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.

A207059 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+289)^2 = y^2.

Original entry on oeis.org

119, 231, 300, 476, 867, 1496, 2120, 2511, 3519, 5780, 9435, 13067, 15344, 21216, 34391, 55692, 76860, 90131, 124355, 201144, 325295, 448671, 526020, 725492, 1173051, 1896656, 2615744, 3066567, 4229175, 6837740, 11055219, 15246371, 17873960, 24650136
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 14 2012

Keywords

Comments

For the generic case x^2 + (x + p^2)^2 = y^2 with p = 2*m^2 - 1 a prime number in A066436, m>=3, (0; p^2) and (4*m^3 + 2*m^2 - 2*m - 1; 4*m^4 + 4*m^3 - 2*m - 1) are solutions. - Mohamed Bouhamida, Aug 24 2019

Links

Crossrefs

Cf. A204765, A205644, A205672, A207058.

Programs

  • Mathematica
    LinearRecurrence[ {1, 0, 0, 0, 6, -6, 0, 0, 0, -1, 1}, {119, 231, 300, 476, 867, 1496, 2120, 2511, 3519, 5780, 9435}, 60]
  • PARI
    Vec(x*(85*x^9+48*x^8+23*x^7+48*x^6+85*x^5-391*x^4-176*x^3-69*x^2-112*x-119)/((x-1)*(x^10-6*x^5+1))+O(x^60)) \\ Stefano Spezia, Aug 24 2019

Formula

G.f.: x*(85*x^9+48*x^8+23*x^7+48*x^6+85*x^5-391*x^4-176*x^3-69*x^2-112*x-119)/((x-1)*(x^10-6*x^5+1)). - Colin Barker, Aug 05 2012
a(n) = 6*a(n-5) - a(n-10) + 578 with a(1) = 119, a(2) = 231, a(3) = 300, a(4) = 476, a(5) = 867, a(6) = 1496, a(7) = 2120, a(8) = 2511, a(9) = 3519, a(10) = 5780. - Mohamed Bouhamida, Aug 24 2019

A207058 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+263)^2 = y^2.

Original entry on oeis.org

0, 145, 364, 789, 1564, 2805, 5260, 9765, 16992, 31297, 57552, 99673, 183048, 336073, 581572, 1067517, 1959412, 3390285, 6222580, 11420925, 19760664, 36268489, 66566664, 115174225, 211388880, 387979585, 671285212, 1232065317, 2261311372, 3912537573
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 14 2012

Keywords

Links

Crossrefs

Cf. A204765, A205644, A205672

Programs

  • Mathematica
    LinearRecurrence[ {1, 0, 6, -6, 0, -1, 1}, {0, 145, 364, 789, 1564, 2805, 5260}, 40]

Formula

G.f.: x^2*(95*x^5+73*x^4+95*x^3-425*x^2-219*x-145)/((x-1)*(x^6-6*x^3+1)). [Colin Barker, Aug 05 2012]

A207060 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+401)^2 = y^2.

Original entry on oeis.org

0, 259, 496, 1203, 2596, 3939, 8020, 16119, 23940, 47719, 94920, 140503, 279096, 554203, 819880, 1627659, 3231100, 4779579, 9487660, 18833199, 27858396, 55299103, 109768896, 162371599, 322307760, 639780979, 946372000, 1878548259, 3728917780, 5515861203
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 14 2012

Keywords

Links

Crossrefs

Cf. A204765, A205644, A205672, A207058, A207059.

Programs

  • Mathematica
    LinearRecurrence[ {1, 0, 6, -6, 0, -1, 1}, {0, 259, 496, 1203, 2596, 3939, 8020}, 50]

Formula

G.f.: x^2*(161*x^5+79*x^4+161*x^3-707*x^2-237*x-259)/((x-1)*(x^6-6*x^3+1)). - Colin Barker, Aug 05 2012

A207061 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+433)^2 = y^2.

Original entry on oeis.org

0, 92, 935, 1299, 1775, 6552, 8660, 11424, 39243, 51527, 67635, 229772, 301368, 395252, 1340255, 1757547, 2304743, 7812624, 10244780, 13434072, 45536355, 59711999, 78300555, 265406372, 348028080, 456370124, 1546902743, 2028457347, 2659921055, 9016010952
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 14 2012

Keywords

Links

Crossrefs

Cf. A205644, A205672, A207058, A207059, A207060.

Programs

  • Mathematica
    LinearRecurrence[ {1, 0, 6, -6, 0, -1, 1}, {0, 92, 935, 1299, 1775, 6552, 8660}, 50]

Formula

G.f.: x^2*(76*x^5+281*x^4+76*x^3-364*x^2-843*x-92)/((x-1)*(x^6-6*x^3+1)). - Colin Barker, Aug 05 2012

A207075 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+479)^2 = y^2.

Original entry on oeis.org

0, 280, 637, 1437, 2937, 4960, 9580, 18300, 30081, 57001, 107821, 176484, 333384, 629584, 1029781, 1944261, 3670641, 6003160, 11333140, 21395220, 34990137, 66055537, 124701637, 203938620, 385001040, 726815560, 1188642541, 2243951661, 4236192681, 6927917584
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 14 2012

Keywords

Links

Crossrefs

Cf. A205672, A207058, A207059, A207060, A207061

Programs

  • Mathematica
    LinearRecurrence[ {1, 0, 6, -6, 0, -1, 1}, {0, 280, 637, 1437, 2937, 4960, 9580}, 50]

Formula

G.f.: x^2*(180*x^5 + 119*x^4 + 180*x^3 - 800*x^2 - 357*x - 280)/((x-1)*(x^6-6*x^3+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.

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