A000621 -id:A000621 - cp's OEIS frontend

A239804 Decimal expansion of a constant related to A000621 and A000624.

1, 6, 8, 1, 3, 6, 7, 5, 2, 4, 4, 4, 1, 8, 8, 0, 2, 5, 5, 5, 9, 1, 7, 0, 2, 3, 7, 0, 3, 9, 2, 0, 0, 6, 3, 2, 2, 8, 6, 4, 0, 4, 3, 4, 7, 5, 3, 6, 4, 3, 1, 3, 5, 5, 2, 7, 5, 2, 9, 3, 3, 0, 1, 9, 5, 8, 4, 1, 3, 9, 4, 5, 5, 2, 8, 2, 7, 9, 0, 6, 8, 0, 6, 4, 2, 1, 2, 2, 1, 3, 2, 7, 8, 9, 8, 9, 2, 3, 4, 1, 9, 1, 4, 1, 1

Offset: 1

Vaclav Kotesovec, Mar 27 2014

			1.681367524441880255591702370392006322864...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6., p.300 and p.560.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 283

Cf. A000621, A000624.

Equals lim n->infinity A000621(n)^(1/n).

Equals lim n->infinity A000624(n)^(1/n).

A239806 Decimal expansion of a constant related to A000621.

Original entry on oeis.org

2, 1, 4, 5, 3, 6, 1, 3, 9, 1, 3, 4, 6, 4, 8, 5, 5, 5, 6, 3, 0, 5, 7, 2, 4, 4, 8, 4, 3, 6, 1, 9, 3, 3, 0, 8, 0, 9, 5, 1, 7, 8, 0, 4, 6, 5, 0, 1, 8, 9, 3, 5, 9, 5, 7, 8, 2, 8, 3, 4, 4, 7, 7, 1, 3, 8, 7, 7, 4, 1, 2, 7, 9, 8, 6, 9, 2, 9, 5, 8, 2, 9, 7, 6, 8, 2, 6, 4, 1, 2, 3, 0, 1, 3, 5, 0, 9, 5, 5, 1, 3, 2, 5, 4, 1, 9

Offset: 0

Vaclav Kotesovec, Mar 27 2014

In both references is cited a constant K = A239806 * A239804 = 0.360714097160142828... (related to A000621 with offset 0).

			0.214536139134648555630572448436193308095...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6., p.300 and p.548.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 283

Cf. A000621, A239804.

Equals lim n->infinity A000621(n) / A239804^n.

A000625 Number of n-node steric rooted ternary trees; number of n carbon alkyl radicals C(n)H(2n+1) taking stereoisomers into account.

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 28, 74, 199, 551, 1553, 4436, 12832, 37496, 110500, 328092, 980491, 2946889, 8901891, 27012286, 82300275, 251670563, 772160922, 2376294040, 7333282754, 22688455980, 70361242924, 218679264772, 681018679604, 2124842137550, 6641338630714, 20792003301836

Offset: 0

N. J. A. Sloane

Nodes are unlabeled, each node has out-degree <= 3.
Steric, or including stereoisomers, means that the children of a node are taken in a certain cyclic order. If the children are rotated it is still the same tree, but any other permutation yields a different tree. See A000598 for the analogous sequence with stereoisomers not counted.
Other descriptions of this sequence: steric planted trees with n nodes; total number of monosubstituted alkanes C(n)H(2n+1)-X with n carbon atoms.
Let the entries in the nine columns of Blair and Henze's Table I (JACS 54 (1932), p. 1098) be denoted by Ps(n), Pn(n), Ss(n), Sn(n), Ts(n), Tn(n), As(n), An(n), T(n) respectively (here P = Primary, S = Secondary, T = Tertiary, s = stereoisomers, n = non-stereoisomers and the last column T(n) gives total).
Then Ps (and As) = A000620, Pn (and An, Sn) = A000621, Ss = A000622, Ts = A000623, Tn = A000624, T = this sequence. Recurrences generating these sequences are given in the Maple program in A000620.

J. K. Percus, Combinatorial Methods, Lecture Notes, 1967-1968, Courant Institute, New York University, 212pp. See pp. 64-65.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1930 (first 200 terms from T. D. Noe)
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1106.
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1105. (Annotated scanned copy)
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443, Eq. (25).
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443, Eq. (25). (Annotated scanned copy)
R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See p. 44.
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361. (Annotated scanned copy)
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000598, A000602, A000620-A000624, A000628, A010732, A010733, A086194, A086200, A239803, A239810.

