A, x, x^2) - x * subst(A, x, x^4) ); polcoeff(A, n)) };

from sympy import mobius, prime, log
import math
def A(n): return n - 2**int(math.floor(log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a(n): return mobius(b(n)) # Indranil Ghosh, May 30 2017

(define (A132971 n) (cond ((zero? n) 1) ((even? n) (A132971 (/ n 2))) ((= 1 (modulo n 4)) (- (A132971 (/ (- n 1) 4)))) (else 0))) ;; Antti Karttunen, May 30 2017

A024490(n) = number of solutions to 2^n <= k < 2^(n+1) and a(k) = 1.
A005252(n) = number of solutions to 2^n <= k < 2^(n+1) and a(k) = -1.
A027935(n-1) = number of solutions to 2^n <= k < 2^(n+1) and a(k) = 0.
G.f. A(x) satisfies A(x) = A(x^2) - x * A(x^4).
G.f. B(x) of A000621 satisfies B(x) = x * A(x^2) / A(x).
a(n) = A008683(A005940(1+n)). [Analogous to Moebius mu] - Antti Karttunen, May 30 2017

A299759 Triangle read by rows in which row n lists in order all FDH numbers of strict integer partitions of n.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 7, 10, 12, 9, 14, 15, 24, 11, 18, 20, 21, 30, 13, 22, 27, 28, 40, 42, 16, 26, 33, 35, 36, 54, 56, 60, 17, 32, 39, 44, 45, 66, 70, 72, 84, 120, 19, 34, 48, 52, 55, 63, 78, 88, 90, 105, 108, 168, 23, 38, 51, 64, 65, 77, 96, 104, 110, 126

Offset: 1

Gus Wiseman, Feb 18 2018

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. This determines a unique strict integer partition (s_k...s_1) whose FDH number is then defined to be n.

This sequence is a permutation of the positive integers.

			Triangle of strict partitions begins:
                  0
                 (1)
                 (2)
               (3) (21)
               (4) (31)
             (5) (41) (32)
          (6) (51) (42) (321)
        (7) (61) (43) (52) (421)
     (8) (71) (62) (53) (431) (521)
(9) (81) (72) (54) (63) (621) (531) (432).

Cf. A050376, A056239, A064547, A106400, A122111, A213925, A215366, A246867, A279065, A279614, A299755, A299757, A299758.

Row lengths give A000009.

nn=25;
FDprimeList=Select[Range[nn],MatchQ[FactorInteger[#],{{?PrimeQ,?(MatchQ[FactorInteger[2#],{{2,_}}]&)}}]&];
Table[Sort[Times@@FDprimeList[[#]]&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,Length[FDprimeList]}]

A005599 Running sum of every third term in the {+1,-1}-version of Thue-Morse sequence A010060.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 6, 7, 8, 9, 10, 11, 12, 11, 12, 13, 14, 15, 16, 17, 18, 19, 18, 19, 20, 21, 20, 19, 18, 19, 18, 19, 20, 21, 22, 23, 24, 25, 24, 25, 26, 27, 28, 29, 30, 29, 30, 31, 32, 33, 34, 35, 36, 35, 36, 35, 34, 33, 34, 35, 36, 35, 36, 37, 38, 39, 40, 41, 42, 43, 42, 43, 44, 45, 46, 47, 48, 47, 48, 49, 50

Offset: 0

M. R. Schroeder

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 98.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
J. Coquet, A summation formula related to the binary digits, Invent. Math. 73 (1983) 107-115.
P. Flajolet et al., Mellin Transforms And Asymptotics: Digital Sums, Theoret. Computer Sci. 23 (1994), 291-314.
P. J. Grabner and H.-K. Hwang, Digital sums and divide-and-conquer recurrences: Fourier expansions and absolute convergence, Constructive Approximation, Jan. 2005, Volume 21, Issue 2, pp 149-179.
Roswitha Hofer, Coquet-type formulas for the rarefied weighted Thue-Morse sequence Discrete Mathematics 311.16 (2011): 1724-1734.
D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc., 21 (1969), 719-721.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
M. R. Schroeder & N. J. A. Sloane, Correspondence, 1991

See A000120 for "wt" (the binary weight of n).

