Sequence (put there some numbers as 1, 2, 3, 5, 8, 13... or a sequence named EJS_SS_TR with P=+ N=-, example EJS_N3.14159P1N2.718_P1P0.577N1) :


Work memorization field (not used for the request) :


Number :  Digits after the comma :   


Choose Database and/or Calculs:
  


EJS_P3.14159P0.5772156000000001P2.7182818_P3.14159P0.5772156000000001P2.7182818
Linear   Recurrence   Equation:
a(0)=3.14159   a(1)=0.5772156000000001   a(2)=2.7182818  
a(n)=3.14159.a(n-3)   +   0.5772156000000001.a(n-2)   +   2.7182818.a(n-1)  

3.14159,   0.5772156000000001,   2.7182818,   17.5918215211946001,   51.2019376868716762,   157.8752960734686318,   513.9703917419812401,   1649.0999407770219462,   5275.3695348382877314,   16906.5114495936476811,   54182.4838491411179463,   173615.0100531509316920,   556322.8242636984231880,   1782674.6497530686334581,   5712357.4479790846951659,   18304523.1249765925782012,   58654546.7529704566505316,   187951128.2591425416397494,   602265757.4372319678060892,   1929884908.0072846938274474,   6184073597.0971022409026911,   19816086464.8631892638318495,   63498158064.6031151337622361,   203471865410.9060442108528429,   652000015041.8197004613376011,   2089252087758.3566303185621993,   6694745683304.1649942302617756,   21452470971196.1986070940750348,   68741746518847.5507582098522938,   220274285456729.7845468019580597,   705841257905358.0336271588926889,   2261779582343864.7209314600595474,   7247588351931532.5875835009777898,   23223985807060737.1311871263054248,   74418343120000346.6731416416989596,   238464225681810791.7591369608179527,   764128634768579686.9933979170268979,   2448554154418101113.2068731926589512,   7846083989423960473.4611399708221923,   25141789842800103627.5661774279256918,   80563705072692490788.8402540785648157,   258156265549187864701.4196604138507474,   827229301112324002696.6025322423083447,   2650752307572403923022.6135640952219802,   8494002553647871719402.1555905148883714,   27217963434482789206162.2884309397414762,   87216542359371836065514.0112802595421125,   279474446331537185307531.7949520467428156,   895540731602121321286190.0101179619024766,   2869647699407436258796303.8412394634301362,   9195425320277956182229445.5492311009896009,   29465584516966730804638900.6746993612543300,   94418761578313296792062970.2388048578060695,   302553052455111998378389937.8300539103736717,   969493223801516840019837265.7126993723983155,   3106619164374512284154899005.5308587898444120,   9954770590985427349551733151.7425510489371567,   31898810982549471562414391900.2969004136148230,   102215729915649679228811767208.1958722607451571,   327537457364938852878058350704.3707031431630152,   1049552608640756525464508176568.8688303833571034

\(EJSGF(x) = hidden\)


First derivative sequence of EJS_P3.14159P0.5772156000000001P2.7182818_P3.14159P0.5772156000000001P2.7182818 (by sum of 3 elements starting in 4):
3.1415900000000000, 0.5772156000000001, 2.7182818000000000, 0.5772156000000001, 2.7182818000000000, 17.5918215211946001, 2.7182818000000000, 17.5918215211946001, 51.2019376868716762, 17.5918215211946001, 51.2019376868716762, 157.8752960734686318, 51.2019376868716762, 157.8752960734686318, 513.9703917419812401, 157.8752960734686318, 513.9703917419812401, 1649.0999407770219462, 513.9703917419812401, 1649.0999407770219462, 5275.3695348382877314, 1649.0999407770219462, 5275.3695348382877314, 16906.5114495936476811, 5275.3695348382877314, 16906.5114495936476811, 54182.4838491411179463, 16906.5114495936476811, 54182.4838491411179463, 173615.0100531509316920, 54182.4838491411179463, 173615.0100531509316920, 556322.8242636984231880, 173615.0100531509316920, 556322.8242636984231880, 1782674.6497530686334581, 556322.8242636984231880, 1782674.6497530686334581, 5712357.4479790846951659, 1782674.6497530686334581, 5712357.4479790846951659, 18304523.1249765925782012, 5712357.4479790846951659, 18304523.1249765925782012, 58654546.7529704566505316, 18304523.1249765925782012, 58654546.7529704566505316, 187951128.2591425416397494, 58654546.7529704566505316, 187951128.2591425416397494, 602265757.4372319678060892, 187951128.2591425416397494, 602265757.4372319678060892, 1929884908.0072846938274474, 602265757.4372319678060892, 1929884908.0072846938274474, 6184073597.0971022409026911, 1929884908.0072846938274474, 6184073597.0971022409026911, 19816086464.8631892638318495, 6184073597.0971022409026911, 19816086464.8631892638318495, 63498158064.6031151337622361, 19816086464.8631892638318495, 63498158064.6031151337622361, 203471865410.9060442108528429, 63498158064.6031151337622361, 203471865410.9060442108528429, 652000015041.8197004613376011, 203471865410.9060442108528429, 652000015041.8197004613376011, 2089252087758.3566303185621993, 652000015041.8197004613376011, 2089252087758.3566303185621993, 6694745683304.1649942302617756, 2089252087758.3566303185621993, 6694745683304.1649942302617756