A := 1; f := proc(n) global A; coeff(series( 1+(1/3)*x*(A^3+2*subs(x=x^3,A)), x, n+1), x, n); end; for n from 1 to 50 do A := series(A+f(n)*x^n,x,n +1); od: A;
A000625 := proc(n)
    local j,i,a ;
    option remember;
    if n <= 1 then
        1 ;
    else
        a :=0 ;
        for j from 1 to n-1 do
            a := a+ j*procname(j)*add(procname(i)*procname(n-j-i-1),i=0..n-j-1) ;
        end do:
        if modp(n-1,3) = 0 then
            a := a+2*(n-1)*procname((n-1)/3)/3 ;
        end if;
        a/ (n-1) ;
    end if;
end proc:
seq(A000625(n),n=0..30) ;

m = 31; c[0] = 1; gf[x_] = Sum[c[k] x^k, {k, 0, m}]; cc = Array[c, m]; coes = CoefficientList[ Series[gf[x] - 1 - (x*(gf[x]^3 + 2*gf[x^3])/3), {x, 0, m}], x] // Rest; Prepend[cc /. Solve[ Thread[ coes == 0], cc][[1]], 1]
(* Jean-François Alcover, Jun 24 2011 *)
a[0] = a[1] = 1; a[n_Integer] := a[n] = (Sum[j*a[j]*Sum[a[i]*a[n-i-j-1], {i, 0, n-j-1}], {j, 1, n-1}] + (2/3)*(n-1)*a[(n-1)/3])/(n-1); a[] = 0; Table[a[n], {n, 0, 31}] (* _Jean-François Alcover, Apr 21 2016, after Emeric Deutsch *)
terms = 32; A[] = 0; Do[A[x] = Normal[1 + x*(A[x]^3 + 2*A[x^3])/3 + O[x]^terms], terms]; CoefficientList[A[x], x] (* Jean-François Alcover, Apr 22 2016, updated Jan 11 2018 *)

a(n) = if(n, my(v=vector(n+1)); v[1]=1; v[2]=1; for(k=1, n-1, v[k+2] = sum(j=1, k, j*v[j+1]*(sum(i=0, k-j, v[i+1]*v[k-j-i+1])))/k + (2/3)*if(k%3, 0, v[k/3+1])); v[n+1], 1) \\ Jianing Song, Feb 17 2019

G.f. A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 28*x^6 + ... satisfies A(x) = 1 + x*(A(x)^3 + 2*A(x^3))/3.
a(0) = a(1) = 1; a(n+1) = 2*a(n/3)/3 + (Sum_{j=1..n} j*a(j)*(Sum_{i=1..n-j} a(i)*a(n-j-i)))/n for n >= 1, where a(k) = 0 if k not an integer (essentially eq (4) in the Robinson et al. paper). - Emeric Deutsch, May 16 2004
a(n) ~ c * b^n / n^(3/2), where b = 3.287112055584474991259... (see A239803), c = 0.346304267394183622435... (see A239810). - Vaclav Kotesovec, Mar 27 2014

Additional comments from Bruce Corrigan, Nov 04 2002

A085357 Common residues of binomial(3n,n)/(2n+1) modulo 2: relates ternary trees (A001764) to the infinite Fibonacci word (A003849).

Original entry on oeis.org

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, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, Jun 25 2003

nonn

The n-th runs of ones is given by: 3 - A003849(n) (infinite Fibonacci word) = A076662(n+1). Runs of zeros are given by: A085358 and are also directly related to the Fibonacci sequence. Coefficients of A(x)^3 are found in A085359.
a(n) = 0 iff some binary digit of n is 1 while the corresponding binary digit of 3*n is 0. - Robert Israel, Jul 12 2016
The Run Length Transform of [0,1,0,0,0,...], A063524, the characteristic function of 1. (See A227349 for the definition). - Antti Karttunen, Oct 15 2016

Robert Israel, Table of n, a(n) for n = 0..10000
J.-P. Allouche, F. v. Haeseler, H.-O. Peitgen, A. Petersen and G. Skordev, Automaticity of double sequences generated by one-dimensional linear cellular automata, Theoret. Comput. Sci. 186 (1997), 195-209.
J.-P. Allouche, F. v. Haeseler, H.-O. Peitgen and G. Skordev, Linear cellular automata, finite automata and Pascal's triangle, Discrete Appl. Math. 66 (1996), 1-22.
R. Bacher and C. Reutenauer, The number of right ideals of given codimension over a finite field, in Noncommutative Birational Geometry, Representations and Combinatorics, edited by Arkady. Berenstein and Vladimir. Retakha, Contemporary Mathematics, Vol. 592, 2013.
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 characteristic functions
Index entries for sequences computed with run length transform

Cf. A001764 (ternary trees), A085358 (runs of zeros), A076662 (runs of ones), A003849 (infinite Fibonacci word), A085359 (A(x)^3).
Cf. also A003714, A005940, A008966, A063524, A227349, A277010, A324964.
Absolute values of A132971.

