A129993 -id:A129993 - cp's OEIS frontend

A118673 Positive solutions x to the equation x^2 + (x+71)^2 = y^2.

0, 13, 160, 213, 280, 1113, 1420, 1809, 6660, 8449, 10716, 38989, 49416, 62629, 227416, 288189, 365200, 1325649, 1679860, 2128713, 7726620, 9791113, 12407220, 45034213, 57066960, 72314749, 262478800, 332610789, 421481416, 1529838729, 1938597916, 2456573889

Offset: 0

Mohamed Bouhamida, May 19 2006

Consider all Pythagorean triples (x,x+71,y) ordered by increasing y; sequence gives x values.
For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, m>=2 the associated value in A066049, the x values are given by the sequence defined by: a(n) = 6*a(n-3) -a(n-6) + 2*p with a(0)=0, a(1)=2m+1, a(2)=6m^2-10m+4, a(3)=3p, a(4)=6m^2+10m+4, a(5)=40m^2-58m+21.
For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, m>=2, Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(0)=p, b(1)=2m^2+2m+1, b(2)=10m^2-14m+5, b(3)=5p, b(4)=10m^2+14m+5, b(5)=58m^2-82m+29. - Mohamed Bouhamida, Sep 09 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A076296 (p=7), A118120 (p=17), A118674 (p=31), A129836 (p=97), A129992 (p=127), A129993 (p=199), A129991 (p=241), A129999 (p=337), A130004 (p=449), A130005 (p=577), A130013 (p=647), A130014 (p=881), A130017 (p=967).

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(13+147*x+53*x^2-11*x^3-49*x^4-11*x^5)/((1-x)*(1 - 6*x^3 +x^6)))); // G. C. Greubel, May 07 2018

Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+71)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,13,160,213,280,1113,1420},100] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 1,-1,0,-6,6,0,1]^n*[0;13;160;213;280;1113;1420])[1,1] \\ Charles R Greathouse IV, Apr 22 2016

x='x+O('x^30); concat([0], Vec(x*(13+147*x+53*x^2-11*x^3 -49*x^4 -11*x^5)/((1-x)*(1-6*x^3+x^6)))) \\ G. C. Greubel, May 07 2018

a(n) = 6*a(n-3) -a(n-6) +142 with a(0)=0, a(1)=13, a(2)=160, a(3)=213, a(4)=280, a(5)=1113.

O.g.f.: x*(13+147*x+53*x^2-11*x^3-49*x^4-11*x^5)/((1-x)*(1-6*x^3+x^6)). - R. J. Mathar, Jun 10 2008

Edited by R. J. Mathar, Jun 10 2008

A159549 Decimal expansion of (201+20*sqrt(2))/199.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 14 2009

lim_{n -> infinity} b(n)/b(n-1) = (201+20*sqrt(2))/199 for n mod 3 = {1, 2}, b = A129993.

lim_{n -> infinity} b(n)/b(n-1) = (201+20*sqrt(2))/199 for n mod 3 = {0, 2}, b = A159548.

			(201+20*sqrt(2))/199 = 1.15218226757518543204...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A129993, A159548, A002193 (decimal expansion of sqrt(2)), A159550 (decimal expansion of (91443+58282*sqrt(2))/199^2).

(201 + 20*Sqrt(2))/199 // G. C. Greubel, Mar 30 2018

with(MmaTranslator[Mma]): Digits:=100:
RealDigits(evalf((201+20*sqrt(2))/199))[1]; # Muniru A Asiru, Mar 31 2018

RealDigits[(201+20*Sqrt[2])/199, 10, 100][[1]] (* G. C. Greubel, Mar 30 2018 *)

(201+20*sqrt(2))/199 \\ G. C. Greubel, Mar 30 2018

(201+20*sqrt(2))/199 = (20+sqrt(2))/(20-sqrt(2)).

