|
| |
|
|
A274850
|
|
G.f. A(x) satisfies: 1 = ...((((A(x) - 2*x)^2 - 4*x^2)^2 - 8*x^3)^2 - 16*x^4)^2 -... - 2^n*x^n)^2 -...., a series of infinite nested squares.
|
|
1
|
|
|
|
1, 2, 2, 2, 0, -2, -4, -2, -6, -2, 4, 50, 78, 34, -248, -146, 248, 1478, 992, -2570, -13918, -19026, -12744, 68034, 193088, 380810, 86880, -1362006, -5050422, -6966190, 1464256, 39349110, 99348446, 130130462, -62116516, -664338662, -1811216390, -2530859462, -458383728, 9647822334, 29564081352, 51363216966, 35437251200, -72752696138, -343482383882, -735851485042, -1167503794928, -1270647820502, -634163168566, 3532238523874, 18635012627076
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: S(0) where S(k) = (2*x)^(k+1) + S(k+1)^(1/2). - Michael Somos, Jul 06 2017
|
|
|
EXAMPLE
|
G.f.: A(x) = 1 + 2*x + 2*x^2 + 2*x^3 - 2*x^5 - 4*x^6 - 2*x^7 - 6*x^8 - 2*x^9 + 4*x^10 + 50*x^11 + 78*x^12 + 34*x^13 - 248*x^14 - 146*x^15 + 248*x^16 +...
such that A(x) satisfies the series of infinite nested squares given by:
1 = (((((((((A - 2*x)^2 - 2^2*x^2)^2 - 2^3*x^3)^2 - 2^4*x^4)^2 - 2^5*x^5)^2 - 2^6*x^6)^2 - 2^7*x^7)^2 - 2^8*x^8)^2 - 2^9*x^9)^2 ...
ILLUSTRATION.
Equivalently, we may start with B0 = A(x) - 2*x, and then continue
B1 = ( B0^2 - 2^2*x^2 )^(1/2)
B2 = ( B1^4 - 2^3*x^3 )^(1/4)
B3 = ( B2^8 - 2^4*x^4 )^(1/8)
B4 = ( B3^16 - 2^5*x^5 )^(1/16)
B5 = ( B4^32 - 2^6*x^6 )^(1/32)
B6 = ( B5^64 - 2^7*x^7 )^(1/64)
B7 = ( B6^128 - 2^8*x^8 )^(1/128)
B8 = ( B7^256 - 2^9*x^9 )^(1/256)
B9 = ( B8^512 - 2^10*x^10 )^(1/512)
...
The above series converge to 1, as can be seen by:
B0 = 1 + 2*x^2 + 2*x^3 - 2*x^5 - 4*x^6 - 2*x^7 - 6*x^8 - 2*x^9 + 4*x^10 +...
B1 = 1 + 2*x^3 + 2*x^4 + 2*x^5 - 4*x^6 - 10*x^7 - 24*x^8 - 10*x^9 + 16*x^10 +...
B2 = 1 + 2*x^4 + 2*x^5 + 2*x^6 + 2*x^7 - 12*x^8 - 26*x^9 - 56*x^10 - 86*x^11 +...
B3 = 1 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 2*x^9 - 28*x^10 - 58*x^11 - 120*x^12 +...
B4 = 1 + 2*x^6 + 2*x^7 + 2*x^8 + 2*x^9 + 2*x^10 + 2*x^11 - 60*x^12 - 122*x^13 +...
B5 = 1 + 2*x^7 + 2*x^8 + 2*x^9 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^13 - 124*x^14 +...
B6 = 1 + 2*x^8 + 2*x^9 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^13 + 2*x^14 + 2*x^15 - 252*x^16 - 506*x^17 - 1016*x^18 - 1526*x^19 - 2548*x^20 +...
B7 = 1 + 2*x^9 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^13 + 2*x^14 + 2*x^15 + 2*x^16 + 2*x^17 - 508*x^18 - 1018*x^19 - 2040*x^20 - 3062*x^21 +...
B8 = 1 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^13 + 2*x^14 + 2*x^15 + 2*x^16 + 2*x^17 + 2*x^18 + 2*x^19 - 1020*x^20 - 2042*x^21 - 4088*x^22 +...
B9 = 1 + 2*x^11 + 2*x^12 + 2*x^13 + 2*x^14 + 2*x^15 + 2*x^16 + 2*x^17 + 2*x^18 + 2*x^19 + 2*x^20 + 2*x^21 - 2044*x^22 - 4090*x^23 +...
...
|
|
|
MATHEMATICA
|
a[ n_] := If[ n < 0, 0, Module[{A = 1 + O[x]^(n + 1), B = (2 x)^(n + 1)}, Do[ A = Sqrt[A] + (B /= 2 x), n]; SeriesCoefficient[A, n]]]; (* Michael Somos, Jul 06 2017 *)
|
|
|
PROG
|
(PARI) {a(n) = my(A=[1], B=1, m); for(i=1, n, A=concat(A, 0); B = Ser(A);
for(m=0, #A-1, B = (B^(2^m) - 2^(m+1)*x^(m+1))^(1/2^m); A[#A] = -Vec(B)[#A]) ); A[n+1]}
for(n=0, 50, print1(a(n), ", "))
(PARI) {a(n) = my(A); if( n<0, 0, A = 1 + x * O(x^n); for(k=1, n, A = sqrt(A) + (2*x)^(n+1-k)); polcoeff(A, n))}; /* Michael Somos, Jul 06 2017 */
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
sign
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