Second derivative sequence of EJS_P3.14159P0.5772156000000001P2.7182818_P3.14159P0.5772156000000001P2.7182818 (by sum of 3 elements starting in 4):
9.8695877281000000, 0.3331778488833601, 7.3890559442112400, 1.8133747568040003, 1.5690346601560802, 47.8195282699115957, 8.5397269200620000, 10.1542738144492555, 139.1812953389573763, 55.2662905727697437, 29.5545571830902518, 429.1495439861212446, 160.8554954176991892, 91.1280837482248561, 1397.1163616110979009, 495.9794513914483189, 296.6717280515827980, 4482.7183553952566145, 1614.6842429926908440, 951.8862117755733537, 14339.9409948253834834, 5180.7958829456843759, 3045.0255912734036833, 45956.6623749220298871, 16573.0481769526163540, 9758.7021502840687930, 147283.2597259142465450, 53113.3273049289076184, 31274.9749244723052982, 471934.5220342972102714, 170219.1494356232447289, 100213.2921968355642886, 1512242.2081206098244406, 545427.1794328784354942, 321118.2128010652991917, 4845812.0557451409604800, 1747738.2214785923293031, 1028987.6175620075413701, 15527797.2859359927075280, 5600432.8529177428881856, 3297261.8317497167310067, 49756852.0683029970313393, 17945885.0349966126874862, 10565656.2982972407014342, 159439586.9258486882508290, 57505306.8041952134777511, 33856319.3967438997832652, 510904131.2362928545450729, 184268537.5336644569087435, 108488323.2687779364532257, 1637128047.2048423004654782, 590465385.0276396173900203, 347637190.5385863728629482, 5245971021.5308762506497226, 1892072080.9072335777399317, 1113959675.1063698319069171, 16810054708.8495858541850007, 6062907128.1466055212813704, 3569543751.7925627466513510, 53865707184.6639468658295147, 19427823771.9042854289974853, 11438154238.4678866904449057, 172605887400.5338720914104519, 62254019077.1495467593615000, 36652127406.1547322136194558, 553093868558.5154214883566453, 199485178394.1765004730761033, 117447134876.2753991999804913, 1772319774487.9047306455056047, 639225177656.2483194323731827, 376344579882.3730486936127644, 5679175925765.5436261042758285, 2048316727255.2303527723335942, 1205948897386.6926863085158468, 18198205346554.2755680132255938, 6563573466380.7756062424918196

EJS_P3.14159P0.5772156000000001P2.7182818_P3.14159P0.5772156000000001P2.7182818 (chart roots 3, -0.1837, -1.6968, 1.4256, 1.1477, -1.9844, -0.1747, 2.1145, -0.8691, -1.7256, 1.7422, 0.8962, -2.2214, 0.1872, 2.173, -1.2682, -1.5876, 2.0806, 0.5876)

The 3 roots are :

Root(0) = 0.312073406009750
Root(1) = -0.247903496571190+0.979044807292145i
Root(2) = -0.247903496571190-0.979044807292145i

Some examples

Fibonacci numbers Fn
EJS_P1P1_P1P1
Linear   Recurrence   Equation:
a(0)=1   a(1)=1  
a(n)=1.a(n-2)   +   1.a(n-1)  

1,   1,   2,   3,   5,   8,   13,   21,   34,   55,   89,   144,   233,   377,   610,   987,   1597,   2584,   4181,   6765,   10946,   17711,   28657,   46368,   75025,   121393,   196418,   317811,   514229,   832040,   1346269,   2178309,   3524578,   5702887,   9227465,   14930352,   24157817,   39088169,   63245986,   102334155,   165580141,   267914296,   433494437,   701408733,   1134903170,   1836311903,   2971215073,   4807526976,   7778742049,   12586269025,   20365011074,   32951280099,   53316291173,   86267571272,   139583862445,   225851433717,   365435296162,   591286729879,   956722026041,   1548008755920,   2504730781961
\(EJSGF(x) = hidden\)
Lucas numbers Ln
EJS_P1P3_P1P1
Linear   Recurrence   Equation:
a(0)=1   a(1)=3  
a(n)=1.a(n-2)   +   1.a(n-1)  

