cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A204765 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+239)^2 = y^2.

Original entry on oeis.org

0, 217, 220, 717, 1900, 1917, 4780, 11661, 11760, 28441, 68544, 69121, 166344, 400081, 403444, 970101, 2332420, 2352021, 5654740, 13594917, 13709160, 32958817, 79237560, 79903417, 192098640, 461830921, 465711820, 1119633501, 2691748444, 2714367981
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 09 2012

Keywords

Links

Crossrefs

Cf. A185394, A198294, A203619, A206426

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,217,220,717,1900,1917,4780}, 70]

Formula

G.f.: x^2*(119*x^5+x^4+119*x^3-497*x^2-3*x-217)/((x-1)*(x^6-6*x^3+1)). [Colin Barker, Aug 05 2012]

A205644 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+287)^2 = y^2.

Original entry on oeis.org

0, 25, 36, 205, 252, 273, 328, 705, 748, 861, 988, 1045, 1968, 2233, 2352, 2665, 4836, 5085, 5740, 6477, 6808, 12177, 13720, 14413, 16236, 28885, 30336, 34153, 38448, 40377, 71668, 80661, 84700, 95325, 169048, 177505, 199752, 224785, 236028, 418405, 470820
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 09 2012

Keywords

Links

Crossrefs

Cf. A185394, A198294, A203619, A204765, A206426.

Programs

  • Mathematica
    LinearRecurrence[ {1,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,-1,1}, {0,25,36,205,252,273,328,705,748,861,988,1045,1968,2233,2352,2665,4836,5085,5740}, 70]

Formula

G.f.: x^2*(23*x^17 +9*x^16 +91*x^15 +17*x^14 +7*x^13 +17*x^12 +91*x^11 +9*x^10 +23*x^9 -113*x^8 -43*x^7 -377*x^6 -55*x^5 -21*x^4 -47*x^3 -169*x^2 -11*x -25)/((x -1)*(x^18 -6*x^9 +1)). - Colin Barker, Aug 05 2012

A205672 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+329)^2 = y^2.

Original entry on oeis.org

0, 87, 112, 184, 235, 376, 451, 595, 660, 987, 1440, 1575, 1971, 2256, 3055, 3484, 4312, 4687, 6580, 9211, 9996, 12300, 13959, 18612, 21111, 25935, 28120, 39151, 54484, 59059, 72487, 82156, 109275, 123840, 151956, 164691, 228984, 318351, 345016, 423280
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 09 2012

Keywords

Links

Crossrefs

Cf. A185394, A198294, A203619, A204765, A205644, A206426.

Programs

  • Mathematica
    LinearRecurrence[ {1,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,-1,1}, {0,87,112,184,235,376,451,595,660,987,1440,1575,1971,2256,3055,3484,4312,4687,6580}, 120]

Formula

G.f.: x^2*(69*x^17 +15*x^16 +36*x^15 +21*x^14 +47*x^13 +21*x^12 +36*x^11 +15*x^10 +69*x^9 -327*x^8 -65*x^7 -144*x^6 -75*x^5 -141*x^4 -51*x^3 -72*x^2 -25*x -87)/((x-1)*(x^18 -6*x^9 +1)). - Colin Barker, Aug 05 2012

A185394 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+193)^2 = y^2.

Original entry on oeis.org

0, 152, 203, 579, 1403, 1692, 3860, 8652, 10335, 22967, 50895, 60704, 134328, 297104, 354275, 783387, 1732115, 2065332, 4566380, 10095972, 12038103, 26615279, 58844103, 70163672, 155125680, 342969032, 408944315, 904139187, 1998970475, 2383502604, 5269709828
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 09 2012

Keywords

Links

Crossrefs

Cf. A206426.

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,152,203,579,1403,1692,3860},70]
  • PARI
    concat(0, Vec(x^2*(88*x^5+17*x^4+88*x^3-376*x^2-51*x-152)/((x-1)*(x^6-6*x^3+1)) + O(x^100))) \\ Colin Barker, May 18 2015

Formula

G.f.: x^2*(152+51*x+376*x^2-88*x^3-17*x^4-88*x^5)/((1-x)*(1-6*x^3+x^6)). - Colin Barker, Aug 04 2012

A198294 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+217)^2 = y^2.

Original entry on oeis.org

0, 63, 68, 135, 155, 248, 276, 407, 420, 651, 980, 1007, 1376, 1488, 2015, 2175, 2928, 3003, 4340, 6251, 6408, 8555, 9207, 12276, 13208, 17595, 18032, 25823, 36960, 37875, 50388, 54188, 72075, 77507, 103076, 105623, 151032, 215943, 221276, 294207, 316355
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 09 2012

Keywords

Links

Crossrefs

Cf. A185394, A206426.