A159550 Decimal expansion of (91443+58282*sqrt(2))/199^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 14 2009

lim_{n -> infinity} b(n)/b(n-1) = (91443+58282*sqrt(2))/199^2 for n mod 3 = 0, b = A129993.

lim_{n -> infinity} b(n)/b(n-1) = (91443+58282*sqrt(2))/199^2 for n mod 3 = 1, b = A159548.

			(91443+58282*sqrt(2))/199^2 = 4.39044960587431442726...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A129993, A159548, A002193 (decimal expansion of sqrt(2)), A159549 (decimal expansion of (201+20*sqrt(2))/199).

(91443 + 58282*Sqrt(2))/199^2; // G. C. Greubel, Mar 30 2018

with(MmaTranslator[Mma]): Digits:=100:
RealDigits(evalf((91443+58282*sqrt(2))/199^2))[1]; # Muniru A Asiru, Mar 31 2018

RealDigits[(91443+58282Sqrt[2])/199^2,10,120][[1]] (* Harvey P. Dale, Sep 23 2012 *)

(91443 + 58282*sqrt(2))/199^2 \\ G. C. Greubel, Mar 30 2018

(91443+58282*sqrt(2))/199^2 = (362+161*sqrt(2))/(362-161*sqrt(2))

= (3+2*sqrt(2))*(20-sqrt(2))^2/(20+sqrt(2))^2.

A159548 Positive numbers y such that y^2 is of the form x^2+(x+199)^2 with integer x.

Original entry on oeis.org

181, 199, 221, 865, 995, 1145, 5009, 5771, 6649, 29189, 33631, 38749, 170125, 196015, 225845, 991561, 1142459, 1316321, 5779241, 6658739, 7672081, 33683885, 38809975, 44716165, 196324069, 226201111, 260624909, 1144260529, 1318396691

Offset: 1

Klaus Brockhaus, Apr 14 2009

(-19,a(1)) and (A129993(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+199)^2 = y^2.

			(-19, a(1)) = (-19, 181) is a solution: (-19)^2+(-19+199)^2 = 361+32400 = 32761 = 181^2.
(A129993(1), a(2)) = (0, 199) is a solution: 0^2+(0+199)^2 = 39601 = 199^2.
(A129993(3), a(4)) = (504, 865) is a solution: 504^2+(504+199)^2 = 254016+494209 = 748225 = 865^2.

Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A129993, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159549 (decimal expansion of (201+20*sqrt(2))/199), A159550 (decimal expansion of (91443+58282*sqrt(2))/199^2).

LinearRecurrence[{0,0,6,0,0,-1},{181,199,221,865,995,1145},30] (* Harvey P. Dale, Aug 09 2025 *)

{forstep(n=-20, 50000000, [1, 3], if(issquare(2*n^2+398*n+39601, &k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=181, a(2)=199, a(3)=221, a(4)=865, a(5)=995, a(6)=1145.
G.f.: x*(1-x)*(181+380*x+601*x^2+380*x^3+181*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 199*A001653(k) for k >= 1.
Limit_{n -> oo} a(n)/a(n-3) = 3+2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = (201+20*sqrt(2))/199 for n mod 3 = {0, 2}.
Limit_{n -> oo} a(n)/a(n-1) = (91443+58282*sqrt(2))/199^2 for n mod 3 = 1.

A159565 Positive numbers y such that y^2 is of the form x^2+(x+241)^2 with integer x.

Original entry on oeis.org

221, 241, 265, 1061, 1205, 1369, 6145, 6989, 7949, 35809, 40729, 46325, 208709, 237385, 270001, 1216445, 1383581, 1573681, 7089961, 8064101, 9172085, 41323321, 47001025, 53458829, 240849965, 273942049, 311580889, 1403776469, 1596651269

Offset: 1

Klaus Brockhaus, Apr 16 2009

(-21,a(1)) and (A129991(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+241)^2 = y^2.

