A156567 -id:A156567 - cp's OEIS frontend

A118337 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 23)^2 = y^2.

0, 12, 33, 69, 133, 252, 460, 832, 1525, 2737, 4905, 8944, 16008, 28644, 52185, 93357, 167005, 304212, 544180, 973432, 1773133, 3171769, 5673633, 10334632, 18486480, 33068412, 60234705, 107747157, 192736885, 351073644, 627996508, 1123352944, 2046207205

Offset: 1

Mohamed Bouhamida, May 14 2006

Also values x of Pythagorean triples (x, x+23, y).
Corresponding values y of solutions (x, y) are in A156567.
For the generic case x^2 + (x + p)^2 = y^2 with p = m^2 - 2 a (prime) number in A028871, m>=5, 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 + 2, a(3) = 3*m^2 - 10m + 8, a(4)=3p, a(5) = 3*m^2 + 10m + 8, a(6) = 20*m^2 - 58m + 42. Pairs (p, m) are (23, 5), (47, 7), (79, 9), (167, 13), (223, 15), (359, 19), (439, 21), (727, 27), (839, 29), ...
Limit_{n -> oo} a(n)/a(n-3) = 3 + 2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = (27 + 10*sqrt(2))/23 for n mod 3 = {1, 2}.
Limit_{n -> oo} a(n)/a(n-1) = (3 + 2*sqrt(2))/((27 + 10*sqrt(2))/23)^2 for n mod 3 = 0.
For the generic case x^2 + (x + p)^2=y^2 with p = m^2 - 2 a prime number in A028871, m>=5, Y values are given by the sequence defined by b(n) = 6*b(n-3) - b(n-6) with b(1) = p, b(2) = m^2 + 2m + 2, b(3) = 5m^2 - 14m + 10, b(4) = 5p, b(5) = 5m^2 + 14m + 10, b(6) = 29m^2 - 82m + 58. - Mohamed Bouhamida, Sep 09 2009
For the generic case x^2 + (x + p)^2 = y^2 with p = m^2 - 2 a prime number, m>=5, the first three consecutive solutions are: (0;p), (2m+2; m^2+2m+2), (3*m^2-10m+8; 5*m^2-14m+10) and the other solutions are defined by: (X(n); Y(n))= (3*X(n-3)+2*Y(n-3)+p; 4*X(n-3)+3*Y(n-3)+2p). - Mohamed Bouhamida, Aug 19 2019
X(n) = 6*X(n-3) - X(n-6) + 2*p, and Y(n) = 6*Y(n-3) - Y(n-6) (can be easily proved using X(n) = 3*X(n-3) + 2*Y(n-3) + p, and Y(n) = 4*X(n-3) + 3*Y(n-3) + 2*p). - Mohamed Bouhamida, Aug 20 2019

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

Cf. A156567, A028871 (primes of form n^2 - 2), A156035 (decimal expansion of 3 + 2*sqrt(2)), A156571 (decimal expansion of (27 + 10*sqrt(2))/23).

Cf. A118675 (p=47), A118676 (p=79), A130608 (p=167), A130609 (p=223), A130610 (p=359), A130645 (p=439), A130646 (p=727), A130647 (p=839).

I:=[0,12,33,69,133,252,460]; [n le 7 select I[n] else Self(n-1) +6*Self(n-3) -6*Self(n-4) -Self(n-6) +Self(n-7): n in [1..30]]; // G. C. Greubel, May 04 2018

Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+23)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,12,33,69,133,252,460},50] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

forstep(n=0, 1124000000, [1, 3], if(issquare(2*n*(n+23)+529), print1(n, ",")))

x='x+O('x^30); concat([0], Vec(x*(12+21*x+36*x^2-8*x^3-7*x^4-8*x^5)/((1-x)*(1-6*x^3+x^6)))) \\ G. C. Greubel, May 04 2018

a(n) = 6*a(n-3) - a(n-6) + 46 for n > 6; a(1)=0, a(2)=12, a(3)=33, a(4)=69, a(5)=133, a(6)=252.

G.f.: x*(12 + 21*x + 36*x^2 - 8*x^3 - 7*x^4 - 8*x^5)/((1-x)*(1 - 6*x^3 + x^6)).

