This file contains articles about one-dimensional cellular automata , and ge\ neralizations to mod 3 and mod 5 3 2 n On the sequence, (x + x + x + 1) , modulo , 2, evaluated at , {x = 1} By Shalosh B. Ekhad The first, 41, terms staring at n=0 are [1, 4, 4, 4, 4, 16, 4, 8, 4, 16, 16, 4, 4, 16, 8, 16, 4, 16, 16, 16, 16, 64, 4, 8, 4, 16, 16, 8, 8, 32, 16, 32, 4, 16, 16, 16, 16, 64, 16, 32, 16] Just for kicks, the googol-th term of our sequence is 10384593717069655257060992658440192 i The first , 40, terms of the sparse subsequence at the, 2 - 1, places are [1, 4, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736, 137438953472, 274877906944, 549755813888, 1099511627776] Using the found enumerative automaton with, 3, states, that we omit, it follows that the (rigorously) PROVED rational generating function for that sparse subsequ\ ence is 2 4 t - 2 t - 1 -------------- 2 t - 1 and in Maple notation (4*t^2-2*t-1)/(2*t-1) This ends this article, that took, 0.016, seconds. 3 2 n On the sequence, (x + x + x + 1) , modulo , 3, evaluated at , {x = 1} By Shalosh B. Ekhad The first, 41, terms staring at n=0 are [1, 4, 7, 4, 16, 22, 7, 22, 19, 4, 16, 28, 16, 46, 61, 22, 67, 40, 7, 28, 37, 22, 76, 76, 19, 64, 61, 4, 16, 28, 16, 64, 88, 28, 88, 76, 16, 64, 94, 46, 139] Just for kicks, the googol-th term of our sequence is 7028155797200644969861037281269047423748688600147272572543696896000000 i The first , 40, terms of the sparse subsequence at the, 3 - 1, places are [1, 7, 19, 61, 181, 547, 1639, 4921, 14761, 44287, 132859, 398581, 1195741, 3587227, 10761679, 32285041, 96855121, 290565367, 871696099, 2615088301, 7845264901, 23535794707, 70607384119, 211822152361, 635466457081, 1906399371247 , 5719198113739, 17157594341221, 51472783023661, 154418349070987, 463255047212959, 1389765141638881, 4169295424916641, 12507886274749927, 37523658824249779, 112570976472749341, 337712929418248021, 1013138788254744067, 3039416364764232199, 9118249094292696601, 27354747282878089801] Using the found enumerative automaton with, 12, states, that we omit, it follows that the (rigorously) PROVED rational generating function for that sparse subsequ\ ence is 2 3 t - 4 t - 1 - ------------------------- (t - 1) (3 t - 1) (t + 1) and in Maple notation -(3*t^2-4*t-1)/(t-1)/(3*t-1)/(t+1) This ends this article, that took, 0.014, seconds. 3 2 n On the sequence, (x + x + x + 1) , modulo , 5, evaluated at , {x = 1} By Shalosh B. Ekhad The first, 41, terms staring at n=0 are [1, 4, 16, 14, 16, 4, 16, 64, 56, 34, 16, 64, 81, 74, 86, 14, 56, 104, 106, 74, 16, 64, 111, 114, 91, 4, 16, 64, 56, 64, 16, 64, 256, 224, 121, 64, 256, 294, 281, 284, 56] Just for kicks, the googol-th term of our sequence is 1920803561069772108988416 i The first , 40, terms of the sparse subsequence at the, 5 - 1, places are [1, 16, 91, 466, 2341, 11716, 58591, 292966, 1464841, 7324216, 36621091, 183105466, 915527341, 4577636716, 22888183591, 114440917966, 572204589841, 2861022949216, 14305114746091, 71525573730466, 357627868652341, 1788139343261716, 8940696716308591, 44703483581542966, 223517417907714841, 1117587089538574216, 5587935447692871091, 27939677238464355466, 139698386192321777341, 698491930961608886716, 3492459654808044433591, 