A155923 -id:A155923 - cp's OEIS frontend

A118120 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+17)^2 = y^2.

0, 7, 28, 51, 88, 207, 340, 555, 1248, 2023, 3276, 7315, 11832, 19135, 42676, 69003, 111568, 248775, 402220, 650307, 1450008, 2344351, 3790308, 8451307, 13663920, 22091575, 49257868, 79639203, 128759176, 287095935, 464171332, 750463515, 1673317776

Offset: 0

Mohamed Bouhamida, May 12 2006

Also values x of Pythagorean triples (x, x+17, y).
Corresponding values y of solutions (x, y) are in A155923.
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)+2p 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 (cf. A118673).
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)=2*m^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. A155923, A118673, A066436 (primes of the form 2*n^2-1), A066049 (2*n^2-1 is prime), A118554, A118611, A118630.

Cf. A155464 (first trisection), A155465 (second trisection), A155466 (third trisection).

[ n: n in [0..25000000] | IsSquare(2*n*(n+17)+289) ];

Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+17)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,7,28,51,88,207,340},50] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

m=32; v=concat([0, 7, 28, 51, 88, 207], vector(m-6)); for(n=7, m, v[n]=6*v[n-3]-v[n-6]+34); v

a(n) = 6*a(n-3) -a(n-6) +34 for n > 5; a(0)=0, a(1)=7, a(2)=28, a(3)=51, a(4)=88, a(5)=207.

G. f.: x*(7 +21*x +23*x^2 -5*x^3 -7*x^4 -5*x^5)/((1-x)*(1-6*x^3+x^6)).

Edited and 248755 changed to 248775 by Klaus Brockhaus, Feb 01 2009

A156163 Decimal expansion of (19+6*sqrt(2))/17.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Feb 09 2009

lim_{n -> infinity} b(n)/b(n-1) = (19+6*sqrt(2))/17 for n mod 3 = {0, 2}, b = A155923.

lim_{n -> infinity} b(n)/b(n-1) = ((19+6*sqrt(2))/17)^2 for n mod 3 = {0, 2}, b = A156159.

			(19+6*sqrt(2))/17 = 1.61678125730815119369...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2

Cf. A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A156164 (decimal expansion of 17+12*sqrt(2)).

RealDigits[(19 + 6*Sqrt[2])/17, 10, 100][[1]] (* G. C. Greubel, Jul 05 2017 *)

(6*sqrt(2)+19)/17 \\ Charles R Greathouse IV, Apr 25 2016

A156156 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1) = 13, a(2) = 53.

Original entry on oeis.org

13, 53, 305, 1777, 10357, 60365, 351833, 2050633, 11951965, 69661157, 406014977, 2366428705, 13792557253, 80388914813, 468540931625, 2730856674937, 15916599117997, 92768738033045, 540695829080273, 3151406236448593

Offset: 1

Klaus Brockhaus, Feb 09 2009

nonn

lim_{n -> infinity} a(n)/a(n-1) = 3+2*sqrt(2).

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

First trisection of A155923.

Cf. A156035 (decimal expansion of 3+2*sqrt(2)), A156157, A156158.

{m=20; v=concat([13, 53], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-v[n-2]); v}

a(n) = ((50+31*sqrt(2))*(3-2*sqrt(2))^n+(50-31*sqrt(2))*(3+2*sqrt(2))^n)/4.

G.f.: x*(13-25*x)/(1-6*x+x^2).

Replaced abbreviation by sqrt(2) Klaus Brockhaus, Feb 12 2009

G.f. corrected by Klaus Brockhaus, Sep 23 2009

A156157 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1) = 17, a(2) = 85.

Original entry on oeis.org

17, 85, 493, 2873, 16745, 97597, 568837, 3315425, 19323713, 112626853, 656437405, 3825997577, 22299548057, 129971290765, 757528196533, 4415197888433, 25733659134065, 149986756915957, 874186882361677, 5095134537254105

Offset: 1

Klaus Brockhaus, Feb 09 2009

nonn

lim_{n -> infinity} a(n)/a(n-1) = 3+2*sqrt(2).

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

Second trisection of A155923. Equals 17*A001653.

Cf. A156035 (decimal expansion of 3+2*sqrt(2)), A156156, A156158.

{m=20; v=concat([17, 85], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-v[n-2]); v}

a(n) = ((2+sqrt(2))*(3-2*sqrt(2))^n+(2-sqrt(2))*(3+2*sqrt(2))^n)*17/4.

G.f.: 17*x*(1-x)/(1-6*x+x^2).

Replaced abbreviation by sqrt(2) Klaus Brockhaus, Feb 12 2009

G.f. corrected by Klaus Brockhaus, Sep 23 2009

A156158 a(n) = 6*a(n-1) - a(n-2) for n > 2; a(1) = 25, a(2) = 137.

