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2019-02-24, 00:12
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#1 |
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Jan 2018
43 Posts |
Curious Fibonacci with three seeds..
If we use three Fibonacci seeds such as: Fn=2*(Fn-1+Fn-2)-Fn-3, we can create number of OEIS sequences
Code:
F0 0 0 0 1 0 0 0 1 1 1 F1 0 1 1 1 0 2 2 1 2 2 F2 1 0 1 1 2 0 2 2 1 2 2*(F1+F2)-F0 2 2 4 3 4 4 8 5 5 7 6 3 9 7 12 6 18 13 10 16 15 10 25 19 30 20 50 34 29 44 40 24 64 49 80 48 128 89 73 113 104 65 169 129 208 130 338 233 194 298 273 168 441 337 546 336 882 610 505 778 714 442 1156 883 1428 884 2312 1597 1325 2039 1870 1155 3025 2311 3740 2310 6050 4181 3466 5336 4895 3026 7921 6051 9790 6052 15842 10946 9077 13972 12816 7920 20736 15841 25632 15840 41472 28657 23761 36577 33552 20737 54289 41473 67104 41474 108578 75025 62210 95762 87841 54288 142129 108577 175682 108576 284258 196418 162865 250706 229970 142130 372100 284259 459940 284260 744200 514229 426389 656359 602070 372099 974169 744199 1204140 744198 1948338 1346269 1116298 1718368 1576239 974170 2550409 1948339 3152478 1948340 5100818 3524578 2922509 4498748 oeis A001654 A059929 A007598 A061646 A079472 MISSING A175395 A001519 A069921 A192921 
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2019-02-24, 03:52
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#2 |
Aug 2006
10111010110112 Posts |
These are linear recurrences (in particular, third-order linear recurrence relations with constant coefficients). There are infinitely many, which leads to a problem: which ones should be in the OEIS? I discuss this issue in a short essay here:
https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Imp |
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2019-02-24, 03:57
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#3 |
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Aug 2006
175B16 Posts |
FWIW, in the OEIS we would write
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3) and describe the sequence as having signature (2,2,-1), this being taken from the coefficients above. An example of such a sequence is A001654. But note that this sequence has much more going on than merely being arbitrary numbers following this progression: it has a plethora of interesting comments, references, and the like. |
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2019-02-24, 12:47
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#4 | |
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Jan 2018
43 Posts |
Quote:
It could just act as a formula for various sequences which can be created with it.. Further we may predict the next number easily in those sequences..  Code:
These 76 OEIS sequences can be generated with this formula: A061646 1 1 1 3 7 19 49 129 337 883 2311 6051 A128535 0 -1 2 2 9 20 56 143 378 986 2585 6764 A292612 4 -3 5 0 13 21 68 165 445 1152 3029 7917 A007598 0 1 1 4 9 25 64 169 441 1156 3025 7921 A001519 1 1 2 5 13 34 89 233 610 1597 4181 10946 A011783 1 1 2 5 13 34 89 233 610 1597 4181 10946 A128534 0 2 1 6 12 35 88 234 609 1598 4180 10947 A226205 1 0 3 5 16 39 105 272 715 1869 4896 12815 A001654 0 1 2 6 15 40 104 273 714 1870 4895 12816 A260259 -1 2 1 7 14 41 103 274 713 1871 4894 12817 A180666 0 1 2 6 15 41 106 279 729 1911 5001 13095 A192878 3 0 4 5 18 42 115 296 780 2037 5338 13970 A192921 1 2 2 7 16 44 113 298 778 2039 5336 13972 A175395 0 2 2 8 18 50 128 338 882 2312 6050 15842 A001906 0 1 3 8 21 55 144 377 987 2584 6765 17711 A192883 2 0 5 8 26 63 170 440 1157 3024 7922 20735 A059929 0 2 3 10 24 65 168 442 1155 3026 7920 20737 A192914 1 0 5 9 28 69 185 480 1261 3297 8636 22605 A192879 0 1 4 10 27 70 184 481 1260 3298 8635 22606 A099016 1 2 4 11 28 74 193 506 1324 3467 9076 23762 A236428 1 1 5 11 31 79 209 545 1429 3739 9791 25631 A079472 0 2 4 12 30 80 208 546 1428 3740 9790 25632 A192916 1 0 6 11 34 84 225 584 1534 4011 10506 27500 A122367 1 2 5 13 34 89 233 610 1597 4181 10946 28657 