[Binomial(3*n,n) mod 2: n in [0..100]]; // Vincenzo Librandi, Jul 09 2016

f:= proc(n) local L,Lp;
  L:= convert(n,base,2);
  Lp:= convert(3*n,base,2);
  if has(L-Lp[1..nops(L)],1) then 0 else 1 fi
end proc:
map(f, [$0..100]); # Robert Israel, Jul 12 2016

Table[Mod[Binomial[3 n, n], 2], {n, 0, 120}] (* Michael De Vlieger, Jul 08 2016 *)

A085357(n) = !bitand(n,n<<1); \\ Antti Karttunen, Aug 22 2019

def A085357(n): return int(not n&(n<<1)) # Chai Wah Wu, Jun 25 2025

G.f.: 1 + x*A(x)^3 = A(x) (Mod 2); a(n) = A001764(n) (Mod 2).
a(n) = binomial(3n, n) (mod 2). Characteristic function of Fibbinary numbers (i.e. a(n)=1 iff n is in A003714). - Benoit Cloitre, Nov 15 2003
Recurrence: a(0) = 1, a(2n) = a(4n+1) = a(n), a(4n+3) = 0.
a(n-2) = A000256(n)(mod 2), for n>2. - John M. Campbell, Jul 08 2016
a(n) = A000621(n+1)(mod 2). - John M. Campbell, Jul 15 2016
a(n) = A000625(n)(mod 2). - John M. Campbell, Jul 15 2016
a(n) = A008966(A005940(1+n)). [Follows from the Run Length Transform interpretation, see also A277010.] - Antti Karttunen, Oct 15 2016
a(n) = abs(A132971(n)) = abs(A008683(A005940(1+n))). - Antti Karttunen, May 30 2017

A000620 Number of monosubstituted alkanes C(n-1)H(2n-1)-X with n-1 carbon atoms that are stereoisomers.

Original entry on oeis.org

0, 0, 0, 0, 2, 6, 20, 60, 176, 512, 1488, 4326, 12648, 37186, 109980, 327216, 979020, 2944414, 8897732, 27005290, 82288516, 251650788, 772127678, 2376238138, 7333188770, 22688297950, 70360977228, 218678818026, 681017928476, 2124840874610, 6641336507270, 20791999731518

Offset: 1

N. J. A. Sloane

nonn

Also number of monosubstituted alkanes C(n)H(2n+1)-X of the form R-CH2-X (primary) that are stereoisomers.
Let the entries in the nine columns of Blair and Henze's Table I (JACS 54 (1932), p. 1098) be denoted by Ps(n), Pn(n), Ss(n), Sn(n), Ts(n), Tn(n), As(n), An(n), T(n) respectively (here P = Primary, S = Secondary, T = Tertiary, s = stereoisomers, n = non-stereoisomers and the last column T(n) gives total).
Then Ps (and As) = this sequence, Pn (and An, Sn) = A000621, Ss = A000622, Ts = A000623, Tn = A000624, T = A000625. Recurrences generating these sequences are given in the Maple program in A000620.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..1930
Jean-François Alcover, Mathematica program
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1106.
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1105. (Annotated scanned copy)

Cf. A000621, A000622, A000623, A000624, A000625, A239803, A239805.