Original entry on oeis.org

25, 137, 797, 4645, 27073, 157793, 919685, 5360317, 31242217, 182092985, 1061315693, 6185801173, 36053491345, 210135146897, 1224757390037, 7138409193325, 41605697769913, 242495777426153, 1413368966787005

Offset: 1

Klaus Brockhaus, Feb 09 2009

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

Third trisection of A155923.

Cf. A156035 (decimal expansion of 3+2*sqrt(2)), A156156, A156157.

LinearRecurrence[{6,-1},{25,137},30] (* Harvey P. Dale, Jan 02 2019 *)

{m=19; v=concat([25,137], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-v[n-2]); v}

a(n) = ((26+7*sqrt(2))*(3-2*sqrt(2))^n+(26-7*sqrt(2))*(3+2*sqrt(2))^n)/4.
G.f.: x*(25-13*x)/(1-6*x+x^2).
Limit_{n -> oo} a(n)/a(n-1) = 3+2*sqrt(2).

A156159 Squares of the form k^2+(k+17)^2 with integer k.

Original entry on oeis.org

169, 289, 625, 2809, 7225, 18769, 93025, 243049, 635209, 3157729, 8254129, 21576025, 107267449, 280395025, 732947329, 3643933225, 9525174409, 24898630849, 123786459889, 323575532569, 845820499225, 4205095700689

Offset: 1

Klaus Brockhaus, Feb 09 2009

Square roots of k^2+(k+17)^2 are in A155923, values k (except for -5) are in A118120.

			625 = 25^2 is of the form k^2+(k+17)^2 with k = 7: 7^2+24^2 = 625. Hence 625 is in the sequence.

Index entries for linear recurrences with constant coefficients, signature (1,0,34,-34,0,-1,1).

Equals A155923^2. Cf. A156160 (first trisection), A156161 (second trisection), A156162 (third trisection).

Cf. A118120, A156035 (decimal expansion of 3+2*sqrt(2)), A156164 (decimal expansion of 17+12*sqrt(2)), A156163 (decimal expansion of (19+6*sqrt(2))/17), A157649 (decimal expansion of (387+182*sqrt(2))/17^2).

LinearRecurrence[{1,0,34,-34,0,-1,1},{169,289,625,2809,7225,18769,93025},30] (* Harvey P. Dale, Apr 22 2022 *)

{forstep(n=-5, 1600000, [1, 3], if(issquare(a=2*n*(n+17)+289), print1(a, ",")))}

a(n) = 34*a(n-3)-a(n-6)-2312 for n > 6; a(1)=169, a(2)=289, a(3)=625, a(4)=2809, a(5)=7225, a(6)=18769.
G.f.: x*(169+120*x+336*x^2-3562*x^3+336*x^4+120*x^5+169*x^6)/((1-x)*(1-34*x^3+x^6)).
Limit_{n -> oo} a(n)/a(n-3) = (17+12*sqrt(2)).
Limit_{n -> oo} a(n)/a(n-1) = ((19+6*sqrt(2))/17)^2 for n mod 3 = {0, 2}.
Limit_{n -> oo} a(n)/a(n-1) = ((387+182*sqrt(2))/17^2)^2 for n mod 3 = 1.

G.f. corrected, fourth comment and cross-references edited by Klaus Brockhaus, Sep 23 2009

A157649 Decimal expansion of (387 + 182*sqrt(2))/17^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Mar 11 2009

Lim_{n -> infinity} b(n)/b(n-1) = (387 + 182*sqrt(2))/17^2 for n mod 3 = 1, b = A155923.

			(387 + 182*sqrt(2))/17^2 = 2.22971234723841971931...

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

Cf. A118120, A155923, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A156163 (decimal expansion of (19+6*sqrt(2))/17).

SetDefaultRealField(RealField(100)); (387 + 182*Sqrt(2))/17^2; // G. C. Greubel, Aug 17 2018

RealDigits[(387 + 182*Sqrt[2])/17^2, 10, 100][[1]] (* G. C. Greubel, Aug 17 2018 *)

(387 + 182*sqrt(2))/17^2 \\ G. C. Greubel, Aug 17 2018

Equals (26 + 7*sqrt(2))/(26 - 7*sqrt(2)) = (3 + 2*sqrt(2))/((19 + 6*sqrt(2))/17)^2 = (3 + 2*sqrt(2))*(6 - sqrt(2))^2/(6 + sqrt(2))^2.

cp's OEIS Frontend

A118120 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+17)^2 = y^2.

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A156163 Decimal expansion of (19+6*sqrt(2))/17.

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A156156 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1) = 13, a(2) = 53.

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A156157 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1) = 17, a(2) = 85.

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A156158 a(n) = 6*a(n-1) - a(n-2) for n > 2; a(1) = 25, a(2) = 137.

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A156159 Squares of the form k^2+(k+17)^2 with integer k.

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A157649 Decimal expansion of (387 + 182*sqrt(2))/17^2.

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