Cf. A010060, A106400.

a005599 n = a005599_list !! n
a005599_list = scanl (+) 0 $ f a106400_list
   where f (x:::xs) = x : f xs
-- Reinhard Zumkeller, May 26 2013

A000120 := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: wt := A000120;
f:=n->add( (-1)^wt(3*k),k=0..n-1);
[seq(f(n),n=0..50)]; # N. J. A. Sloane, Jul 22 2012
A005599 := proc(n)
        add( A106400(3*i),i=0..n-1) ;
end proc: # R. J. Mathar, Jul 22 2012

wt[n_] := DigitCount[n, 2, 1]; a[n_] := Sum[(-1)^wt[3*k], {k, 0, n-1}]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 03 2014, after N. J. A. Sloane *)

a(n) = sum(k=0, n-1, (-1)^hammingweight(3*k)); \\ Michel Marcus, Jul 03 2017

a(n) = Sum( (-1)^wt(3*k),k=0..n-1). See Allouche-Shallit for asymptotics. - From N. J. A. Sloane, Jul 22 2012

The generating function -(2*z^4+z^3+z+1)*(z^3-z^2-1)/(z^6+z^5+z^4+z^3+z^2+z+1)/(z-1)^2 proposed in the Plouffe thesis is wrong.

A343029 Number of 1-bits in the binary expansion of n which have an even number of 0-bits at less significant bit positions.

Original entry on oeis.org

0, 1, 0, 2, 1, 1, 0, 3, 0, 2, 1, 2, 2, 1, 0, 4, 1, 1, 0, 3, 1, 2, 1, 3, 0, 3, 2, 2, 3, 1, 0, 5, 0, 2, 1, 2, 2, 1, 0, 4, 1, 2, 1, 3, 2, 2, 1, 4, 2, 1, 0, 4, 1, 3, 2, 3, 0, 4, 3, 2, 4, 1, 0, 6, 1, 1, 0, 3, 1, 2, 1, 3, 0, 3, 2, 2, 3, 1, 0, 5, 1, 2, 1, 3, 2, 2, 1

Offset: 0

Kevin Ryde, Apr 03 2021

Each term of Per Nørgård's infinity sequence (A004718) is a sum of +1 or -1 for each 1-bit of n according as that bit has an even or odd number of 0-bits below it.  The present sequence counts the "+1" bits and A343030 counts the "-1" bits so that A004718(n) = a(n) - A343030(n).
a(n) and A343030(n) can be iterated together by pair [a(n-1)-t, A343030(n-1)+1] -> [a(n), A343030(n)] if t odd or [A343030(n), a(n)] if t even, where t = A007814(n) is the 2-adic valuation of n.
In the generating function sum below, k is a bit position (0 for the least significant bit).  1/2 of each sum term gives 1*x^n at those n where bit k of n is 1 and has an even number of 0 bits below.  The product part is like the +-1 Thue-Morse sequence A106400, but only the k lowest bits, and each product term negated so parity of 0-bits.  These +-1 are turned into 2 or 0 and shifted and repeated in blocks which are where bit k of n is 1.

			n = 860 = binary 1101011100
                 ^^   ^^^    a(n) = 5

Kevin Ryde, Table of n, a(n) for n = 0..8192

Cf. A343030, A004718, A000225 (indices of new highs).