# Blair and Henze's recurrences for A000620-A000625 (see comments lines for relationship between the sequences and their symbols).
Ps := [0,0,0]; Pn := [1,1,1]; Ss := [0,0,0]; Sn := [0,0,1]; Ts := [0,0,0]; Tn := [0,0,0]; As := [0,0,0]; An := [1,1,2]; T := [1,1,2];
for n from 4 to 60 do Ps := [op(Ps),As[n-1]]; Pn := [op(Pn),An[n-1]]; t1 := add( 2*T[n-1-j]*T[j],j=1..floor((n-2)/2) ); if n mod 2 = 1 then i := (n-1)/2; t1 := t1+T[i]^2-An[i]; fi; Ss := [op(Ss),t1];
t2 := 0; if n mod 2 = 1 then i := (n-1)/2; t2 := An[i]; fi; Sn := [op(Sn),t2]; t3 := 0; for i from 1 to (n-1)/3 do for j from i+1 to (n-2)/2 do k := n-1-i-j; if j 0 and i <> j then t4 := t4+(T[i]^2-An[i])*T[j]+An[i]*As[j]; t5 := t5+An[i]*An[j]; fi; od; t6 := 0; t7 := 0; if n mod 3 = 1 then i := (n-1)/3; t6 := (2*T[i]+T[i]^3)/3-An[i]^2; t7 := An[i]^2; fi;
Ts := [op(Ts), t3+t4+t6]; Tn := [op(Tn), t5+t7]; As := [op(As), Ps[n]+Ss[n]+Ts[n]]; An := [op(An), Pn[n]+Sn[n]+Tn[n]]; T := [op(T),As[n]+An[n]]; od: Ps; Pn; Ss; Ts; Tn; T;

Mathematica
```
(* See links *)
```

See Maple program for recurrences for this sequence and A000621-A000625.

a(n) ~ c * b^n / n^(3/2), where b = 3.287112055584474991259... (see A239803), c = 0.105352133282419523497... (see A239805). - Vaclav Kotesovec, Mar 27 2014

Additional comments from Bruce Corrigan, Nov 04 2002

A000622 Number of monosubstituted alkanes C(n)H(2n+1)-X of the form shown in the Comments lines that are stereoisomers.

Original entry on oeis.org

0, 0, 0, 2, 4, 14, 34, 98, 270, 768, 2192, 6360, 18576, 54780, 162658, 486154, 1461174, 4413988, 13393816, 40807290, 124783604, 382842018, 1178140170, 3635626680, 11247841040, 34880346840, 108402132234, 337576497920, 1053229357732, 3291813720292, 10305275270364

Offset: 1

N. J. A. Sloane

nonn

R-CH-X (secondary)
.....|
.....R'
Let the entries in the nine columns of Blair and Henze's Table I (JACS 54 (1932), p. 1098) be denoted by Ps(n), Pn(n), Ss(n), Sn(n), Ts(n), Tn(n), As(n), An(n), T(n) respectively (here P = Primary, S = Secondary, T = Tertiary, s = stereoisomers, n = non-stereoisomers and the last column T(n) gives total).
Then Ps (and As) = A000620, Pn (and An, Sn) = A000621, Ss = this sequence, Ts = A000623, Tn = A000624, T = A000625. Recurrences generating these sequences are given in the Maple program in A000620.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..1930
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1106.
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1105. (Annotated scanned copy)

Cf. A000620-A000625, A239803, A239807.

a(n) ~ c * b^n / n^(3/2), where b = 3.287112055584474991259... (see A239803), c = 0.171310881484463744523... (see A239807). - Vaclav Kotesovec, Mar 27 2014

Additional comments from Bruce Corrigan, Nov 04 2002

A000623 Number of monosubstituted alkanes C(n)H(2n+1)-X of the form shown in the Comments lines that are stereoisomers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 6, 18, 66, 208, 646, 1962, 5962, 18014, 54578, 165650, 504220, 1539330, 4713742, 14475936, 44578668, 137634872, 425970290, 1321323952, 4107268140, 12792332438, 39915708564, 124762612530, 390593588402, 1224681912368, 3845387953884, 12090382743374

Offset: 1

N. J. A. Sloane

nonn

....X
....|
R-C-R' (tertiary)
....|
....R"
Let the entries in the nine columns of Blair and Henze's Table I (JACS 54 (1932), p. 1098) be denoted by Ps(n), Pn(n), Ss(n), Sn(n), Ts(n), Tn(n), As(n), An(n), T(n) respectively (here P = Primary, S = Secondary, T = Tertiary, s = stereoisomers, n = non-stereoisomers and the last column T(n) gives total).
Then Ps (and As) = A000620, Pn (and An, Sn) = A000621, Ss = A000622, Ts = this sequence, Tn = A000624, T = A000625. Recurrences generating these sequences are given in the Maple program in A000620.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..1930
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1106.
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1105. (Annotated scanned copy)

Cf. A000620-A000625, A239803, A239808.