1,   3,   4,   7,   11,   18,   29,   47,   76,   123,   199,   322,   521,   843,   1364,   2207,   3571,   5778,   9349,   15127,   24476,   39603,   64079,   103682,   167761,   271443,   439204,   710647,   1149851,   1860498,   3010349,   4870847,   7881196,   12752043,   20633239,   33385282,   54018521,   87403803,   141422324,   228826127,   370248451,   599074578,   969323029,   1568397607,   2537720636,   4106118243,   6643838879,   10749957122,   17393796001,   28143753123,   45537549124,   73681302247,   119218851371,   192900153618,   312119004989,   505019158607,   817138163596,   1322157322203,   2139295485799,   3461452808002,   5600748293801
\(EJSGF(x) = hidden\)
Pell numbers Pn
EJS_P0P1_P1P2
Linear   Recurrence   Equation:
a(0)=0   a(1)=1  
a(n)=1.a(n-2)   +   2.a(n-1)  

0,   1,   2,   5,   12,   29,   70,   169,   408,   985,   2378,   5741,   13860,   33461,   80782,   195025,   470832,   1136689,   2744210,   6625109,   15994428,   38613965,   93222358,   225058681,   543339720,   1311738121,   3166815962,   7645370045,   18457556052,   44560482149,   107578520350,   259717522849,   627013566048,   1513744654945,   3654502875938,   8822750406821,   21300003689580,   51422757785981,   124145519261542,   299713796309065,   723573111879672,   1746860020068409,   4217293152016490,   10181446324101389,   24580185800219268,   59341817924539925,   143263821649299118,   345869461223138161,   835002744095575440,   2015874949414289041,   4866752642924153522,   11749380235262596085,   28365513113449345692,   68480406462161287469,   165326326037771920630,   399133058537705128729,   963592443113182178088,   2326317944764069484905,   5616228332641321147898,   13558774610046711780701,   32733777552734744709300
\(EJSGF(x) = hidden\)
Pell-Lucas numbers Qn
EJS_P2P2_P1P2
Linear   Recurrence   Equation:
a(0)=2   a(1)=2  
a(n)=1.a(n-2)   +   2.a(n-1)  

2,   2,   6,   14,   34,   82,   198,   478,   1154,   2786,   6726,   16238,   39202,   94642,   228486,   551614,   1331714,   3215042,   7761798,   18738638,   45239074,   109216786,   263672646,   636562078,   1536796802,   3710155682,   8957108166,   21624372014,   52205852194,   126036076402,   304278004998,   734592086398,   1773462177794,   4281516441986,   10336495061766,   24954506565518,   60245508192802,   145445522951122,   351136554095046,   847718631141214,   2046573816377474,   4940866263896162,   11928306344169798,   28797478952235758,   69523264248641314,   167844007449518386,   405211279147678086,   978266565744874558,   2361744410637427202,   5701755387019728962,   13765255184676885126,   33232265756373499214,   80229786697423883554,   193691839151221266322,   467613464999866416198,   1128918769150954098718,   2725451003301774613634,   6579820775754503325986,   15885092554810781265606,   38350005885376065857198,   92585104325562912980002
\(EJSGF(x) = hidden\)
Padovan numbers Pn
EJS_P1P1P1_P1P1P0
Linear   Recurrence   Equation:
a(0)=1   a(1)=1   a(2)=1  
a(n)=1.a(n-3)   +   1.a(n-2)   +   0.a(n-1)  

1,   1,   1,   2,   2,   3,   4,   5,   7,   9,   12,   16,   21,   28,   37,   49,   65,   86,   114,   151,   200,   265,   351,   465,   616,   816,   1081,   1432,   1897,   2513,   3329,   4410,   5842,   7739,   10252,   13581,   17991,   23833,   31572,   41824,   55405,   73396,   97229,   128801,   170625,   226030,   299426,   396655,   525456,   696081,   922111,   1221537,   1618192,   2143648,   2839729,   3761840,   4983377,   6601569,   8745217,   11584946,   15346786
\(EJSGF(x) = hidden\)
Perrin Fn numbers
EJS_P3P0P2_P1P1P0
Linear   Recurrence   Equation:
a(0)=3   a(1)=0   a(2)=2  
a(n)=1.a(n-3)   +   1.a(n-2)   +   0.a(n-1)  

