Table of Cumulative Filling of the Integers, Fibonacci-Wise

This is a companion table to the one giving cardinalities of the appearance of each {a,b} pair in 2^a*3^b*n+c formulas.  Here we present the contribution to the total density of odd integers provided by each formula's complete set of instantiations, and several running sums for a from 3 to 22 and b from 1 to 11 (which are the extremes of a and b completely covered in the file which was the basis for the cardinality table).  The table is annotated below it,  and additional indicators of the fraction of the integers reached at various depths into the cardinality table appear beneath the annotation.


    a--->       3          4          5          6          7          8          9         10         11         12         13         14         15         16         17         18         19         20         21         22    23     sum
   \  |__________________________________________________________________________________________________________________________________________________________________________________________________________________________________
   b  |          1          1          2          3          5          8         13         21         34         55         89        144        233        377        610        987       1597       2584       4181       6765    *
   _  | .083333333 .041666666 .020833333 .010416666 .005208333 .002604167 .001302083 .000651042 .000325521 .000162760 .000081380 .000040960 .000020345 .000010173 .000005086 .000002543 .000001272 .000000636 .000000318 .000000159          1/3
   1  | .083333333 .041666666 .041666666 .031250000 .026041666 .020833333 .016927083 .013671875 .011067708 .008951822 .007242838 .005859375 .004740397 .003835042 .003102620 .002510070 .002030690 .001642862 .001329104 .001075267      .333333333
      | .083333333 .125000000 .166666666 .197691666 .223958333 .244791666 .261718750 .275390625 .286458333 .295410156 .302652994 .308512369 .313252766 .317087809 .320190429 .322700500 .324731190 .326374053 .327703158 .328778425      .333333333

      |          2          2          4          6         10         16         26         42         68        110        178        288        466        754       1220       1974       3194      *5168      *8362     *13530
      | .027777777 .013888888 .006944444 .003472222 .001736111 .000868056 .000434028 .000217014 .000108507 .000054253 .000027127 .000013563 .000006782 .000003391 .000001695 .000000848 .000000424
   2  | .055555555 .027777777 .027777777 .020833333 .017361111 .013888888 .011284722 .009114583 .007378472 .005967881 .004828559 .003906250 .003160264 .002556695 .002068413 .001673380 .001353793 .001095241 .000886069 .000716845          2/9
      | .055555555 .083333333 .111111111 .131944444 .149305555 .163194444 .174479166 .183593749 .190972222 .196940104 .201768663 .205674913 .208835177 .211391873 .213460286 .215133666 .216487460 .217582702 .218468772 .219185617      .222222222
      | .138888888 .208333333 .277777777 .329636111 .373263888 .407986111 .436197916 .458984375 .477430555 .492350260 .504421657 .514187282 .522087944 .528479682 .533650716 .537834167 .541218651 .543956756 .546171930 .547964043      .555555555

      |          4          4          8         12         20         32         52         84        136        220        356        576        932       1508       2440       3948      *6388     *10336     *16724     *27060
      | .009259259 .004629630 .002314815 .001157407 .000578704 .000289352 .000144676 .000072338 .000036169 .000018084 .000009042 .000004521 .000002261 .000001130 .000000565 .000000283
   3  | .037037037 .018518518 .018518518 .013888888 .011574074 .009259259 .007523148 .006076388 .004918981 .003978587 .003219039 .002604166 .002106843 .001704463 .001378942 .001115587 .000902529 .000730161 .000590712 .000477896          4/27
      | .037037037 .055555555 .074074074 .087962962 .099537037 .108796296 .116319444 .122395833 .127314814 .131293402 .134512442 .137116608 .139223451 .140927915 .142306857 .143422444 .144324973 .145055135 .145645848 .146123744      .148148148
      | .175925925 .263888888 .351851851 .417824074 .472800925 .516782407 .552517361 .581380208 .604745370 .623643663 .638934100 .651303891 .661311396 .669407597 .675957573 .681256612 .685543625 .689011891 .691817778 .694087787      .703703703

      |          8          8         16         24         40         64        104        168        272        440        712       1152       1864       3016      *4880      *7896     *12776     *20672     *33448     *54120
      | .003086420 .001543210 .000771605 .000385802 .000192901 .000096451 .000048225 .000024113 .000012056 .000006028 .000003014 .000001507 .000000754 .000000377
   4  | .024691358 .012345679 .012345679 .009259259 .007716049 .006172839 .005015432 .004050925 .003279320 .002652391 .002146026 .001736111 .001404562 .001136308 .000919294 .000743724 .000601686 .000486774 .000393808 .000318597          8/81
      | .024691358 .037037037 .049382716 .058641975 .066358024 .072530864 .077546296 .081597222 .084876543 .087528935 .089674961 .091411072 .092815634 .093951943 .094871238 .095614963 .096216649 .096703423 .097097232 .097415829      .098765432
      | .200617283 .300925925 .401234567 .476466049 .539158950 .589313257 .630063657 .662977430 .689621913 .711172598 .728609616 .742714964 .754127031 .763359540 .770828812 .776871575 .781760274 .785715315 .788915010 .791503617      .802469135