			(-21, a(1)) = (-21, 221) is a solution: (-21)^2+(-21+241)^2 = 441+48400 = 48841 = 221^2.
(A129993(1), a(2)) = (0, 241) is a solution: 0^2+(0+241)^2 = 58081= 241^2.
(A129993(3), a(4)) = (620, 1061) is a solution: 620^2+(620+241)^2 = 384400+741321 = 1125721 = 1061^2.

Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A129991, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159566 (decimal expansion of (243+22*sqrt(2))/241), A159567 (decimal expansion of (137283+87958*sqrt(2))/241^2).

LinearRecurrence[{0,0,6,0,0,-1},{221,241,265,1061,1205,1369},30] (* Harvey P. Dale, Nov 21 2011 *)

{forstep(n=-24, 50000000, [3, 1], if(issquare(2*n^2+482*n+58081, &k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=221, a(2)=241, a(3)=265, a(4)=1061, a(5)=1205, a(6)=1369.
G.f.: x*(1-x)*(221+462*x+727*x^2+462*x^3+221*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 241*A001653(k) for k >= 1.
Limit_{n -> oo} a(n)/a(n-3) = 3+2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = (243+22*sqrt(2))/241 for n mod 3 = {0, 2}.
Limit_{n -> oo} a(n)/a(n-1) = (137283+87958*sqrt(2))/241^2 for n mod 3 = 1.

A159574 Positive numbers y such that y^2 is of the form x^2+(x+337)^2 with integer x.

Original entry on oeis.org

313, 337, 365, 1513, 1685, 1877, 8765, 9773, 10897, 51077, 56953, 63505, 297697, 331945, 370133, 1735105, 1934717, 2157293, 10112933, 11276357, 12573625, 58942493, 65723425, 73284457, 343542025, 383064193, 427133117, 2002309657

Offset: 1

Klaus Brockhaus, Apr 16 2009

(-25,a(1)) and (A129999(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+337)^2 = y^2.

			(-25, a(1)) = (-25, 313) is a solution: (-25)^2+(-25+337)^2 = 625+97344 = 97969 = 313^2.
(A129993(1), a(2)) = (0, 337) is a solution: 0^2+(0+337)^2 = 113569 = 337^2.
(A129993(3), a(4)) = (888, 1513) is a solution: 888^2+(888+337)^2 = 788544+1500625 = 2289169 = 1513^2.

Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A129999, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159575 (decimal expansion of (339+26*sqrt(2))/337), A159576(decimal expansion of (278307+179662*sqrt(2))/337^2).

{forstep(n=-28, 50000000, [3, 1], if(issquare(2*n^2+674*n+113569, &k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6)for n > 6; a(1)=313, a(2)=337, a(3)=365, a(4)=1513, a(5)=1685, a(6)=1877.
G.f.: x*(1-x)*(313+650*x+1015*x^2+650*x^3+313*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 337*A001653(k) for k >= 1.
Limit_{n -> oo} a(n)/a(n-3) = 3+2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = (339+26*sqrt(2))/337 for n mod 3 = {0, 2}.
Limit_{n -> oo} a(n)/a(n-1) = (278307+179662*sqrt(2))/337^2 for n mod 3 = 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

T. D. Noe, Feb 09 2012

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.

Colin Barker, Table of n, a(n) for n = 1..1000

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).

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

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

cp's OEIS Frontend

A118673 Positive solutions x to the equation x^2 + (x+71)^2 = y^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

Extensions

A159549 Decimal expansion of (201+20*sqrt(2))/199.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A159550 Decimal expansion of (91443+58282*sqrt(2))/199^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A159548 Positive numbers y such that y^2 is of the form x^2+(x+199)^2 with integer x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A159565 Positive numbers y such that y^2 is of the form x^2+(x+241)^2 with integer x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A159574 Positive numbers y such that y^2 is of the form x^2+(x+337)^2 with integer x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views