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 2019-02-24, 00:12 #1 MARTHA   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 Am I the first one to notice that
 2019-02-24, 03:52 #2 CRGreathouse     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
 2019-02-24, 03:57 #3 CRGreathouse     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.
2019-02-24, 12:47   #4
MARTHA

Jan 2018

43 Posts

Quote:
 Originally Posted by CRGreathouse 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.
No one is disagreeing with you Sire, In fact I am not suggesting to create any sequence..
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

 2019-02-24, 16:21 #5 Dr Sardonicus     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.
 2019-02-24, 23:27 #6 MARTHA   Jan 2018 538 Posts

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