Cf. A000120, A007814, A106400.

a(n) = my(t=1,ret=0); for(i=0,if(n,logint(n,2)), if(bittest(n,i), ret+=t, t=!t)); ret;

def a(n):
  b = bin(n)[2:]
  return sum(bi=='1' and b[i:].count('0')%2==0 for i, bi in enumerate(b))
print([a(n) for n in range(87)]) # Michael S. Branicky, Apr 03 2021

a(n) = A004718(n) + A343030(n).
a(n) = A000120(n) - A343030(n), where A000120 is the number of 1-bits in n (binary weight).
a(2*n) = A000120(n) - a(n).
a(2*n+1) = a(n) + 1.
G.f. satisfies g(x) = (x-1)*g(x^2) + A000120(x^2) + x/(1-x^2).
G.f.: (1/2) * Sum_{k>=0} x^(2^k)*( (1-x^(2^k))/(1-x) + Prod_{j=0..k-1} x^(2^j)-1 )/( 1-x^(2*2^k) ).
a(2^n - 1) = n. - Michael S. Branicky, Apr 03 2021

A106407 Expansion of x((1-x)(1-x^2)(1-x^4)(1-x^8)...)^2.

Original entry on oeis.org

1, -2, -1, 4, -3, 2, 3, -8, 1, 6, -1, -4, 5, -6, -5, 16, -7, -2, 7, -12, 5, 2, -5, 8, 1, -10, -1, 12, -11, 10, 11, -32, 9, 14, -9, 4, 5, -14, -5, 24, -7, -10, 7, -4, -3, 10, 3, -16, 9, -2, -9, 20, -11, 2, 11, -24, 1, 22, -1, -20, 21, -22, -21, 64, -23, -18, 23, -28, 5, 18, -5, -8, 9, -10, -9, 28, -19, 10, 19, -48, 17, 14, -17, 20

Offset: 1

Michael Somos, May 02 2005

The Stern polynomial B(n,x) evaluated at x=-2. See A125184. - T. D. Noe, Feb 28 2011

Self-convolution of the signed Thue-Morse sequence A106400. - Vladimir Reshetnikov, Apr 13 2018

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Maciej Gawron, Piotr Miska, Maciej Ulas, Arithmetic properties of coefficients of power series expansion of Prod_{n>=0} (1-x^(2^n))^t, arXiv:1703.01955 [math.NT], 2017.
Eric Rowland, A matrix generalization of a theorem of Fine, arXiv:1704.05872 [math.NT], 2017.
Eric Rowland, A matrix generalization of a theorem of Fine, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A18.

Table[Sum[(-1)^(DigitCount[k, 2, 1] + DigitCount[n-k-1, 2, 1]), {k, 0, n-1}], {n, 1, 80}] (* Vladimir Reshetnikov, Apr 13 2018 *)

{a(n)=local(A,m); if(n<1,0,n--; A=1+x*O(x^n); m=1; while(m<=n, A*=(1-x^m); m*=2;); polcoeff(A^2,n))}

Euler transform of sequence b(n) where b(2^k)=-2 and zero otherwise.
G.f.: A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=u1^3*u6 + 6*u1^2*u2*u6 + 9*u1*u2^2*u6 - u3*u2^3.
G.f.: x(Product_{k>=0} (1-x^(2^k)))^2.
G.f.: A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=-v^3+4uvw+u^2w.

A292118 G.f.: 1 + 2Sum_{k >= 1} (-1)^kq^A003159(k).

Original entry on oeis.org

1, -2, 0, 2, -2, 2, 0, -2, 0, 2, 0, -2, 2, -2, 0, 2, -2, 2, 0, -2, 2, -2, 0, 2, 0, -2, 0, 2, -2, 2, 0, -2, 0, 2, 0, -2, 2, -2, 0, 2, 0, -2, 0, 2, -2, 2, 0, -2, 2, -2, 0, 2, -2, 2, 0, -2, 0, 2, 0, -2, 2, -2, 0, 2, -2, 2, 0, -2, 2, -2, 0, 2, 0, -2, 0, 2, -2

Offset: 0

N. J. A. Sloane, Sep 09 2017

sign

Sean A. Irvine, Table of n, a(n) for n = 0..10000
George E. Andrews and David Newman, Binary Representations and Theta Functions, 2017.