17462298274040222167966, 87311491370201110839841, 436557456851005554199216, 2182787284255027770996091, 10913936421275138854980466, 54569682106375694274902341, 272848410531878471374511716, 1364242052659392356872558591, 6821210263296961784362792966, 34106051316484808921813964841] Using the found enumerative automaton with, 100, states, that we omit, it follows that the Guessed (but absolutely certain!) rational generating function for that \ sparse subsequence is 1 + 10 t ----------------- (5 t - 1) (t - 1) and in Maple notation (1+10*t)/(5*t-1)/(t-1) This ends this article, that took, 0.011, seconds. 4 2 n On the sequence, (x + x + x + 1) , modulo , 2, evaluated at , {x = 1} By Shalosh B. Ekhad The first, 41, terms staring at n=0 are [1, 4, 4, 8, 4, 12, 8, 14, 4, 16, 12, 24, 8, 24, 14, 30, 4, 16, 16, 32, 12, 36, 24, 44, 8, 32, 24, 48, 14, 46, 30, 60, 4, 16, 16, 32, 16, 48, 32, 56, 12] Just for kicks, the googol-th term of our sequence is 74055541156789762609567360141472128384892928000 i The first , 40, terms of the sparse subsequence at the, 2 - 1, places are [1, 4, 8, 14, 30, 60, 118, 238, 476, 950, 1902, 3804, 7606, 15214, 30428, 60854 , 121710, 243420, 486838, 973678, 1947356, 3894710, 7789422, 15578844, 31157686 , 62315374, 124630748, 249261494, 498522990, 997045980, 1994091958, 3988183918, 7976367836, 15952735670, 31905471342, 63810942684, 127621885366, 255243770734, 510487541468, 1020975082934, 2041950165870] Using the found enumerative automaton with, 7, states, that we omit, it follows that the (rigorously) PROVED rational generating function for that sparse subsequ\ ence is 2 3 t + 3 t + 1 - ---------------------- 2 (2 t - 1) (t + t + 1) and in Maple notation -(3*t^2+3*t+1)/(2*t-1)/(t^2+t+1) This ends this article, that took, 0.009, seconds. 4 2 n On the sequence, (x + x + x + 1) , modulo , 3, evaluated at , {x = 1} By Shalosh B. Ekhad The first, 41, terms staring at n=0 are [1, 4, 10, 4, 16, 22, 10, 34, 34, 4, 16, 40, 16, 52, 61, 22, 67, 70, 10, 40, 52 , 34, 112, 85, 34, 106, 109, 4, 16, 40, 16, 64, 88, 40, 136, 118, 16, 64, 136, 52, 160] Just for kicks, the googol-th term of our sequence is 4848912751072160383596084844126621622790754586013645424480738644852736000 i The first , 40, terms of the sparse subsequence at the, 3 - 1, places are [1, 10, 34, 109, 307, 928, 2788, 8371, 25093, 75286, 225862, 677593, 2032759, 6098284, 18294856, 54884575, 164653705, 493961122, 1481883370, 4445650117, 13336950331, 40010851000, 120032553004, 360097659019, 1080292977037, 3240878931118, 9722636793358, 29167910380081, 87503731140223, 262511193420676, 787533580262032, 2362600740786103, 7087802222358289, 21263406667074874, 63790220001224626, 191370660003673885, 574111980011021635, 1722335940033064912, 5167007820099194740, 15501023460297584227, 46503070380892752661] Using the found enumerative automaton with, 54, states, that we omit, it follows that the (rigorously) PROVED rational generating function for that sparse subsequ\ ence is 4 3 2 21 t - 7 t - 4 t - 7 t - 1 - ---------------------------------- 2 (t - 1) (t + 1) (t + 1) (3 t - 1) and in Maple notation -(21*t^4-7*t^3-4*t^2-7*t-1)/(t-1)/(t+1)/(t^2+1)/(3*t-1) This ends this article, that took, 0.102, seconds.