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 1-4 of 4 results.

A161480 Decimal expansion of (129 +44*sqrt(2))/113.

Original entry on oeis.org

1, 6, 9, 2, 2, 6, 0, 1, 4, 8, 1, 8, 0, 6, 7, 4, 1, 7, 8, 2, 9, 4, 4, 6, 2, 8, 6, 6, 0, 6, 3, 9, 5, 3, 2, 3, 4, 1, 3, 3, 3, 2, 3, 5, 6, 2, 0, 9, 3, 6, 7, 8, 8, 9, 5, 7, 5, 0, 2, 1, 1, 5, 7, 9, 3, 9, 4, 6, 2, 1, 4, 9, 6, 0, 3, 8, 3, 4, 2, 5, 6, 3, 8, 0, 0, 1, 5, 0, 8, 9, 8, 3, 2, 2, 2, 6, 7, 5, 1, 5, 0, 4, 7, 2, 6
Offset: 1

Views

Author

Klaus Brockhaus, Jun 13 2009

Keywords

Comments

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A161478.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A161479.

Examples

			(129 +44*sqrt(2))/113 = 1.69226014818067417829...

Links

Crossrefs

Cf. A161478, A161479, A002193 (decimal expansion of sqrt(2)), A161481 (decimal expansion of (16131+6970*sqrt(2))/113^2).

Programs

  • Magma
    (129 +44*Sqrt(2))/113; // G. C. Greubel, Apr 07 2018
  • Maple
    with(MmaTranslator[Mma]): Digits:=150:
RealDigits(evalf((129+44*sqrt(2))/113))[1]; # Muniru A Asiru, Apr 08 2018
  • Mathematica
    RealDigits[(129 +44*Sqrt[2])/113, 10, 100][[1]] (* G. C. Greubel, Apr 07 2018 *)
  • PARI
    (129 +44*sqrt(2))/113 \\ G. C. Greubel, Apr 07 2018

Formula

Equals (11 +2*sqrt(2))/(11 -2*sqrt(2)).

A161481 Decimal expansion of (16131 + 6970*sqrt(2))/113^2.

Original entry on oeis.org

2, 0, 3, 5, 2, 4, 6, 9, 6, 7, 6, 3, 5, 7, 1, 7, 1, 6, 5, 8, 0, 3, 7, 2, 5, 4, 6, 0, 7, 0, 4, 9, 5, 6, 9, 7, 4, 5, 1, 8, 8, 0, 4, 3, 6, 7, 7, 1, 6, 9, 1, 8, 1, 6, 6, 6, 5, 6, 8, 6, 0, 9, 7, 3, 7, 4, 8, 5, 6, 3, 3, 4, 6, 7, 6, 8, 0, 2, 3, 0, 2, 9, 2, 7, 3, 0, 2, 0, 6, 8, 4, 0, 2, 0, 9, 6, 1, 8, 4, 2, 1, 1, 4, 7, 3
Offset: 1

Views

Author

Klaus Brockhaus, Jun 13 2009

Keywords

Comments

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A161478.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A161479.

Examples

			(16131+6970*sqrt(2))/113^2 = 2.03524696763571716580...

Links

Crossrefs

Cf. A161478, A161479, A002193 (decimal expansion of sqrt(2)), A161480 (decimal expansion of (129+44*sqrt(2))/113).

Programs

  • Magma
    (16131 + 6970*Sqrt(2))/113^2; // G. C. Greubel, Apr 07 2018
  • Maple
    with(MmaTranslator[Mma]): Digits:=150:
RealDigits(evalf((16131+6970*sqrt(2))/113^2))[1]; # Muniru A Asiru, Apr 08 2018
  • Mathematica
    RealDigits[(16131+6970Sqrt[2])/113^2,10,120][[1]] (* Harvey P. Dale, Sep 02 2015 *)
  • PARI
    (16131 + 6970*sqrt(2))/113^2 \\ G. C. Greubel, Apr 07 2018

Formula

Equals (170 +41*sqrt(2))/(170 -41*sqrt(2)).
Equals (3 +2*sqrt(2))*(11 -2*sqrt(2))^2/(11 +2*sqrt(2))^2.

A161479 Positive numbers y such that y^2 is of the form x^2+(x+113)^2 with integer x.

Original entry on oeis.org

85, 113, 173, 337, 565, 953, 1937, 3277, 5545, 11285, 19097, 32317, 65773, 111305, 188357, 383353, 648733, 1097825, 2234345, 3781093, 6398593, 13022717, 22037825, 37293733, 75901957, 128445857, 217363805, 442389025, 748637317, 1266889097
Offset: 1

Views

Author

Klaus Brockhaus, Jun 13 2009

Keywords

Comments

(-36, a(1)) and (A161478(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+113)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (129+44*sqrt(2))/113 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (16131+6970*sqrt(2))/113^2 for n mod 3 = 1.

Examples

			(-36, a(1)) = (-36, 85) is a solution: (-36)^2+(-36+113)^2 = 1296+5929 = 7225 = 85^2.
(A161478(1), a(2)) = (0, 113) is a solution: 0^2+(0+113)^2 = 12769 = 113^2.
(A161478(3), a(4)) = (175, 337) is a solution: 175^2+(175+113)^2 = 30625+82944 = 113569 = 337^2.

Crossrefs

Cf. A161478, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A161480 (decimal expansion of (129+44*sqrt(2))/113), A161481 (decimal expansion of (16131+6970*sqrt(2))/113^2).

Programs

  • PARI
    {forstep(n=-36, 10000000, [3, 1], if(issquare(2*n^2+226*n+12769, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=85, a(2)=113, a(3)=173, a(4)=337, a(5)=565, a(6)=953.
G.f.: (1-x)*(85+198*x+371*x^2+198*x^3+85*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 113*A001653(k) for k >= 1.

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 1-4 of 4 results.

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