A155923 -id:A155923

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A118120

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

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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).

FORMULA

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)).

MATHEMATICA

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 *)

PROG

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

(PARI) 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

CROSSREFS

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).

KEYWORD

nonn,easy

AUTHOR

Mohamed Bouhamida, May 12 2006

EXTENSIONS

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

STATUS

approved

A156163

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

+10
10

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

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

Index entries for algebraic numbers, degree 2

EXAMPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

Klaus Brockhaus, Feb 09 2009

STATUS

approved

A156156

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..20.

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

FORMULA

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).

PROG

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

CROSSREFS

First trisection of A155923.

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

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, Feb 09 2009

EXTENSIONS

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

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

STATUS

approved

A156157

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..20.

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

FORMULA

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).

PROG

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

CROSSREFS

Second trisection of A155923. Equals 17*A001653.

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

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, Feb 09 2009

EXTENSIONS

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

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

STATUS

approved

A156158

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..19.

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

FORMULA

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).

MATHEMATICA

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

PROG

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

CROSSREFS

Third trisection of A155923.

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

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Feb 09 2009

STATUS

approved

A156159

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..22.

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

FORMULA

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.

EXAMPLE

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.

MATHEMATICA

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

PROG

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

CROSSREFS

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).

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Feb 09 2009

EXTENSIONS

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

STATUS

approved

A157649

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

FORMULA

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.

EXAMPLE

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

MATHEMATICA

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

PROG

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

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

CROSSREFS

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).

KEYWORD

cons,nonn

AUTHOR

Klaus Brockhaus, Mar 11 2009

STATUS

approved

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