3,   0,   2,   3,   2,   5,   5,   7,   10,   12,   17,   22,   29,   39,   51,   68,   90,   119,   158,   209,   277,   367,   486,   644,   853,   1130,   1497,   1983,   2627,   3480,   4610,   6107,   8090,   10717,   14197,   18807,   24914,   33004,   43721,   57918,   76725,   101639,   134643,   178364,   236282,   313007,   414646,   549289,   727653,   963935,   1276942,   1691588,   2240877,   2968530,   3932465,   5209407,   6900995,   9141872,   12110402,   16042867,   21252274
\(EJSGF(x) = hidden\)

Sequence with real numbers Pi, e, γ
EJS_P3.14159P0.5772156000000001P2.7182818_P3.14159P0.5772156000000001P2.7182818
Linear   Recurrence   Equation:
a(0)=3.14159   a(1)=0.5772156000000001   a(2)=2.7182818  
a(n)=3.14159.a(n-3)   +   0.5772156000000001.a(n-2)   +   2.7182818.a(n-1)  

3.14159,   0.5772156000000001,   2.7182818,   17.5918215211946001,   51.2019376868716762,   157.8752960734686318,   513.9703917419812401,   1649.0999407770219462,   5275.3695348382877314,   16906.5114495936476811,   54182.4838491411179463,   173615.0100531509316920,   556322.8242636984231880,   1782674.6497530686334581,   5712357.4479790846951659,   18304523.1249765925782012,   58654546.7529704566505316,   187951128.2591425416397494,   602265757.4372319678060892,   1929884908.0072846938274474,   6184073597.0971022409026911,   19816086464.8631892638318495,   63498158064.6031151337622361,   203471865410.9060442108528429,   652000015041.8197004613376011,   2089252087758.3566303185621993,   6694745683304.1649942302617756,   21452470971196.1986070940750348,   68741746518847.5507582098522938,   220274285456729.7845468019580597,   705841257905358.0336271588926889,   2261779582343864.7209314600595474,   7247588351931532.5875835009777898,   23223985807060737.1311871263054248,   74418343120000346.6731416416989596,   238464225681810791.7591369608179527,   764128634768579686.9933979170268979,   2448554154418101113.2068731926589512,   7846083989423960473.4611399708221923,   25141789842800103627.5661774279256918,   80563705072692490788.8402540785648157,   258156265549187864701.4196604138507474,   827229301112324002696.6025322423083447,   2650752307572403923022.6135640952219802,   8494002553647871719402.1555905148883714,   27217963434482789206162.2884309397414762,   87216542359371836065514.0112802595421125,   279474446331537185307531.7949520467428156,   895540731602121321286190.0101179619024766,   2869647699407436258796303.8412394634301362,   9195425320277956182229445.5492311009896009,   29465584516966730804638900.6746993612543300,   94418761578313296792062970.2388048578060695,   302553052455111998378389937.8300539103736717,   969493223801516840019837265.7126993723983155,   3106619164374512284154899005.5308587898444120,   9954770590985427349551733151.7425510489371567,   31898810982549471562414391900.2969004136148230,   102215729915649679228811767208.1958722607451571,   327537457364938852878058350704.3707031431630152,   1049552608640756525464508176568.8688303833571034
\(EJSGF(x) = hidden\)
A299277 - OEIS
EJS_P2P2P4P10P16P28P48P76P110P144P182_P1N3P4N4P4N4P4N4P4N4P3
Linear   Recurrence   Equation:
a(0)=2   a(1)=2   a(2)=4   a(3)=10   a(4)=16   a(5)=28   a(6)=48   a(7)=76   a(8)=110   a(9)=144   a(10)=182  
a(n)=1.a(n-11)   -3.a(n-10)   +   4.a(n-9)   -4.a(n-8)   +   4.a(n-7)   -4.a(n-6)   +   4.a(n-5)   -4.a(n-4)   +   4.a(n-3)   -4.a(n-2)   +   3.a(n-1)  

2,   2,   4,   10,   16,   28,   48,   76,   110,   144,   182,   222,   264,   310,   356,   408,   468,   536,   610,   684,   762,   842,   924,   1010,   1096,   1188,   1288,   1396,   1510,   1624,   1742,   1862,   1984,   2110,   2236,   2368,   2508,   2656,   2810,   2964,   3122,   3282,   3444,   3610,   3776,   3948,   4128,   4316,   4510,   4704,   4902,   5102,   5304,   5510,   5716,   5928,   6148,   6376,   6610,   6844,   7082
\(EJSGF(x) = hidden\)
Copyrightdepot.com (8695)
Contains infinite sequences.

Licence Creative Commons
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