a(n) ~ c * b^n / n^(3/2), where b = 3.287112055584474991259... (see A239803), c = 0.069641252627300354415... (see A239808). - Vaclav Kotesovec, Mar 27 2014

Additional comments from Bruce Corrigan, Nov 04 2002

A000624 Number of monosubstituted alkanes C(n)H(2n+1)-X of the form shown in the Comments lines that are not stereoisomers.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 4, 9, 13, 26, 40, 74, 118, 210, 342, 595, 981, 1684, 2798, 4763, 7951, 13469, 22548, 38082, 63862, 107666, 180740, 304382, 511292, 860504, 1445998, 2432665, 4088805, 6877172, 11560684, 19441791, 32684789, 54961955, 92404472, 155377371, 261235027

Offset: 1

N. J. A. Sloane

nonn

....X
....|
R-C-R' (tertiary)
....|
....R"
Then Ps (and As) = A000620, Pn (and An, Sn) = A000621, Ss = A000622, Ts = A000623, Tn = this sequence, T = A000625. Recurrences generating these sequences are given in the Maple program in A000620.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..3000
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1106.
C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1105. (Annotated scanned copy)

Cf. A000620-A000625, A239804, A239809.

a(n) ~ c * beta^n, where beta = 1.681367524441880255591... (see A239804), c = 0.146177958025494272954... (see A239809). - Vaclav Kotesovec, Mar 27 2014

Additional comments from Bruce Corrigan, Nov 04 2002

False g.f. deleted by N. J. A. Sloane, May 13 2008

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

Original entry on oeis.org

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

A101913 G.f. satisfies: A(x) = 1/(1 + xA(x^3)) and also the continued fraction: 1+xA(x^4) = [1;1/x,1/x^3,1/x^9,1/x^27,...,1/x^(3^(n-1)),...].

Original entry on oeis.org

1, -1, 1, -1, 2, -3, 4, -6, 9, -13, 19, -28, 41, -61, 90, -132, 195, -288, 424, -625, 922, -1359, 2004, -2955, 4356, -6423, 9471, -13963, 20587, -30355, 44755, -65987, 97293, -143449, 211503, -311844, 459785, -677912, 999524, -1473709, 2172854, -3203685, 4723551, -6964461, 10268490, -15139986

Offset: 0

Paul D. Hanna, Dec 20 2004

sign

Cf. A101912, A101914-A101918.

{a(n)=local(A);A=1-x;for(i=1,n\3+1, A=1/(1+x*subst(A,x,x^3)+x*O(x^n)));polcoeff(A,n,x)}

{a(n)=local(M=contfracpnqn(concat(1, vector(ceil(log(n+1)/log(3))+1,n,1/x^(3^(n-1)))))); polcoeff(M[1,1]/M[2,1]+x*O(x^(4*n+1)),4*n+1)}

From Joerg Arndt, Oct 15 2011: (Start)
For the sequence abs(a(n)) we have
g.f. B(x) 1/(1-x/(1-x^3/(1-x^9/(1-x^27(1- ... ))))) and
B(x) satisfies B(x) = 1 + x*B(x)*B(x^3)  (cf. A000621)
(End)
G.f.: T(0), where T(k) = 1 - (-x)^(3^k)/((-x)^(3^k) - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 18 2013
a(0) = 1; a(n) = -Sum_{k=0..floor((n-1)/3)} a(k) * a(n-3*k-1). - Ilya Gutkovskiy, Mar 01 2022

cp's OEIS Frontend

A239804 Decimal expansion of a constant related to A000621 and A000624.

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A239806 Decimal expansion of a constant related to A000621.

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A000625 Number of n-node steric rooted ternary trees; number of n carbon alkyl radicals C(n)H(2n+1) taking stereoisomers into account.

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A085357 Common residues of binomial(3n,n)/(2n+1) modulo 2: relates ternary trees (A001764) to the infinite Fibonacci word (A003849).

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A000620 Number of monosubstituted alkanes C(n-1)H(2n-1)-X with n-1 carbon atoms that are stereoisomers.

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A000622 Number of monosubstituted alkanes C(n)H(2n+1)-X of the form shown in the Comments lines that are stereoisomers.

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A000623 Number of monosubstituted alkanes C(n)H(2n+1)-X of the form shown in the Comments lines that are stereoisomers.

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

A101913 G.f. satisfies: A(x) = 1/(1 + xA(x^3)) and also the continued fraction: 1+xA(x^4) = [1;1/x,1/x^3,1/x^9,1/x^27,...,1/x^(3^(n-1)),...].