A118674 -id:A118674 - cp's OEIS frontend

A130014 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+881)^2 = y^2.

0, 43, 2440, 2643, 2860, 16443, 17620, 18879, 97980, 104839, 112176, 573199, 613176, 655939, 3342976, 3575979, 3825220, 19486419, 20844460, 22297143, 113577300, 121492543, 129959400, 661979143, 708112560, 757461019, 3858299320

Offset: 1

Mohamed Bouhamida, Jun 15 2007

Also values x of Pythagorean triples (x, x+881, y).
Corresponding values y of solutions (x, y) are in A159690.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (883+42*sqrt(2))/881 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (2052963+1343918*sqrt(2))/881^2 for n mod 3 = 0.

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

Cf. A159690, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159691 (decimal expansion of (883+42*sqrt(2))/881), A159692 (decimal expansion of (2052963+1343918*sqrt(2))/881^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,43,2440,2643,2860,16443,17620},30] (* Harvey P. Dale, Aug 13 2015 *)

{forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+1762*n+776161), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+1762 for n > 6; a(1)=0, a(2)=43, a(3)=2440, a(4)=2643, a(5)=2860, a(6)=16443.
G.f.: x*(43+2397*x+203*x^2-41*x^3-799*x^4-41*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 881*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 21 2009

A157646 Positive numbers y such that y^2 is of the form x^2 + (x+31)^2 with integer x.

Original entry on oeis.org

25, 31, 41, 109, 155, 221, 629, 899, 1285, 3665, 5239, 7489, 21361, 30535, 43649, 124501, 177971, 254405, 725645, 1037291, 1482781, 4229369, 6045775, 8642281, 24650569, 35237359, 50370905, 143674045, 205378379, 293583149, 837393701

Offset: 1

Klaus Brockhaus, Mar 11 2009

(-7,a(1)) and (A118674(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+31)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (33+8*sqrt(2))/31 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (1539+850*sqrt(2))/31^2 for n mod 3 = 1.
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 x values are given by the sequence defined by: a(n)=6*a(n-3)-a(n-6)+2p with a(1)=0, a(2)=2m+1, a(3)=6m^2-10m+4, a(4)=3p, a(5)=6m^2+10m+4, a(6)=40m^2-58m+21.Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(1)=p, b(2)=2m^2+2m+1, b(3)=10m^2-14m+5, b(4)=5p, b(5)=10m^2+14m+5, b(6)=58m^2-82m+29. - Mohamed Bouhamida, Sep 09 2009

			(-7, a(1)) = (-7, 25) is a solution: (-7)^2+(-7+31)^2 = 49+576 = 625 = 25^2.
(A118674(1), a(2)) = (0, 31) is a solution: 0^2+(0+31)^2 = 961 = 31^2.
(A118674(3), a(4)) = (60, 109) is a solution: 60^2+(60+31)^2 = 3600+8281 = 11881 = 109^2.

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

Cf. A118674, A001653, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A157647 (decimal expansion of (33+8*sqrt(2))/31), A157648 (decimal expansion of (1539+850*sqrt(2))/31^2).

I:=[25,31,41,109,155,221]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..50]]; // G. C. Greubel, Mar 31 2018

LinearRecurrence[{0,0,6,0,0,-1},{25,31,41,109,155,221},40] (* Harvey P. Dale, Oct 12 2017 *)

{forstep(n=-8, 840000000, [1, 3], if(issquare(2*n^2+62*n+961, &k), print1(k, ",")))};

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=25, a(2)=31, a(3)=41, a(4)=109, a(5)=155, a(6)=221.
G.f.: (1-x)*(25 + 56*x + 97*x^2 + 56*x^3 + 25*x^4)/(1 - 6*x^3 + x^6).
a(3*k-1) = 31*A001653(k) for k >= 1.

A157647 Decimal expansion of (33+8*sqrt(2))/31.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Mar 11 2009

lim_{n -> infinity} b(n)/b(n-1) = (33+8*sqrt(2))/31 for n mod 3 = {1, 2}, b = A118674.

lim_{n -> infinity} b(n)/b(n-1) = (33+8*sqrt(2))/31 for n mod 3 = {0, 2}, b = A157646.

			(33+8*sqrt(2))/31 = 1.42947446770918581904...

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

A118674, A157646, A002193 (decimal expansion of sqrt(2)), A157648 (decimal expansion of (1539+850*sqrt(2))/31^2).

(33+8*Sqrt(2))/31; // G. C. Greubel, Mar 30 2018

with(MmaTranslator[Mma]): Digits:=100:
RealDigits(evalf((33+8*sqrt(2))/31))[1]; # Muniru A Asiru, Mar 31 2018

RealDigits[(33+8*Sqrt[2])/31,10,120][[1]] (* Harvey P. Dale, Mar 02 2014 *)

(33+8*sqrt(2))/31 \\ G. C. Greubel, Mar 30 2018

(33+8*sqrt(2))/31 = (8+sqrt(2))/(8-sqrt(2)).

A157648 Decimal expansion of (1539+850*sqrt(2))/31^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Mar 11 2009

Limit_{n -> oo} b(n)/b(n-1) = (1539+850*sqrt(2))/31^2 for n mod 3 = 0, b = A118674.

Limit_{n -> oo} b(n)/b(n-1) = (1539+850*sqrt(2))/31^2 for n mod 3 = 1, b = A157646.

			2.85232208950794046980...

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

Cf. A118674, A157646, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A157647 (decimal expansion of (33+8*sqrt(2))/31).

(1539+850*Sqrt(2))/31^2; // G. C. Greubel, Mar 30 2018

with(MmaTranslator[Mma]): Digits:=100:
RealDigits(evalf((1539+850*sqrt(2))/31^2))[1]; # Muniru A Asiru, Mar 31 2018

RealDigits[(1539+850*Sqrt[2])/31^2, 10, 100][[1]] (* G. C. Greubel, Mar 30 2018 *)

(1539+850*sqrt(2))/31^2 \\ G. C. Greubel, Mar 30 2018

Equals (50+17*sqrt(2))/(50-17*sqrt(2)).

Equals (3+2*sqrt(2))*(8-sqrt(2))^2/(8+sqrt(2))^2.

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

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A130014 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+881)^2 = y^2.

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A157646 Positive numbers y such that y^2 is of the form x^2 + (x+31)^2 with integer x.

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A157647 Decimal expansion of (33+8*sqrt(2))/31.

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A157648 Decimal expansion of (1539+850*sqrt(2))/31^2.

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

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