      |         16         16         32         48         80        128        208        336        544        880       1424       2304      *3728      *6032      *9760     *15792     *25552     *41344     *66896    *108240
      | .001028807 .000514403 .000257202 .000128601 .000064300 .000032150 .000016075 .000008038 .000004019 .000002009 .000001005 .000000502
   5  | .016460905 .008230452 .008230452 .006172839 .051440332 .004115226 .003343621 .002700617 .002186213 .001768261 .001430684 .001157407 .000936374 .000757539 .000612863 .000495816 .000401124 .000324516 .000262539 .000212398         16/243
      | .016460905 .024691358 .032921810 .039094650 .044238683 .048353909 .051697530 .054398148 .056584362 .058352623 .059783307 .060940715 .061877089 .062634628 .063247492 .063743308 .064144432 .064468948 .064731488 .064943886      .065843621
      | .217078189 .325617283 .434156378 .515560699 .583397633 .637667188 .681761188 .717375578 .746206275 .769525221 .788392369 .803655679 .816004121 .825994169 .834076304 .840614883 .845904707 .850184264 .853646498 .856447504      .868312756

      |         32         32         64         96        160        256        416        672       1088       1760      *2848      *4608      *7456     *12064     *19520     *31584     *51104     *82688    *133792    *216480
      | .000342936 .000171468 .000085734 .000042867 .000021433 .000010717 .000005358 .000002679 .000001340 .000000670
   6  | .010973036 .005486968 .005486968 .004115226 .003429355 .002743484 .002229080 .001800411 .001457475 .001178840 .000953789 .000771604 .000624249 .000505026 .000408575 .000330544 .000267416 .000216344 .000175026 .000141599         32/729
      | .010973036 .016460905 .021947873 .026063100 .029492455 .032235939 .034465020 .036265432 .037722908 .038901748 .059783307 .040627143 .041251393 .041756419 .042164994 .042495539 .042762955 .042979299 .043154325 .043295924      .043895748
      | .228052126 .342078189 .456104252 .541623799 .612890089 .669903120 .716226208 .753641010 .783929183 .808426970 .828247907 .844282822 .857255514 .867750589 .876241299 .883110423 .888667662 .893163563 .893163563 .896800823      .912208504

      |         64         64        128        192        320        512        832       1344      *2176      *3520      *5696      *9216     *14912     *24128     *39040     *63168    *102208    *165376    *267584    *432960
      | .000114312 .000057156 .000028578 .000014289 .000007144 .000003572 .000001786 .000000893
   7  | .007315957 .003657978 .003657978 .002743484 .002286236 .001828989 .001486053 .001200274 .000971650 .000785893 .000635859 .000514403 .000416166 .000336684 .000272383 .000220362 .000178277 .000144229 .000116684 .000094399         64/2187
      | .007315957 .010973936 .014631915 .017375400 .019661636 .021490626 .022976680 .024176954 .025148605 .025934499 .026570358 .027084762 .027500928 .027837612 .028109996 .028330359 .028508636 .028652866 .028769550 .028863949      .029263832
      | .235368084 .353052126 .470736168 .558999199 .632551726 .691393747 .7392022889.777917965 .809077789 .834361470 .854818266 .871367584 .884756443 .895588201 .904351295 .911440782 .917176299 .921816429 .925570373 .928607378      .941472336

      |        128        128        256        384        640       1024      *1664      *2688      *4352      *7040     *11392     *18432     *29824     *48256     *78080    *126336    *204416    *330752    *535168    *865920
      | .000038104 .000019052 .000009526 .000004763 .000002381 .000001691
   8  | .004877305 .002438652 .002438652 .001828989 .001524157 .001219326 .000990702 .000800182 .000647767 .000523929 .000423906 .000342935 .000277444 .000224456 .000181589 .000146908 .000118851 .000096152 .000077789 .000062932        128/6561
      | .004877305 .007315957 .009754610 .011583600 .013197757 .014327084 .015317786 .016117969 .016765736 .017289666 .017713572 .018056508 .018333952 .018558408 .018739997 .018886906 .019005757 .019101910 .019179700 .019242633      .019509221
      | .240245389 .360368084 .480490778 .570582799 .645659484 .705720831 .754520676 .793935935 .825843526 .851651136 .872531839 .889424093 .903090395 .914146610 .923091293 .930327688 .936182057 .940918340 .944750073 .947850011      .960981557

