[math-fun] Help with some math
Let f(x)^2 = f(x^2) - 1 and assume f is of the form f(x) = x + a x^-1 + b x^-3 + c x^-5 + ... What are the first few coefficients (say 20) as rational numbers? With the tools I have, it would take me a while to power through this. I assume it would be easier with a symbolic math program.
Maxima works: a=-1/2,b=-3/8,c=-3/16,d=-45/128,e=-63/256,f=-375/1024, ... At 03:01 PM 3/21/2017, David Wilson wrote:
Let
f(x)^2 = f(x^2) - 1
and assume f is of the form
f(x) = x + a x^-1 + b x^-3 + c x^-5 + ...
What are the first few coefficients (say 20) as rational numbers?
With the tools I have, it would take me a while to power through this. I assume it would be easier with a symbolic math program.
On 2017-03-21 15:58, Henry Baker wrote:
Maxima works:
a=-1/2,b=-3/8,c=-3/16,d=-45/128,e=-63/256,f=-375/1024, ...
At 03:01 PM 3/21/2017, David Wilson wrote:
Let
f(x)^2 = f(x^2) - 1
and assume f is of the form
f(x) = x + a x^-1 + b x^-3 + c x^-5 + ...
What are the first few coefficients (say 20) as rational numbers?
With the tools I have, it would take me a while to power through this. I assume it would be easier with a symbolic math program.
OEIS A088674 Out[8]= 1 - 1/2 x (1 + (3 x)/4 + (3 x^2)/8 + (45 x^3)/64 + (63 x^4)/128 + ( 375 x^5)/512 + (699 x^6)/1024 + (19389 x^7)/16384 + (37947 x^8)/ 32768 + (224781 x^9)/131072 + (506949 x^10)/262144 + ( 6105897 x^11)/2097152 + (14473899 x^12)/4194304) In[9]:= %^2 - (% /. x -> x^2) // Expand Out[9]= -x + (83196831 x^14)/16777216 + (31624083 x^15)/8388608 + . . . --rwg
On 2017-03-21 19:37, rwg wrote:
On 2017-03-21 15:58, Henry Baker wrote:
Maxima works:
a=-1/2,b=-3/8,c=-3/16,d=-45/128,e=-63/256,f=-375/1024, ...
At 03:01 PM 3/21/2017, David Wilson wrote:
Let
f(x)^2 = f(x^2) - 1
and assume f is of the form
f(x) = x + a x^-1 + b x^-3 + c x^-5 + ...
What are the first few coefficients (say 20) as rational numbers?
With the tools I have, it would take me a while to power through this. I assume it would be easier with a symbolic math program.
OEIS A088674 Out[8]= 1 - 1/2 x (1 + (3 x)/4 + (3 x^2)/8 + (45 x^3)/64 + (63 x^4)/128 + ( 375 x^5)/512 + (699 x^6)/1024 + (19389 x^7)/16384 + (37947 x^8)/ 32768 + (224781 x^9)/131072 + (506949 x^10)/262144 + ( 6105897 x^11)/2097152 + (14473899 x^12)/4194304)
In[9]:= %^2 - (% /. x -> x^2) // Expand
Out[9]= -x + (83196831 x^14)/16777216 + (31624083 x^15)/8388608 + . . . --rwg
Oops, make that In[24]:= x - {1, 3, 6, 45, 126, 750, 2796, 19389, 75894, 449562, 2027796, 12211794, 57895596}.(2 x)^-Range[1, 25, 2] Out[24]= -(14473899/(8388608 x^25)) - 6105897/( 4194304 x^23) - 506949/(524288 x^21) - 224781/(262144 x^19) - 37947/( 65536 x^17) - 19389/(32768 x^15) - 699/(2048 x^13) - 375/( 1024 x^11) - 63/(256 x^9) - 45/(128 x^7) - 3/(16 x^5) - 3/( 8 x^3) - 1/(2 x) + x In[25]:= %^2 - (% /. x -> x^2) // Expand Out[25]= -1 + 330909617204793/(70368744177664 x^50) + 88376136482403/( 17592186044416 x^48) + 121592195378505/( 17592186044416 x^46) + 1587208967343/( 274877906944 x^44) + 3516540856089/( 549755813888 x^42) + 185059976877/(34359738368 x^40) + 192862771689/( 34359738368 x^38) + 2359404189/(536870912 x^36) + 4836713079/( 1073741824 x^34) + 4196659491/(1073741824 x^32) + 4414827525/( 1073741824 x^30) + 31624083/(8388608 x^28) + 83196831/(16777216 x^26) --rwg
participants (3)
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David Wilson -
Henry Baker -
rwg