need some mathematical series help...

[sao123]

given a series Y1, Y2... Yi defined by a function f(X1,X2...Xi)....

Function F defined as...

F(Xi) = Yi = N
F(X[i+j8]) = Y[i+j8] = (2^j)N where X,Y,i,j & N are real positive integers.

Given some aribtrary ordered pair (Xa, Ya), what formula can i derive to calcluate the ration of Y[a+1] / Ya


there is a known base ordered pair or
(X2,Y2) = (2,83)

 

[CycloWizard]

I'm not sure I understand your notation... What is X[i+j8]?

 

[sao123]

Originally posted by: CycloWizard
I'm not sure I understand your notation... What is X[i+j8]?

essentially... (i & j are the index of the ith & jth terms of the series)
for every 8 iterations of X -> Y doubles.

so starting with
F( X{i} ) = Y{i}
F( X{i+8} ) = 2 * Y{i} -> X{i+8} = 2^1 * Y{i}
F( X{i+16} ) = 2 * 2 * Y{i} -> X{i+2*8} = 2^2 * Y{i}
F( X{i+24} ) = 2 * 2 * 2 * Y{i} -> X{i+3*8} = 2^3 * Y{i}

pattern reduces to
F( X{i+j*8} ) = 2^j * Y{i}




this is the beginning of a series i am going to do calculations on.... the above formula is an approximation of the higher terms in the series.
X{i}-Y{i}
2-----83
3-----174
4-----276
5-----388
6-----512
7-----650
8-----801
9-----969
10----1,154
11----1,358
12----1,584
13----1,833
14----2,107
15----2,411
16----2,746

 

[CycloWizard]

Ah, I gotcha. I'll take a crack at it and report back.

 

[CycloWizard]

Just plotting it in Excel, I see that the ration is pretty much constant at 1.104... Do you still need a formula for a more general case?

 

[sao123]

the goal of this eventually is to find a curve which will approximate the first 100 terms of this series.... for both columns


x{i}--Y{i}------------{i+1} - Y{i]
1-----0........................83
2-----83.......................91
3-----174......................102
4-----276......................112
5-----388......................124
6-----512......................138
7-----650......................151
8-----801......................168
9-----969......................185
10----1,154....................204
11----1,358....................226
12----1,584....................249
13----1,833....................274
14----2,107....................304
15----2,411....................335
16----2,746....................369
17----3,115....................408
18----3,523....................450
19----3,973....................497
20----4,470....................548
21----5,018....................606
22----5,624....................667
23----6,291....................737
24----7,028....................814
25----7,842....................898
26----8,740....................990
27----9,730....................1,094
28----10,824...................1,207
29----12,031...................1,332
30----13,363...................1,470
31----14,833...................1,573
32----16,456...................1,701
33----18,247...................1,977
34----20,224...................2,182
35----22,406...................2,409
36----24,815...................2,658
37----27,473...................2,935
38----30,408...................3,240
39----33,648...................3,576
40----37,224...................3,947
41----41,171...................4,358
42----45,529...................4,870
43----50,399...................5,250
44----55,649...................5,863
45----61,512...................6,471
46----67,983...................7,144
47----75,127...................7,887
48----83,014...................8,707
49----91,721...................9,612
50----101,333..................10,612
51----111,945..................11,715
52----123,660..................12,935
53----136,594..................14,278
54----150,872..................15,764
55----166,636..................17,404
56----184,040..................19,214
57----203,254..................21,212
58----224,466..................23,420
59----247,886..................25,856
60----273,742..................28,546
61----302,288..................31,516
62----333,804..................34,795
63----368,599..................38,416
64----407,015..................42,413
65----449,428..................46,826
66----496,254..................51,699
67----547,953..................57,097
68----605,032..................63,019
69----668,051..................69,576
70----737,627..................76,818
71----814,445..................84,812
72----899,257..................93,638
73----992,895..................103,383
74----1,096,278................114,143
75----1,210,421................126,022
76----1,336,443................139,138
77----1,475,581................153,619
78----1,629,200................169,608
79----1,798,808................187,260
80----1,986,068................206,750
81----2,192,818................228,269
82----2,421,087................252,027
83----2,673,114................278,259
84----2,951,373................307,221
85----3,258,594................339,198
86----3,597,792................374,502
87----3,972,294................413,482
88----4,385,776................456,519
89----4,842,295................504,037
90----5,346,332................556,499
91----5,902,831................614,422
92----6,517,253................678,376
93----7,195,629................748,985
94----7,944,614................826,944
95----8,771,558................913,019
96----9,684,577................1,008,052
97----10,692,629...............1,112,977
98----11,805,606...............1,228,825
99----13,034,431...............1,356,729
100-14,391,160...............N/A