Edited by Klaus Brockhaus, Feb 10 2009

A156571 Decimal expansion of (27 + 10*sqrt(2))/23.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Feb 10 2009

Lim_{n -> infinity} a(n)/a(n-1) = (27+10*sqrt(2))/23 for n mod 3 = {1, 2}, b = A118337, A156567.

Lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/((27+10*sqrt(2))/23)^2 for n mod 3 = 0, b = A118337, A156567.

			(27 + 10*sqrt(2))/23 = 1.78878850537960654295...

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)).

(27+10*Sqrt(2))/23; // G. C. Greubel, Jan 27 2018

RealDigits[(27 + 10*Sqrt[2])/23, 10, 100][[1]] (* G. C. Greubel, Jan 28 2018 *)

(27+10*sqrt(2))/23 \\ G. C. Greubel, Jan 27 2018

A156572 Squares of the form k^2+(k+23)^2 with integer k.

Original entry on oeis.org

289, 529, 1369, 4225, 13225, 42025, 139129, 444889, 1423249, 4721929, 15108769, 48344209, 160402225, 513249025, 1642275625, 5448949489, 17435353849, 55789022809, 185103876169, 592288777609, 1895184495649, 6288082836025

Offset: 1

Klaus Brockhaus, Feb 11 2009

Square roots of k^2+(k+17)^2 are in A156567, values k are in A118337.

			4225 = 65^2 is of the form k^2+(k+23)^2 with k = 33: 33^2+56^2 = 4225. Hence 4225 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,34,-34,0,-1,1).

Cf. A156567, A156575 (first trisection), A156573 (second trisection), A156574 (third trisection).

Cf. A118337, A156035 (decimal expansion of 3+2*sqrt(2)), A156164 (decimal expansion of 17+12*sqrt(2)), A156571 (decimal expansion of (27+10*sqrt(2))/23), A157472 (decimal expansion of (627+238*sqrt(2))/23^2).

LinearRecurrence[{1,0,34,-34,0,-1,1}, {289,529,1369,4225,13225,42025,139129}, 30] (* Harvey P. Dale, Mar 21 2020 *)

{forstep(n=-8, 1800000, [1, 3], if(issquare(a=2*n*(n+23)+529), print1(a, ",")))}

def f(n,p,q): return p*chebyshev_U(n,17) - q*chebyshev_U(n-1,17)
def a(n):
    if (n%3==0): return -289*bool(n==0) + (1/4)*(529 + 3*f(n/3, 209, 5457))
    elif (n%3==1): return (1/4)*(529 + 3*f((n-1)/3, 209, 1649))
    else: return (1/4)*(529 + 3*f((n-2)/3, 529, 529))
[a(n) for n in (1..30)] # G. C. Greubel, Jan 04 2022

a(n) = 34*a(n-3) - a(n-6) - 4232 for n > 6; a(1)=289, a(2)=529, a(3)=1369, a(4)=4225, a(5)=13225, a(6)=42025.
a(n) = A156567(n)^2.
G.f.: x*(289 +240*x +840*x^2 -6970*x^3 +840*x^4 +240*x^5 +289*x^6)/((1-x)*(1 -34*x^3 +x^6)).
Limit_{n -> infinity} a(n)/a(n-3) = 17 + 12*sqrt(2).
Limit_{n -> infinity} a(n)/a(n-1) = ((627 + 238*sqrt(2))/23^2)^2 for n mod 3 = 1.
Limit_{n -> infinity} a(n)/a(n-1) = ((27 + 10*sqrt(2))/23)^2 for n mod 3 = {0, 2}.
a(n) = -289*[n=0] + (529/4) + (3/4)*( f(n/3, 209, 5457)*(n mod 3 = 1) + f((n-1)/3, 209, 1649)*(n mod 3 = 1) + f((n-2)/2, 529, 529)*(n mod 3 = 2) ), where f(n, p, q) = p*ChebyshevU(n, 17) - q*ChebyshevU(n-1, 17). - G. C. Greubel, Jan 04 2022

Revised by Klaus Brockhaus, Feb 16 2009

G.f. corrected, third comment and cross-references edited by Klaus Brockhaus, Sep 22 2009

A156568 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1)=23, a(2)=115.

