1. ## Golden Ratio/Formula

Since my professor could not figure this out, I thought I would look here for help. I have an idea of what is going on, but how to prove it.

I need to find a formula that given any pair of numbers (such as in the coordinate system below), I can tell if it's one of the below.

(1,2)
(3,5)
(4,7)
(6,10)
(8,13)
(9,15)
(11,18)
(12,20)
(14,23)
(16,26)
(17,28)
(19,31)
(21,34)
(22,36)
(24,39)
(25,41)
(27,44)
(29,47)
(30,49)
(32,52)
(33,54)
(35,57)
(37,60)
(38,62)
(40,65)
(42,68)
(43,70)
(45,73)
(46,75)
(48,78)
(50,81)
(51,83)
(53,86)
(55,89)
(56,91)
(58,94)
(59,96)
(61,99)
(63,102)
(64,104)
(66,107)
(67,109)
(69,112)
(71,115)
(72,117)
(74,120)
(76,123)
(77,125)
(79,128)
(80,130)
(82,133)
(84,136)
(85,138)
(87,141)
(88,143)
(90,146)
(92,149)
(93,151)
(95,154)
(97,157)
(98,159)
(100,162)
(101,164)
(103,167)
(105,170)
(106,172)
(108,175)
(110,178)
(111,180)
(113,183)
(114,185)
(116,188)
(118,191)
(119,193)
(121,196)
(122,198)
(124,201)
(126,204)
(127,206)
(129,209)
(131,212)
(132,214)
(134,217)
(135,219)
(137,222)
(139,225)
(140,227)
(142,230)
(144,233)
(145,235)
(147,238)
(148,240)
(150,243)
(152,246)
(153,248)
(155,251)
(156,253)
(158,256)
(160,259)
(161,261)
(163,264)
(165,267)
(166,269)
(168,272)
(169,274)
(171,277)
(173,280)
(174,282)
(176,285)
(177,287)
(179,290)
(181,293)
(182,295)
(184,298)
(186,301)
(187,303)
(189,306)
(190,308)
(192,311)
(194,314)
(195,316)
(197,319)
(199,322)
(200,324)
(202,327)
(203,329)
(205,332)
(207,335)
(208,337)
(210,340)
(211,342)
(213,345)
(215,348)
(216,350)
(218,353)
(220,356)
(221,358)
(223,361)
(224,363)
(226,366)
(228,369)
(229,371)
(231,374)
(232,376)
(234,379)
(236,382)
(237,384)
(239,387)
(241,390)
(242,392)
(244,395)
(245,397)
(247,400)
(249,403)
(250,405)
(252,408)
(254,411)
(255,413)
(257,416)
(258,418)
(260,421)
(262,424)
(263,426)
(265,429)
(266,431)
(268,434)
(270,437)
(271,439)
(273,442)
(275,445)
(276,447)
(278,450)
(279,452)
(281,455)
(283,458)
(284,460)
(286,463)
(288,466)
(289,468)
(291,471)
(292,473)
(294,476)
(296,479)
(297,481)
(299,484)
(300,486)
(302,489)
(304,492)
(305,494)
(307,497)
(309,500)
(310,502)
(312,505)
(313,507)
(315,510)
(317,513)
(318,515)
(320,518)
(321,520)
(323,523)
(325,526)
(326,528)
(328,531)
(330,534)
(331,536)
(333,539)
(334,541)
(336,544)
(338,547)
(339,549)
(341,552)
(343,555)
(344,557)
(346,560)
(347,562)
(349,565)
(351,568)
(352,570)
(354,573)
(355,575)
(357,578)
(359,581)
(360,583)
(362,586)
(364,589)
(365,591)
(367,594)
(368,596)
(370,599)
(372,602)
(373,604)
(375,607)
(377,610)
(378,612)
(380,615)
(381,617)
(383,620)
(385,623)
(386,625)
(388,628)
(389,630)
(391,633)
(393,636)
(394,638)
(396,641)
(398,644)
(399,646)
(401,649)
(402,651)
(404,654)
(406,657)
(407,659)
(409,662)
(410,664)
(412,667)
(414,670)
(415,672)
(417,675)
(419,678)
(420,680)
(422,683)
(423,685)
(425,688)
(427,691)
(428,693)
(430,696)
(432,699)
(433,701)
(435,704)
(436,706)
(438,709)
(440,712)
(441,714)
(443,717)
(444,719)
(446,722)
(448,725)
(449,727)
(451,730)
(453,733)
(454,735)
(456,738)
(457,740)
(459,743)
(461,746)
(462,748)
(464,751)
(465,753)
(467,756)
(469,759)
(470,761)
(472,764)
(474,767)
(475,769)
(477,772)
(478,774)
(480,777)
(482,780)
(483,782)
(485,785)
(487,788)
(488,790)
(490,793)
(491,795)
(493,798)
(495,801)
(496,803)
(498,806)
(499,808)
(501,811)
(503,814)
(504,816)
(506,819)
(508,822)
(509,824)
(511,827)
(512,829)
(514,832)
(516,835)
(517,837)
(519,840)
(521,843)
(522,845)
(524,848)
(525,850)
(527,853)
(529,856)
(530,858)
(532,861)
(533,863)
(535,866)
(537,869)
(538,871)
(540,874)
(542,877)
(543,879)
(545,882)
(546,884)
(548,887)
(550,890)
(551,892)
(553,895)
(554,897)
(556,900)
(558,903)
(559,905)
(561,908)
(563,911)
(564,913)
(566,916)
(567,918)
(569,921)
(571,924)
(572,926)
(574,929)
(576,932)
(577,934)
(579,937)
(580,939)
(582,942)
(584,945)
(585,947)
(587,950)
(588,952)
(590,955)
(592,958)
(593,960)
(595,963)
(597,966)
(598,968)
(600,971)
(601,973)
(603,976)
(605,979)
(606,981)
(608,984)
(609,986)
(611,989)
(613,992)
(614,994)
(616,997)
.
.
.
...and so forth. (And the converses, so if you have (1,2), also (2,1) for all the above).