 

[sao123]

Originally posted by: CycloWizard
Just plotting it in Excel, I see that the ration is pretty much constant at 1.104... Do you still need a formula for a more general case?

wow... thanks. a few more steps and i might have it.

 

[eilute]

Originally posted by: sao123
what formula can i derive to calcluate the ration of Y[a+1] / Ya

I believe this equation is called 'the ratio test'

 

[DrPizza]

I discovered one of the nerdiest sites for math people about 2 1/2 weeks ago. I saw this thread, thought Hmmmm...
Went to that site, did a search, found it.
Ever been to the online encyclopedia of integer sequences??

your sequence

 

[CycloWizard]

Originally posted by: DrPizza

I discovered one of the nerdiest sites for math people about 2 1/2 weeks ago. I saw this thread, thought Hmmmm...
Went to that site, did a search, found it.
Ever been to the online encyclopedia of integer sequences??

your sequence

Wow... Every random combination I can think of has some sequence on there. Creepy.

 

[sao123]

Originally posted by: DrPizza

I discovered one of the nerdiest sites for math people about 2 1/2 weeks ago. I saw this thread, thought Hmmmm...
Went to that site, did a search, found it.
Ever been to the online encyclopedia of integer sequences??

your sequence

that site isnt working here at work, but ill try it again when i get home.
thanks.

 

[TuxDave]

Concerning the popular MMORPG "Runescape" by JAGeX corporation, this sequence gives the amount of experience points needed for a given level in a skill.

lmao.. is this what you're trying to do?

 

[CSMR]

Originally posted by: sao123
given a series Y1, Y2... Yi defined by a function f(X1,X2...Xi)....

Function F defined as...

F(Xi) = Yi = N

This doesn't make sense. Is F a function on any number of integers or on one integer? I presume f maps from an integer to an integer since that is what happens in your formula. F(xi)=yi=n doesn't make sense either unless f is constant and equal to N. Ignoring these then...

F(X[i+j8]) = Y[i+j8] = (2^j)N where X,Y,i,j & N are real positive integers.

I assume you mean Y(I) instead of N.

Given some aribtrary ordered pair (Xa, Ya), what formula can i derive to calcluate the ration of Y[a+1] / Ya


there is a known base ordered pair or
(X2,Y2) = (2,83)

Obviously you can't work that out. Your equation defines a sequence when f(X1) to f(X8) are given, but for any f(X1) to f(X8) there is a sequence that works. (Assuming the function X is one-to-one.) So just knowing f(X2) isn't enough.

 

[sao123]

Originally posted by: CSMR

Originally posted by: sao123
given a series Y1, Y2... Yi defined by a function f(X1,X2...Xi)....

Function F defined as...

F(Xi) = Yi = N

This doesn't make sense. Is F a function on any number of integers or on one integer? I presume f maps from an integer to an integer since that is what happens in your formula. F(xi)=yi=n doesn't make sense either unless f is constant and equal to N. Ignoring these then...

F(X[i+j8]) = Y[i+j8] = (2^j)N where X,Y,i,j & N are real positive integers.

I assume you mean Y(I) instead of N.

Given some aribtrary ordered pair (Xa, Ya), what formula can i derive to calcluate the ration of Y[a+1] / Ya


there is a known base ordered pair or
(X2,Y2) = (2,83)

Obviously you can't work that out. Your equation defines a sequence when f(X1) to f(X8) are given, but for any f(X1) to f(X8) there is a sequence that works. (Assuming the function X is one-to-one.) So just knowing f(X2) isn't enough.

obviosly you didnt read, because the answer is already there above you...

xp(n) = floor( sum( floor(x + 300*2^(x/7), x, 1, n-1) ) / 4 )
Notation: sum(equation, variable, first index, last index)