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Old 2019-02-24, 00:12   #1
MARTHA
 
Jan 2018

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Lightbulb 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
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Old 2019-02-24, 03:52   #2
CRGreathouse
 
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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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Old 2019-02-24, 03:57   #3
CRGreathouse
 
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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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Old 2019-02-24, 12:47   #4
MARTHA
 
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Quote:
Originally Posted by CRGreathouse View Post
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
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Old 2019-02-24, 16:21   #5
Dr Sardonicus
 
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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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Old 2019-02-24, 23:27   #6
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