Original entry on oeis.org

23, 115, 667, 3887, 22655, 132043, 769603, 4485575, 26143847, 152377507, 888121195, 5176349663, 30169976783, 175843511035, 1024891089427, 5973503025527, 34816127063735, 202923259356883, 1182723429077563

Offset: 1

Klaus Brockhaus, Feb 11 2009, Feb 16 2009

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

Second trisection of A156567. Equals 23*A001653.

Cf. A156035 (decimal expansion of 3+2*sqrt(2)), A156569, A156570.

{m=19; v=concat([23, 115], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-v[n-2]); v}

a(n) = 23*((2+sqrt(2))*(3-2*sqrt(2))^n +(2-sqrt(2))*(3+2*sqrt(2))^n)/4.
G.f.: 23*x*(1-x)/(1-6*x+x^2). [corrected by Klaus Brockhaus, Sep 22 2009]
Limit_{n -> oo} a(n)/a(n-1) = 3+2*sqrt(2).

A156569 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1)=37, a(2)=205.

Original entry on oeis.org

37, 205, 1193, 6953, 40525, 236197, 1376657, 8023745, 46765813, 272571133, 1588660985, 9259394777, 53967707677, 314546851285, 1833313400033, 10685333548913, 62278687893445, 362986793811757, 2115642074977097

Offset: 1

Klaus Brockhaus, Feb 11 2009, Feb 16 2009

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

Third trisection of A156567.

Cf. A156035 (decimal expansion of 3+2*sqrt(2)), A156568, A156570.

LinearRecurrence[{6,-1},{37,205},30] (* Harvey P. Dale, Aug 18 2014 *)

{m=19; v=concat([37, 205], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-v[n-2]); v}

a(n) = ((34+7*sqrt(2))*(3-2*sqrt(2))^n+(34-7*sqrt(2))*(3+2*sqrt(2))^n)/4.
G.f.: x*(37-17*x)/(1-6*x+x^2). [corrected by Klaus Brockhaus, Sep 22 2009]
Limit_{n -> oo} a(n)/a(n-1) = 3+2*sqrt(2).

A156570 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1)=17, a(2)=65.

Original entry on oeis.org

17, 65, 373, 2173, 12665, 73817, 430237, 2507605, 14615393, 85184753, 496493125, 2893773997, 16866150857, 98303131145, 572952636013, 3339412684933, 19463523473585, 113441728156577, 661186845465877, 3853679344638685

Offset: 1

Klaus Brockhaus, Feb 11 2009, Feb 16 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 A156567.

Cf. A156035 (decimal expansion of 3+2*sqrt(2)), A156568, A156569.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((74+47*r2)*(3-2*r2)^n+(74-47*r2)*(3+2*r2)^n)/4: n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ];

{m=20; v=concat([17, 65], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-v[n-2]); v}

a(n) = ((74+47*sqrt(2))*(3-2*sqrt(2))^n+(74-47*sqrt(2))*(3+2*sqrt(2))^n)/4.

G.f.: x*(17-37*x)/(1-6*x+x^2).

G.f. corrected by Klaus Brockhaus, Sep 22 2009

A157472 Decimal expansion of (627 + 238*sqrt(2))/23^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Mar 12 2009

			(627 + 238*sqrt(2))/23^2 = 1.82151763297693123178...

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

Cf. A118337, A156567, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A156571 (decimal expansion of (27+10*sqrt(2))/23).

SetDefaultRealField(RealField(100)); (627+238*sqrt(2))/23^2; // G. C. Greubel, Aug 17 2018

RealDigits[(627+238Sqrt[2])/23^2,10,120][[1]] (* Harvey P. Dale, Feb 03 2015 *)

(627 + 238*sqrt(2))/23^2 \\ G. C. Greubel, Aug 17 2018

Equals (34 + 7*sqrt(2))/(34 - 7*sqrt(2)) = (3+2*sqrt(2))*(10- 2*sqrt(2) )^2/(10+2*sqrt(2))^2.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, where b is A118337.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, where b is A156567.

cp's OEIS Frontend

A118337 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 23)^2 = y^2.

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A156571 Decimal expansion of (27 + 10*sqrt(2))/23.

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A156572 Squares of the form k^2+(k+23)^2 with integer k.

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A156568 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1)=23, a(2)=115.

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A156569 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1)=37, a(2)=205.

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A156570 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1)=17, a(2)=65.

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A157472 Decimal expansion of (627 + 238*sqrt(2))/23^2.

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