Cf. A003159, A292118.

First differences of A106400.

Join[{1},Differences[(-1)^ThueMorse[Range[0,100]]]] (* Paolo Xausa, Dec 18 2023 *)

Andrews-Newman (2017) give many properties of this series.

A357761 a(n) = A227872(n) - A356018(n).

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 2, 4, -1, 0, 2, 0, 2, 4, -2, 5, 0, -2, 2, 0, 2, 4, 0, 0, 1, 4, -2, 6, 0, -4, 2, 6, 0, 0, 2, -3, 2, 4, 0, 0, 2, 4, 0, 6, -4, 0, 2, 0, 3, 2, -2, 6, 0, -4, 2, 8, 0, 0, 2, -6, 2, 4, 0, 7, 0, 0, 2, 0, 0, 4, 0, -4, 2, 4, -2, 6, 2, 0, 2, 0, -1, 4, 0

Offset: 1

Amiram Eldar, Oct 12 2022

The excess of the number of odious (A000069) divisors of n over the number of evil (A001969) divisors of n.

Every integer occurs in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000069, A000290 (positions of odd terms), A001969, A027697, A027699, A106400, A227872, A230851 (positions of 0's), A356018, A357762.

Similar sequences: A046660, A048272.

a[n_] := -DivisorSum[n, (-1)^DigitCount[#, 2, 1] &]; Array[a, 100]

a(n) = -sumdiv(n, d, (-1)^hammingweight(d));

a(n) = -Sum_{d|n} A106400(d).
a(n) = A000005(n) - 2*A356018(n).
a(n) = 2*A227872(n) - A000005(n).
a(n) = 0 iff n is in A230851.
a(n) == 1 (mod 2) iff n is a square (A000290).
a(2^n) = n + 1.
a(p*2^n) = 0 when p is an evil prime (A027699).
a(p^2*2^n) = n + 1 when p is an evil prime (A027699) and p^2 is odious, and when p is an odd odious prime (A027697) and p^2 is evil.
a(p^2*2^n) = -(n+1) when p is an evil prime and p^2 is also evil.
a(p^2*2^n) = 3*(n+1) when p is an odd odious prime and p^2 is also odious.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = -Sum_{k>=1} A106400(k)/k = 1.196283264... (A357762).

A373308 Expansion of Product_{n>=0} (1 - x^(2^n))^3.

Original entry on oeis.org

1, -3, 0, 8, -9, 3, 8, -24, 15, 19, -24, 0, 17, -27, 0, 64, -57, -21, 64, -72, 33, 53, -72, 24, 17, -51, 24, 64, -81, 27, 64, -192, 135, 107, -192, 120, 1, -171, 120, 152, -183, -27, 152, -192, 87, 163, -192, 0, 89, -75, 0, 136, -129, -21, 136, -216, 111, 179, -216, 0, 145, -219, 0, 512

Offset: 0

Paul D. Hanna, Jun 20 2024

Equals the self-convolution cube of the signed Thue-Morse sequence A106400: [1, -1, -1, 1, -1, 1, 1, -1, ...].
Conjecture: a(3*n) == A001285(n) (mod 3) for n >= 0, where A001285 is the Thue-Morse sequence: [1, 2, 2, 1, 2, 1, 1, 2, ...].
Conjecture: a(3*n+1) and a(3*n+2) are divisible by 3 for n >= 0.
Conjecture: a(n) = 0 iff n = 3*A000695(k) - 1 for k >= 1, where A000695 lists sums of distinct powers of 4.

