845.sagews

sage: [lucas_number1(n, n*2, n) for n in xrange(1, 19)] # nnnnnnn

[1, 4, 33, 448, 8525, 208656, 6242257, 220692480, 9002380905, 416161000000, 21500535742961, 1227679976079360, 76774090373079541, 5218477281302905088, 383076988778810390625, 30203242469020785442816, 2545523388139209022093201, 228373795143170324284916736]

sage: [lucas_number2(n, n*2, n)/2 for n in xrange(1, 19)] #

[1, 6, 81, 1552, 38125, 1142856, 40454449, 1651511296, 76387735017, 3948049900000, 225499521276161, 14104969064976384, 958908081417478933, 70401107481746122368, 5551312967189525390625, 467906620335928341692416, 41981847207112401510178193, 3994909840569244440370705920]

sage: [lucas_number1(n, n+2, n) for n in xrange(1, 19)] #nnnnn

[1, 4, 22, 168, 1691, 21344, 325277, 5819520, 119663824, 2781822912, 72160819061, 2066629591040, 64773461683009, 2205470927608832, 81068087935136434, 3199611863135453184, 134959750240331326711, 6058738873148643783680]

sage: [lucas_number1(n*2, n, n) for n in xrange(1, 19)] #nnnnnnnnnnnn

[1, 0, 0, 1024, 171875, 36391680, 10416995407, 3949625081856, 1928108090927376, 1181184400000000000, 888465061334204807629, 805609217870451588464640, 867014843538402334118853133, 1092939480442258112968753741824, 1595451656240607074227294921875000, 2670487990343629158238318161943330816, 5081040163691790986862745971008008411727, 10905552816990002173789189198764989774561280]

sage: [lucas_number1(n+2, n, n) for n in xrange(1, 19)] #n

[0, 0, 9, 192, 3625, 72576, 1599066, 38961152, 1045816839, 30744000000, 983714163641, 34061680852992, 1269604047146604, 50702279645810688, 2160292443992578125, 97832935572960706560, 4693266672923224844897, 237771449355800744902656]

sage: [lucas_number1(3*n, n, 0) for n in xrange(1, 19)] # nnnnnnnn

[1, 32, 6561, 4194304, 6103515625, 16926659444736, 79792266297612001, 590295810358705651712, 6461081889226673298932241, 100000000000000000000000000000, 2111377674535255285545615254209921, 59066822915424320448445358917464096768, 2137210935411428674141543654682486133398329, 98005277522749794820791054154499091349964324864, 5597774487475881147025802420102991163730621337890625, 392318858461667547739736838950479151006397215279002157056, 33300140732146818380750772381422989832214186835186851059977249, 3384096747052124176096919006498525214763459451708514090419295879168]

sage: [lucas_number1(3*n, 1, 0) for n in xrange(1, 20)] # nem kell

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

sage: [lucas_number1(3*n, 2, 0)/8 for n in xrange(2, 21)] #

[4, 32, 256, 2048, 16384, 131072, 1048576, 8388608, 67108864, 536870912, 4294967296, 34359738368, 274877906944, 2199023255552, 17592186044416, 140737488355328, 1125899906842624, 9007199254740992, 72057594037927936]

sage: [lucas_number1(3*n, 2, 0) for n in xrange(1, 20)] #

[4, 32, 256, 2048, 16384, 131072, 1048576, 8388608, 67108864, 536870912, 4294967296, 34359738368, 274877906944, 2199023255552, 17592186044416, 140737488355328, 1125899906842624, 9007199254740992, 72057594037927936]

sage: [lucas_number1(3*n, 2, 0)/4 for n in xrange(1, 20)] #A001018     Powers of 8.  (Other) sage: [lucas_number1(n, 8, 0) for n in xrange(1, 22)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]

[1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 134217728, 1073741824, 8589934592, 68719476736, 549755813888, 4398046511104, 35184372088832, 281474976710656, 2251799813685248, 18014398509481984]

sage: [lucas_number1(3*n, 2, 1) for n in xrange(1, 20)] #

[3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57]

sage: [lucas_number1(2*n, 2, 1) for n in xrange(1, 26)] #

[2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50]

sage: [lucas_number1(2*n, 2, 0) for n in xrange(1, 26)] # ok 
A004171 		2^(2n+1).

