OFFSET
0,4
COMMENTS
Conjecture: for n > 0, a(n) is odd iff n = A003714(k) for some k > 0, where A003714 lists Fibbinary numbers whose binary representation contains no two adjacent 1's.
Conjectures. For n > 0, we have the following occurrences:
a(n) = 0 iff n = 11 * 2^k or n = 23 * 2^k,
a(n) = 1 iff n = 5 * 2^k,
a(n) = 2 iff n = 3 * 2^k,
a(n) = 5 iff n = 9 * 2^k,
a(n) = 22 iff n = 15 * 2^k,
a(n) = 33 iff n = 21 * 2^k,
a(n) = 42 iff n = 131 * 2^k,
a(n) = 52 iff n = 27 * 2^k,
a(n) = 90 iff n = 35 * 2^k,
a(n) = 125 iff n = 33 * 2^k,
a(n) = 144 iff n = 47 * 2^k,
a(n) = 154 iff n = 39 * 2^k,
a(n) = 256 iff n = 51 * 2^k,
a(n) = 470 iff n = 63 * 2^k,
a(n) = -1 iff n = 2^k,
a(n) = -6 iff n = 7 * 2^k,
a(n) = -8 iff n = 13 * 2^k,
a(n) = -11 iff n = 17 * 2^k,
a(n) = -16 iff n = 25 * 2^k,
a(n) = -30 iff n = 19 * 2^k,
a(n) = -40 iff n = 29 * 2^k,
a(n) = -114 iff n = 31 * 2^k,
a(n) = -123 iff n = 41 * 2^k,
a(n) = -149 iff n = 37 * 2^k,
a(n) = -235 iff n = 65 * 2^k,
a(n) = -360 iff n = 43 * 2^k or n = 53 * 2^k,
etc., each of which hold for k >= 0.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..5000
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n, where B(x) is the g.f. of A374570 and C(x) = x + C(x)^2 is the g.f. of A000108, satisfies the following formulas.
(1) A(x) = A(x^2) - x*A(x^2)^2.
(2) A(x^2) = (1 - sqrt(1 - 4*x*A(x))) / (2*x).
(3) A(x^2) = (1/x) * C(x*A(x)).
(4) x^2 = B( x * C(x*A(x)) ).
(5) A(B(x)) = x / B(x).
(6) A(B(x)^2) = C(x) / B(x).
(7) B(x)^2 = B( B(x)*C(x) ).
EXAMPLE
G.f.: A(x) = 1 - x - x^2 + 2*x^3 - x^4 + x^5 + 2*x^6 - 6*x^7 - x^8 + 5*x^9 + x^10 + 2*x^12 - 8*x^13 - 6*x^14 + 22*x^15 - x^16 - 11*x^17 + 5*x^18 - 30*x^19 + x^20 + ...
where A(x^2) = (1 - sqrt(1 - 4*x*A(x)))/(2*x).
RELATED SERIES.
Let B(x) = Series_Reversion(x*A(x)), then
B(x) = x + x^2 + 3*x^3 + 8*x^4 + 27*x^5 + 90*x^6 + 320*x^7 + 1152*x^8 + 4257*x^9 + 15934*x^10 + 60486*x^11 + 231894*x^12 + ... + A374570(n)*x^n + ...
where B(x)^2 = B( B(x)*C(x) ), and C(x) begins:
C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + ... + A000108(n)*x^n + ,,,
where C(x) = (1 - sqrt(1 - 4*x))/2 is the Catalan function.
SPECIFIC VALUES.
A(t) = 4/5 at t = 0.1786763406278486221896028296025274247659944115...
A(t) = 3/4 at t = 0.2209727374872302749773868295900473238254186343...
A(t) = 2/3 at t = 0.2927920532546611624693565662579476873870699464...
A(t) = 3/5 at t = 0.3532836501852252091389612952989266014287213872...
A(t) = 1/2 at t = 0.4540878993396162878365437853450173746622109652...
A(t) = 2/5 at t = 0.5753264646036491718800481741299163550606457682...
A(t) = 1/3 at t = 0.6711059159867924708010090309770441047524321152...
A(t) = 1/4 at t = 0.8063263233032142016966341297674341884930955548...
A(t) = 1/5 at t = 0.8884702348196434968520432792716046325517863531...
A(1/2) = 0.4596569887547343191321148479065626411948116168891503813...
where A(1/4) = (1 - sqrt(1 - 2*A(1/2))).
A(1/3) = 0.6215166290026409046430206750366100166629591510407086872...
where A(1/9) = (3/2) * (1 - sqrt(1 - (4/3)*A(1/3))).
A(1/4) = 0.7159471484203487850228006105062270686816491955635126263...
where A(1/16) = 2 * (1 - sqrt(1 - A(1/4))).
A(1/5) = 0.7747713037551020088783260174094983351988173792698848600...
where A(1/25) = (5/2) * (1 - sqrt(1 - (4/5)*A(1/5))).
A(1/6) = 0.8141931617547219509824463958597943246122338043286847588...
where A(1/36) = 3 * (1 - sqrt(1 - (2/3)*A(1/6))).
A(1/8) = 0.8630723739180924020163457579861333293488991044015651008...
where A(1/64) = 4 * (1 - sqrt(1 - (1/2)*A(1/8))).
A(1/10) = 0.891911395101161792043000371010714789952867553398091597...
where A(1/100) = 5 * (1 - sqrt(1 - (2/5)*A(1/10))).
PROG
(PARI) {a(n) = my(A = 1+x); for(i=0, #binary(n), A = subst(A, x, x^2) - x*subst(A^2, x, x^2) + x*O(x^n) ); polcoeff(A, n)}
for(n=0, 80, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jul 11 2024
STATUS
approved