			G.f.: A(x) = 1 - 3*x + 8*x^3 - 9*x^4 + 3*x^5 + 8*x^6 - 24*x^7 + 15*x^8 + 19*x^9 - 24*x^10 + 17*x^12 - 27*x^13 + 64*x^15 - 57*x^16 - 21*x^17 + 64*x^18 + ...
where A(x) = (1-x)^3 * (1-x^2)^3 * (1-x^4)^3 * (1-x^8)^3 * (1-x^16)^3 * ...
Notice that a(n) = 0 at n = [2, 11, 14, 47, 50, 59, 62, 191, 194, 203, 206, 239, 242, 251, 254, 767, ...] which appears to equal 3*A000695(k) - 1 for k >= 1.

Paul D. Hanna, Table of n, a(n) for n = 0..16400

Cf. A373309, A106400, A106407, A001285, A000695.

{a(n) = my(A = prod(k=0,#binary(n), (1 - x^(2^k))^3 +x*O(x^n))); polcoeff(A, n)}
for(n=0,60, print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = Product_{n>=0} (1 - x^(2^n))^3.
(2) A(x) = (1-x)^3 * A(x^2).
(3) a(n) = Sum_{k=0..n} A106400(k) * A106407(n-k+1) for n >= 0.

A268676 a(n) = A101080(n,A268823(3+n)), where A101080(x,y) gives the Hamming distance between binary expansions of x and y.

Original entry on oeis.org

1, 3, 3, 1, 3, 1, 1, 3, 3, 1, 1, 3, 1, 3, 3, 1, 3, 1, 1, 3, 1, 3, 3, 1, 1, 3, 3, 1, 3, 1, 1, 3, 3, 1, 1, 3, 1, 3, 3, 1, 1, 3, 3, 1, 3, 1, 1, 3, 1, 3, 3, 1, 3, 1, 1, 3, 3, 1, 1, 3, 1, 3, 3, 1, 3, 1, 1, 3, 1, 3, 3, 1, 1, 3, 3, 1, 3, 1, 1, 3, 1, 3, 3, 1, 3, 1, 1, 3, 3, 1, 1, 3, 1, 3, 3, 1, 1, 3, 3, 1, 3, 1, 1, 3, 3, 1, 1, 3, 1, 3, 3, 1, 3, 1, 1, 3, 1, 3, 3, 1, 1

Offset: 0

Antti Karttunen, Feb 19 2016

nonn

It seems that A001969 gives the positions of 1's, while A000069 gives the positions of 3's.

The above observation follows because by definition, a(n) gives the Hamming distance between binary expansions of n and A003188(3+A006068(n)). To see how this leads to the stated claim, consider the illustration "Visualized as a traversal of vertices of a tesseract" in the Wikipedia article "Gray code". Starting from any vertex with either (A) an even, or (B) an odd number of 1-bits, traverse three edges along the red path, to the direction indicated by the arrow, and then note the Hamming distance between the starting and the ending vertex. It is always 1 in case (A), and 3 in case (B), because the position of the flipped bit is given by sequence A007814, with its every other term zero, so in case (A) the third flip cancels the first flip (both toggling the rightmost bit), which leaves only the second bit-flip effective. Note that the properties of a tesseract generalize to those of an infinite dimensional hypercube. - Antti Karttunen, Mar 11 2024

Antti Karttunen, Table of n, a(n) for n = 0..8191
Wikipedia, Gray code.

Row 3 of A268833.

Cf. A000069, A000120, A001969, A003188, A006068, A007814, A010060, A101080, A106400, A268823.

a := n -> 2-(-1)^add(convert(n, base, 2)):
seq(a(n), n = 0 .. 120); # Lorenzo Sauras Altuzarra, Mar 10 2024

A003188(n) = bitxor(n, n>>1);
A006068(n) = { my(s=1, ns); while(1, ns = n >> s; if(0==ns, return(n)); n = bitxor(n, ns); s <<= 1); };
A268676(n) = hammingweight(bitxor(n,A003188(3+A006068(n)))); \\ Antti Karttunen, Mar 11 2024

A268676(n) = 2-((-1)^hammingweight(n)); \\ Antti Karttunen, Mar 11 2024, after Lorenzo Sauras Altuzarra's Maple-code.