[2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728, 536870912, 2147483648, 8589934592, 34359738368, 137438953472, 549755813888, 2199023255552, 8796093022208, 35184372088832, 140737488355328, 562949953421312]

sage: [lucas_number1(2*n, 2, 0)/2 for n in xrange(1, 26)] #A000302    Powers of 4.    	

(Other) sage: [lucas_number1(n, 4, 0) for n in xrange(1, 26)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]

[1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304, 16777216, 67108864, 268435456, 1073741824, 4294967296, 17179869184, 68719476736, 274877906944, 1099511627776, 4398046511104, 17592186044416, 70368744177664, 281474976710656]

sage: [lucas_number1(3*n, 0, 2) for n in xrange(1, 20)] #nnnnnnnnn

[-2, 0, 16, 0, -128, 0, 1024, 0, -8192, 0, 65536, 0, -524288, 0, 4194304, 0, -33554432, 0, 268435456]

sage: [lucas_number2(3*n, 2, 0)/2 for n in xrange(0, 20)] # ok A092811  	 	 Expansion of (1-4x)/(1-8x).

[1, 4, 32, 256, 2048, 16384, 131072, 1048576, 8388608, 67108864, 536870912, 4294967296, 34359738368, 274877906944, 2199023255552, 17592186044416, 140737488355328, 1125899906842624, 9007199254740992, 72057594037927936]

sage: [lucas_number1(3*n, 2, 0)/2 for n in xrange(1, 20)] # ok A013730  	 	 2^(3n+1).

[2, 16, 128, 1024, 8192, 65536, 524288, 4194304, 33554432, 268435456, 2147483648, 17179869184, 137438953472, 1099511627776, 8796093022208, 70368744177664, 562949953421312, 4503599627370496, 36028797018963968]

sage: [lucas_number2(3*n, 2, 0)/8 for n in xrange(1, 20)] #nem kell A001018  	 	 Powers of 8.

[1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 134217728, 1073741824, 8589934592, 68719476736, 549755813888, 4398046511104, 35184372088832, 281474976710656, 2251799813685248, 18014398509481984]

sage: [lucas_number1(3*n, 2, 0)/8 for n in xrange(2, 21)] #nem kell

[4, 32, 256, 2048, 16384, 131072, 1048576, 8388608, 67108864, 536870912, 4294967296, 34359738368, 274877906944, 2199023255552, 17592186044416, 140737488355328, 1125899906842624, 9007199254740992, 72057594037927936]

sage: [lucas_number1(5*n, 2, 0) for n in xrange(1, 16)] #ok A013825  	 	 2^(5n+4).

[16, 512, 16384, 524288, 16777216, 536870912, 17179869184, 549755813888, 17592186044416, 562949953421312, 18014398509481984, 576460752303423488, 18446744073709551616, 590295810358705651712, 18889465931478580854784]

sage: [lucas_number1(4*n, 2, 0) for n in xrange(1, 17)] #ok A013777  	 	 2^(4n+3).

[8, 128, 2048, 32768, 524288, 8388608, 134217728, 2147483648, 34359738368, 549755813888, 8796093022208, 140737488355328, 2251799813685248, 36028797018963968, 576460752303423488, 9223372036854775808]

sage: [lucas_number1(6*n, 2, 0) for n in xrange(1, 16)] #nnnnnnnnnnn

[32, 2048, 131072, 8388608, 536870912, 34359738368, 2199023255552, 140737488355328, 9007199254740992, 576460752303423488, 36893488147419103232, 2361183241434822606848, 151115727451828646838272, 9671406556917033397649408, 618970019642690137449562112]

sage: [lucas_number2(5*n, 2, 0) for n in xrange(1, 26)] #

[32, 1024, 32768, 1048576, 33554432, 1073741824, 34359738368, 1099511627776, 35184372088832, 1125899906842624, 36028797018963968, 1152921504606846976, 36893488147419103232, 1180591620717411303424, 37778931862957161709568, 1208925819614629174706176, 38685626227668133590597632, 1237940039285380274899124224, 39614081257132168796771975168, 1267650600228229401496703205376, 40564819207303340847894502572032, 1298074214633706907132624082305024, 41538374868278621028243970633760768, 1329227995784915872903807060280344576, 42535295865117307932921825928971026432]

sage: [lucas_number2(3*n, 2, 0)*lucas_number1(3*n, 2, 0)/32 for n in xrange(1, 15)] # ok A089357  	 	 2^(6n).