      |        256        256        512        768      *1280      *2048      *3328      *5376      *8704     *14080     *22784     *36864     *59648     *96512    *156160    *252672    *408832    *661504   *1070336   *1731840
      | .000012701 .000006351 .000003175 .000001588
   9  | .003251536 .001625768 .001625768 .001219326 .001016105 .000812884 .000660468 .000533455 .000431844 .000349286 .000282604 .000228623 .000184962 .000149637 .000121059 .000097939 .000079234 .000064101 .000051859 .000041955       256/19683
      | .003251536 .004877305 .006503073 .007722400 .008738505 .009551389 .010211857 .010745313 .011177157 .011526444 .011809048 .012037672 .012222635 .012372272 .012493331 .012591270 .012670505 .012734607 .012786466 .012828422      .013006147
      | .243496926 .365245389 .486993852 .578305199 .654397989 .715272220 .764732534 .804681248 .837020684 .863177580 .884340887 .901461765 .915313030 .926518882 .935584625 .942918959 .948852562 .953652947 .957536540 .960678433      .973987704

      |        512        512       1024      *1526      *2560      *4096      *6656     *10752     *17408     *28160     *45568     *73728    *119296    *193024    *312320    *505344    *817664   *1323008   *2140672   *3463680
      | .000004234 .000002117 .000001058
  10  | .002167691 .001083845 .001083845 .000812884 .000677403 .000541922 .000440312 .000355636 .000287896 .000232857 .000188402 .000152415 .000123308 .000099758 .000080706 .000065292 .000052822 .000042734 .000034573 .000027970       512/59049
      | .002167691 .003251536 .004335382 .005148266 .005825670 .006367593 .006807905 .007163542 .007451438 .007684296 .007872698 .008025114 .008148423 .008248181 .008328887 .008394180 .008447003 .008489738 .008524311 .008552281      .008670765
      | .245664617 .368496926 .491329235 .583453466 .660223659 .721639813 .771540439 .811844790 .844472122 .870861876 .892213586 .909486879 .923461454 .934767064 .943913513 .951313140 .957299566 .962142685 .966060851 .969230714      .982658469

              1024      *1024      *2048      *3052      *5120      *8192    *13312      *21504     *34816     *56320     *91136    *147456    *238592    *386048    *624640   *1010688   *1635328   *2646016   *4281344   *6927360
      | .000001411
  11  | .001445127 .000722563 .000722563 .000541922 .000451602 .000361281 .000293541 .000237091 .000191930 .000155238 .000125601 .000101610 .000082205 .000066505 .000053804 .000043528 .000035215 .000028489 .000023048 .000018646      1024/177147
      | .001445127 .002167691 .002890254 .003432177 .003883780 .004245062 .004538603 .004775694 .004967625 .005122864 .005248465 .005350076 .005432282 .005498787 .005552591 .005596120 .005631335 .005659825 .005682874 .005701520      .005780513
      | .247109745 .370664617 .494219490 .586885644 .664107439 .725884875 .776079042 .816620485 .849439748 .875984740 .897462052 .914836956 .928893736 .940265852 .949466105 .956909260 .962930901 .967802511 .971743725 .974932235      .988438980

             *
               1/4        1/8       2/16       3/32       5/64      8/128     13/256     21/512    34/1024    55/2048    89/4096   144/8192  233/16384   377/2^15   610/2^16   987/2^17  1597/2^18  2584/2^19  4181/2^20  6765/2^21
      | .250000000 .125000000 .125000000 .093750000 .078125000 .062500000 .050781250 .041015625 .033203125 .026855469 .021728516 .017578125 .014221191 .011505127 .009307861 .007530212 .006092072 .004928589 .003987312 .003225803
      | .250000000 .375000000 .500000000 .593750000 .671875000 .734375000 .785156250 .826171875 .859375000 .886230469 .907958984 .925537109 .939758301 .951263428 .960571289 .968101501 .974193573 .979122162 .983109474 .986335278


  

The zeroth row in each cell gives the cardinality of the appearance of the given coefficient pair in the formulas;  the first row in each cell the contribution of a single formula out of the set sharing and (a,b) pair;  the second row the contribution of all formulas sharing the exponent pair;  the third row a running sum across the rows;  and the fourth row gives the 2-D running sum at its position (e.g. the entry in (x,y) gives the total for the contributions for a<=x and b<=y). The first row of the cells are filled only for those {a,b} pairs which appeared in the cardinality table obtained experimentally. 

An extra row and column of cells have been added to show what the sums at each stage would be if the rows/columns were infinitely long instead of being truncated after a=22 or b=11.  The cell of the final column shows the formula used for the row sum, what the infinite sum across its row would be,  and the running sum of the contributions of all the infinite rows to date.  The final row shows the formula used for the column sum, what the infinite sum down its column would be,  and the running sum of the contributions of all the infinite columns to date. 

This gives an impression of the rate of coverage of the integers by the small left descent assemblages.  We see,  for example,  that over half the odd integers are covered by formulas with a<=8 and b<=3 or a<=6 and b<=5.  Using complete rows (with a to infinity) or complete columns (with b to infinity),  over 98 percent of the odd integers are covered by b<=11 or a<=22.

Three series of single- or double-finite sums provide alternative ways of getting at the coverage of the integers by the abstract predecessor tree or by the cardinality table.


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