A081714 0 3 4 14 33 90 232 611 1596 4182 10945 28658 A192919 3 0 8 13 42 102 275 712 1872 4893 12818 33550 A129905 1 3 6 17 43 114 297 779 2038 5337 13971 36578 A014742 1 2 7 17 46 119 313 818 2143 5609 14686 38447 A120718 0 3 6 18 45 120 312 819 2142 5610 14685 38448 A001254 4 1 9 16 49 121 324 841 2209 5776 15129 39601 A047946 3 2 8 17 48 122 323 842 2208 5777 15128 39602 A005248 2 3 7 18 47 123 322 843 2207 5778 15127 39603 A110391 1 4 6 19 46 124 321 844 2206 5779 15126 39604 A292696 -1 6 4 21 44 126 319 846 2204 5781 15124 39606 A192917 0 5 6 22 51 140 360 949 2478 6494 16995 44500 A128533 0 4 7 22 54 145 376 988 2583 6766 17710 46369 A061647 1 3 9 23 61 159 417 1091 2857 7479 19581 51263 A069959 1 5 8 25 61 164 425 1117 2920 7649 20021 52420 A192920 0 5 8 26 63 170 440 1157 3024 7922 20735 54290 A248161 2 3 11 26 71 183 482 1259 3299 8634 22607 59183 A069921 1 5 10 29 73 194 505 1325 3466 9077 23761 62210 A215602 2 3 12 28 77 198 522 1363 3572 9348 24477 64078 A002878 1 4 11 29 76 199 521 1364 3571 9349 24476 64079 A121801 0 6 10 32 78 210 544 1430 3738 9792 25630 67106 A048575 2 5 13 34 89 233 610 1597 4181 10946 28657 75025 A264080 1 5 13 35 91 239 625 1637 4285 11219 29371 76895 A171089 6 4 16 34 96 244 646 1684 4416 11554 30256 79204 A189316 5 5 15 35 95 245 645 1685 4415 11555 30255 79205 A102714 2 5 14 36 95 248 650 1701 4454 11660 30527 79920 A054486 1 5 14 37 97 254 665 1741 4558 11933 31241 81790 A127546 2 6 14 38 98 258 674 1766 4622 12102 31682 82946 A201157 0 5 15 40 105 275 720 1885 4935 12920 33825 88555 A069960 1 10 13 45 106 289 745 1962 5125 13429 35146 92025 A025169 2 6 16 42 110 288 754 1974 5168 13530 35422 92736 A054492 1 6 17 45 118 309 809 2118 5545 14517 38006 99501 A099921 5 5 20 45 125 320 845 2205 5780 15125 39605 103680 A055267 1 7 20 53 139 364 953 2495 6532 17101 44771 117212 A014717 9 4 25 49 144 361 961 2500 6561 17161 44944 117649 A265802 1 11 19 59 145 389 1009 2651 6931 18155 47521 124421 A055273 1 8 23 61 160 419 1097 2872 7519 19685 51536 134923 A106729 5 10 25 65 170 445 1165 3050 7985 20905 54730 143285 A069961 1 17 20 73 169 464 1193 3145 8212 21521 56321 147472 A055849 1 9 26 69 181 474 1241 3249 8506 22269 58301 152634 A271357 3 10 27 71 186 487 1275 3338 8739 22879 59898 156815 A055850 1 10 29 77 202 529 1385 3626 9493 24853 65066 170345 A056123 1 11 32 85 223 584 1529 4003 10480 27437 71831 188056 A097512 11 15 34 87 227 594 1555 4071 10658 27903 73051 191250 A265804 1 19 29 95 229 619 1601 4211 11005 28831 75461 197579 A271358 4 13 35 92 241 631 1652 4325 11323 29644 77609 203183 A206981 4 16 36 100 256 676 1764 4624 12100 31684 82944 217156 A069962 1 26 29 109 250 689 1769 4666 12181 31925 83546 218761 A155110 8 16 40 104 272 712 1864 4880 12776 33448 87568 229256 A271359 5 16 43 113 296 775 2029 5312 13907 36409 95320 249551 A100545 7 19 50 131 343 898 2351 6155 16114 42187 110447 289154 A069963 1 37 40 153 349 964 2473 6525 17032 44641 116821 305892 A090692 30 42 146 346 942 2430 6398 16714 43794 114618 300110 785662 |
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2019-02-24, 16:21
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#5 |
Feb 2017
Nowhere
578410 Posts |
I point out that the characteristic polynomial f = x^3 - 2*x^2 - 2*x + 1 is reducible, with factorization (x + 1)*(x^2 - 3*x + 1). Thus, for example, the sequence
an = trace(Mod(x,f)^n) may be expressed as an = (-1)^n + trace(Mod(x, x^2 - 3*x + 1)^n). The sequence trace(Mod(x, x^2 - 3*x + 1)^n) consists of the even-order terms of the original Lucas sequence 2, 1, 3, 4, 7, etc., i.e. 2, 3, 7, etc with recurrence xn+1 = 3*xn - xn-1. The occurrence of the polynomial x^2 - 3*x + 1 as a factor may help explain the number of OEIS sequences that crop up. |
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2019-02-24, 23:27
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#6 |
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Jan 2018
538 Posts |
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