[1, 64, 4096, 262144, 16777216, 1073741824, 68719476736, 4398046511104, 281474976710656, 18014398509481984, 1152921504606846976, 73786976294838206464, 4722366482869645213696, 302231454903657293676544]

sage: [lucas_number2(3*n, 2, 0)*lucas_number1(3*n, 2, 0)/2^9 for n in xrange(2, 15)] #ok  A013734  	 	 4^(3n+1).

[4, 256, 16384, 1048576, 67108864, 4294967296, 274877906944, 17592186044416, 1125899906842624, 72057594037927936, 4611686018427387904, 295147905179352825856, 18889465931478580854784]

sage: [lucas_number2(2*n, 2, 0)*lucas_number1(2*n, 2, 0) for n in xrange(1, 15)] #nem kell, már volt másképp A013777   	  	    2^(4n+3).

[8, 128, 2048, 32768, 524288, 8388608, 134217728, 2147483648, 34359738368, 549755813888, 8796093022208, 140737488355328, 2251799813685248, 36028797018963968]

sage: [lucas_number2(4*n, 2, 0)*lucas_number1(4*n, 2, 0) for n in xrange(1, 15)] #nnnnnnnnnn

[128, 32768, 8388608, 2147483648, 549755813888, 140737488355328, 36028797018963968, 9223372036854775808, 2361183241434822606848, 604462909807314587353088, 154742504910672534362390528, 39614081257132168796771975168, 10141204801825835211973625643008, 2596148429267413814265248164610048]

sage: [lucas_number2(n, 2, 0)*lucas_number1(n, 2, 0) for n in xrange(1, 15)] #nem kellA004171  	 	 2^(2n+1).

[2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728]

sage: [lucas_number2(n, 2, 0)*lucas_number1(n, 3, 0)/2 for n in xrange(1, 15)] #van több, nem kell 
A000400 		Powers of 6.

[1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616, 10077696, 60466176, 362797056, 2176782336, 13060694016]

sage: [lucas_number2(n, 3, 0)+lucas_number1(n, 2, 0) for n in xrange(0, 15)] #nnnnnnn

[2, 4, 11, 31, 89, 259, 761, 2251, 6689, 19939, 59561, 178171, 533489, 1598419, 4791161]

sage: [lucas_number2(n, 3, 0)*lucas_number1(n, 2, 0) for n in xrange(0, 15)] #nnnnnnnnnnnnnnnnn

[0, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048]

sage: [lucas_number2(n, 3, 0)*lucas_number1(n, 4, 0)/3 for n in xrange(0, 15)] # A001021  	 	 Powers of 12.  	

(Other) sage: [lucas_number1(n, 12, 0) for n in xrange(1, 19)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 27 2009]

[0, 1, 12, 144, 1728, 20736, 248832, 2985984, 35831808, 429981696, 5159780352, 61917364224, 743008370688, 8916100448256, 106993205379072]

sage: [lucas_number2(n, 2, 0)+lucas_number1(n, 1, 0) for n in xrange(0, 15)] #  	A000051  	 	 2^n + 1. 
sage: [lucas_number2(n, 3, 2) for n in range(37)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
(Other) sage: [sigma(2, n)for n in xrange(0, 32)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 04 2009]

[2, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385]

sage: [lucas_number2(n, 2, 0)-lucas_number1(n, 1, 0) for n in xrange(1, 15)] #  nnnnnnnnnnnnnnnnnn

[1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383]

sage: [lucas_number1(n, 2, 0)-lucas_number2(n, 1, 0) for n in xrange(1, 15)] #A000225  	 	 2^n - 1. (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.)    	

sage: [stirling_number2(i, 2) for i in xrange(1, 30)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 26 2008

(Other) sage: [lucas_number1(n, 3, 2) for n in xrange(0, 33)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]

(Other) sage: [gaussian_binomial(n, 1, 2) for n in xrange(1, 33)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 24 2009]

[0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191]

sage: [lucas_number1(n, 3, 0)-lucas_number2(n, 1, 0) for n in xrange(1, 22)] #ok A024023  	 	 3^n-1.