If you look at (616,997) for instance and divide the second # by the first, you'll notice that it's 1.61851, which is very close to the Golden Ratio. It appears that it's converging toward it.

Is there a formula to determine if, given two numbers, it will be in that pattern? For instance, (3,3) doesn't work.

2. The formula for the fibonacci sequence is:
$a_n=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5} }{2}\right)^n-\left( \frac{1-\sqrt{5}}{2}\right)^n\right)$
I can derive it if you wish.

3. Originally Posted by AfterShock
Since my professor could not figure this out, I thought I would look here for help. I have an idea of what is going on, but how to prove it.

I need to find a formula that given any pair of numbers (such as in the coordinate system below), I can tell if it's one of the below.

(1,2)
(3,5)
(4,7)
(6,10)
(8,13).
.
.
...and so forth. (And the converses, so if you have (1,2), also (2,1) for all the above).

If you look at (616,997) for instance and divide the second # by the first, you'll notice that it's 1.61851, which is very close to the Golden Ratio. It appears that it's converging toward it.

Is there a formula to determine if, given two numbers, it will be in that pattern? For instance, (3,3) doesn't work.
If you tell us how the entries are produced you might get a more usefull answer.

Your list looks familiar but I am not prepared to do the research necessary
to identify how its generated right now , maybe you will get lucky and someone
who recognises it will be able to help.

RonL

4. Hello, AfterShock!

Your wording is baffling . . .
Could you state the original problem?

I need to find a formula that given any pair of numbers
(such as in the coordinate system below), ?
I can tell if it's one of the below. ?

(1,2), (3,5), (4,7), (6,10), (8,13), (9,15), (11,18), (12,20), (14,23), (16,26), . . .

I don't recognize any coordinate system,
. . but you seem to have a set of points.

Given a pair of numbers, is it on the list?
. . I'd look for it on the list . . . what else can I do?

If I knew exactly what pairs appear on the list, it would help.
. . But there is no discernible pattern in the pairs.

You mentioned the Golden Ratio, which suggests a Fibonacci-type sequence.
. . But the pairs on the list follow no recognizable rule.

Where did your professor get that list?

5. Here is one way to get your pairs using the floor function.
$
\begin{array}{l}
\phi = \frac{{1 + \sqrt 5 }}{2} \\
a_n = \left\lfloor {n\phi } \right\rfloor \quad \& \quad b_n = \left\lfloor {n\phi ^2 } \right\rfloor \\
\left( {a_n ,b_n } \right) \\
\end{array}
$