Programs

  • Mathematica
    LinearRecurrence[{1,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,-1,1}, {0,63,68,135,155,248,276,407,420,651,980,1007,1376,1488,2015,2175,2928,3003,4340}, 110]

Formula

G.f.: x^2*(49*x^17 +3*x^16 +33*x^15 +8*x^14 +31*x^13 +8*x^12 +33*x^11 +3*x^10 +49*x^9 -231*x^8 -13*x^7 -131*x^6 -28*x^5 -93*x^4 -20*x^3 -67*x^2 -5*x -63)/((x -1)*(x^18 -6*x^9 +1)). - Colin Barker, Aug 05 2012

A201916 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+2737)^2 = y^2.

Original entry on oeis.org

0, 75, 203, 323, 552, 708, 1020, 1127, 1311, 1428, 1608, 1820, 1955, 2336, 2675, 3128, 3311, 3627, 3927, 4140, 4508, 4743, 5535, 6003, 6800, 7280, 7848, 8211, 8588, 9240, 9860, 11063, 11895, 13583, 14168, 15180, 15827, 16827, 18011, 18768, 20915, 22836
Offset: 1

Views

Author

T. D. Noe, Feb 09 2012

Keywords

Comments

Note that 2737 = 7 * 17 * 23, the product of the first three distinct primes in A058529 (and A001132) and hence the smallest such number. This sequence satisfies a linear difference equation of order 55 whose 55 initial terms can be found by running the Mathematica program.
There are many sequences like this one. What determines the order of the linear difference equation? All primes p have order 7. For those p, it appears that p^2 has order 11, p^3 order 15, and p^i order 3+4*i. It appears that for semiprimes p*q (with p > q), the order is 19. What is the next term of the sequence beginning 3, 7, 19, 55, 163? This could be sequence A052919, which is 1 + 2*3^f, where f is the number of primes.
The crossref list is thought to be complete up to Feb 14 2012.

Links

Crossrefs

Cf. A001652 (1), A076296 (7), A118120 (17), A118337 (23), A118674 (31).
Cf. A129288 (41), A118675 (47), A118554 (49), A118673 (71), A129289 (73).
Cf. A118676 (79), A129298 (89), A129836 (97), A157119 (103), A161478 (113).
Cf. A129837 (119), A129992 (127), A129544 (137), A161482 (151).
Cf. A206426 (161), A130608 (167), A161486 (191), A185394 (193).
Cf. A129993 (199), A198294 (217), A130609 (223), A129625 (233).
Cf. A204765 (239), A129991 (241), A207058 (263), A129626 (281).
Cf. A205644 (287), A207059 (289), A129640 (313), A205672 (329).
Cf. A129999 (337), A118611 (343), A130610 (359), A207060 (401).
Cf. A129641 (409), A207061 (433), A130645 (439), A130004 (449).
Cf. A129642 (457), A129725 (521), A101152 (569), A130005 (577).
Cf. A207075 (479), A207076 (487), A207077 (497), A207078 (511).
Cf. A111258 (601), A115135 (617), A130013 (647), A130646 (727).
Cf. A122694 (761), A123654 (809), A129010 (833), A130647 (839).
Cf. A129857 (857), A130014 (881), A129974 (937), A129975 (953).
Cf. A130017 (967), A118630 (2401), A118576 (16807).

Programs

  • Mathematica
    d = 2737; terms = 100; t = Select[Range[0, 55000], IntegerQ[Sqrt[#^2 + (#+d)^2]] &]; Do[AppendTo[t, t[[-1]] + 6*t[[-27]] - 6*t[[-28]] - t[[-54]] + t[[-55]]], {terms-55}]; t

Formula

a(n) = a(n-1) + 6*a(n-27) - 6*a(n-28) - a(n-54) + a(n-55), where the 55 initial terms can be computed using the Mathematica program.
G.f.: x^2*(73*x^53 +116*x^52 +100*x^51 +171*x^50 +104*x^49 +184*x^48 +57*x^47 +92*x^46 +55*x^45 +80*x^44 +88*x^43 +53*x^42 +139*x^41 +113*x^40 +139*x^39 +53*x^38 +88*x^37 +80*x^36 +55*x^35 +92*x^34 +57*x^33 +184*x^32 +104*x^31 +171*x^30 +100*x^29 +116*x^28 +73*x^27 -363*x^26 -568*x^25 -480*x^24 -797*x^23 -468*x^22 -792*x^21 -235*x^20 -368*x^19 -213*x^18 -300*x^17 -316*x^16 -183*x^15 -453*x^14 -339*x^13 -381*x^12 -135*x^11 -212*x^10 -180*x^9 -117*x^8 -184*x^7 -107*x^6 -312*x^5 -156*x^4 -229*x^3 -120*x^2 -128*x -75) / ((x -1)*(x^54 -6*x^27 +1)). - Colin Barker, May 18 2015
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.