[0, 2, 8, 26, 80, 242, 728, 2186, 6560, 19682, 59048, 177146, 531440, 1594322, 4782968, 14348906, 43046720, 129140162, 387420488, 1162261466, 3486784400]

sage: [(lucas_number1(n, 3, 0)-lucas_number2(n, 1, 0))/2 for n in xrange(1, 22)] #

[0, 1, 4, 13, 40, 121, 364, 1093, 3280, 9841, 29524, 88573, 265720, 797161, 2391484, 7174453, 21523360, 64570081, 193710244, 581130733, 1743392200]

sage: [lucas_number1(n, 5, 0)-1 for n in xrange(1, 22)] #

[0, 4, 24, 124, 624, 3124, 15624, 78124, 390624, 1953124, 9765624, 48828124, 244140624, 1220703124, 6103515624, 30517578124, 152587890624, 762939453124, 3814697265624, 19073486328124, 95367431640624]

sage: [lucas_number1(n, 5, 0)-lucas_number1(n, 2, 0) for n in xrange(1, 22)] #A005057  	 	 5^n - 2^n.     	
(Other) sage: [5^n - 2^n for n in xrange(0, 21)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 04 2009]

[0, 3, 21, 117, 609, 3093, 15561, 77997, 390369, 1952613, 9764601, 48826077, 244136529, 1220694933, 6103499241, 30517545357, 152587825089, 762939322053, 3814697003481, 19073485803837, 95367430592049]

sage: [lucas_number1(n, 5, 1)-lucas_number1(n, 1, 1) for n in xrange(1, 22)] #nnn

[0, 4, 24, 116, 552, 2640, 12648, 60604, 290376, 1391276, 6666000, 31938720, 153027600, 733199284, 3512968824, 16831644836, 80645255352, 386394631920, 1851327904248, 8870244889324, 42499896542376]

sage: [(lucas_number1(n, 5, 1)-lucas_number1(n, 1, 1))/4 for n in xrange(1, 22)] #nnnnnnnnnnnn

[0, 1, 6, 29, 138, 660, 3162, 15151, 72594, 347819, 1666500, 7984680, 38256900, 183299821, 878242206, 4207911209, 20161313838, 96598657980, 462831976062, 2217561222331, 10624974135594]

sage: [lucas_number1(n, 5, 1)-lucas_number1(n, 2, 1) for n in xrange(1, 22)] #nnnnnnnnnnnnnn

[0, 3, 21, 111, 546, 2634, 12642, 60597, 290367, 1391265, 6665988, 31938708, 153027588, 733199271, 3512968809, 16831644819, 80645255334, 386394631902, 1851327904230, 8870244889305, 42499896542355]

sage: [(lucas_number1(n, 5, 1)-lucas_number1(n, 3, 1))/2 for n in xrange(1, 22)] #nnnnnn

[0, 1, 8, 47, 248, 1248, 6136, 29809, 143896, 692255, 3324144, 15946176, 76453104, 366440737, 1756068392, 8414733263, 40319776232, 193189850784, 925644408040, 4435071277585, 21249814314040]

sage: [lucas_number1(n, 5, 1)-lucas_number1(n, 4, 1) for n in xrange(1, 22)] #

[0, 1, 9, 59, 342, 1860, 9738, 49741, 249831, 1239959, 6101280, 29831160, 145162080, 703844761, 3403416249, 16422789059, 79119384822, 380700005580, 1830075269418, 8790928976341, 42203885525271]

sage: [lucas_number2(n, 1, 0) for n in xrange(1, 22)] #

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

sage: [lucas_number1(n, 1, 0) for n in xrange(1, 22)] #

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]