(define (A268676 n) (A101080bi n (A268823 (+ 3 n))))
;; Where A101080bi implements the dyadic function A101080(x,y) which gives the Hamming distance between binary expansions of x and y.

a(n) = A101080(n,A268823(3+n)), where A101080(x,y) gives the Hamming distance between binary expansions of x and y.
a(n) = 2 - (-1)^A000120(n) = 2 - A106400(n). - Lorenzo Sauras Altuzarra, Mar 10 2024
a(n) = 1 + 2 * A010060(n). - Joerg Arndt, Mar 11 2024

A309303 Expansion of g.f. (sqrt(x+1) - sqrt(1-3x))/(2(x+1)^(3/2)).

Original entry on oeis.org

0, 1, -1, 2, -1, 4, 2, 13, 23, 68, 164, 439, 1146, 3067, 8231, 22306, 60791, 166684, 459308, 1271479, 3534116, 9859573, 27598757, 77490472, 218183522, 615902899, 1742738477, 4942022648, 14043034703, 39979680748, 114020882010, 325721340061

Offset: 0

Vladimir Reshetnikov, Jul 21 2019

sign

(-1)^a(n) = (-1)^A010060(n) = A106400(n) (Thue-Morse sequence).

a(n) + a(n+1) = A005043(n) = (-1)^n * hypergeom([-n, 1/2], [2], 4) (Motzkin sums).

Robert Israel, Table of n, a(n) for n = 0..2106
Eric Weisstein's World of Mathematics, Thue-Morse sequence.
Wikipedia, Thue-Morse sequence.

Cf. A005043, A010060, A106400.

f:= gfun:-rectoproc({n*a(n) = (n-4)*a(n-1) + (n-2)*(5*a(n-2) + 3*a(n-3)),a(0)=0,a(1)=1,a(2)=-1},a(n),remember):
map(f, [$0..40]); # Robert Israel, Jul 23 2019

Table[(-1)^n/2 + 3^(n + 3/2)/2^(n + 4) (2 n - 3)!!/n! Hypergeometric2F1[3/2, 3/2, 3/2 - n, 1/4], {n, 0, 31}]

a(n) = (-1)^n/2 + 3^(n+3/2)/2^(n+4) * (2*n-3)!!/n! * hypergeom([3/2, 3/2], [3/2 - n], 1/4).
D-finite with recurrence: n*a(n) = (n-4)*a(n-1) + (n-2)*(5*a(n-2) + 3*a(n-3)).
a(n) ~ c * 3^n / n^(3/2), where c = 3^(3/2) / (32*sqrt(Pi)) = 0.09161297...

cp's OEIS Frontend

A132971 a(2*n) = a(n), a(4*n+1) = -a(n), a(4*n+3) = 0, with a(0) = 1.

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A299759 Triangle read by rows in which row n lists in order all FDH numbers of strict integer partitions of n.

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A005599 Running sum of every third term in the {+1,-1}-version of Thue-Morse sequence A010060.

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A343029 Number of 1-bits in the binary expansion of n which have an even number of 0-bits at less significant bit positions.

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A106407 Expansion of x((1-x)(1-x^2)(1-x^4)(1-x^8)...)^2.

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A292118 G.f.: 1 + 2*Sum_{k >= 1} (-1)^k*q^A003159(k).

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A357761 a(n) = A227872(n) - A356018(n).

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A132971 a(2n) = a(n), a(4n+1) = -a(n), a(4*n+3) = 0, with a(0) = 1.

A292118 G.f.: 1 + 2Sum_{k >= 1} (-1)^kq^A003159(k).

A309303 Expansion of g.f. (sqrt(x+1) - sqrt(1-3x))/(2(x+1)^(3/2)).