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
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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.