A156035 -id:A156035 - cp's OEIS frontend

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

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

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

A157120 Positive numbers y such that y^2 is of the form x^2+(x+103)^2 with integer x.

Original entry on oeis.org

73, 103, 205, 233, 515, 1157, 1325, 2987, 6737, 7717, 17407, 39265, 44977, 101455, 228853, 262145, 591323, 1333853, 1527893, 3446483, 7774265, 8905213, 20087575, 45311737, 51903385, 117078967, 264096157, 302515097, 682386227, 1539265205

Offset: 1

Klaus Brockhaus, Feb 25 2009

(-48, a(1)) and (A157119(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+103)^2 = y^2.

			(-48, a(1)) = (-48, 73) is a solution: (-48)^2+(-48+103)^2 = 2304+3025 = 5329 = 73^2.
(A157119(1), a(2)) = (0, 103) is a solution: 0^2+(0+103)^2 = 10609 = 103^2,
(A157119(3), a(4)) = (105, 233) is a solution: 105^2+(105+103)^2 = 11025+43264 = 54289 = 233^2.

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

Cf. A157119, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157121 (decimal expansion of 11+3*sqrt(2)), A157122 (decimal expansion of 11-3*sqrt(2)), A157123 (decimal expansion of (11+3*sqrt(2))/(11-3*sqrt(2))).

Select[Table[Sqrt[x^2+(x+103)^2],{x,-50,3*10^6}],IntegerQ] (* THe program generates the first 20 terms of the sequence. *) (* or *) LinearRecurrence[ {0,0,6,0,0,-1},{73,103,205,233,515,1157},50](* Harvey P. Dale, Aug 19 2020 *)

{forstep(n=-48, 1100000000, [1, 3], if(issquare(2*n^2+206*n+10609, &k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1) = 73, a(2) = 103, a(3) = 205, a(4) = 233, a(5) = 515, a(6) = 1157.
G.f.: x*(1-x)*(73+176*x+381*x^2+176*x^3+73*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 103*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) = (3+2*sqrt(2))*(11-3*sqrt(2))^2/(11+3*sqrt(2))^2 for n mod 3 = 1.
Limit_{n -> oo} a(n)/a(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {0, 2}.

Typo corrected by Klaus Brockhaus, Mar 01 2009

A157297 Positive numbers y such that y^2 is of the form x^2+(x+233)^2 with integer x.

Original entry on oeis.org

185, 233, 317, 793, 1165, 1717, 4573, 6757, 9985, 26645, 39377, 58193, 155297, 229505, 339173, 905137, 1337653, 1976845, 5275525, 7796413, 11521897, 30748013, 45440825, 67154537, 179212553, 264848537, 391405325, 1044527305, 1543650397

Offset: 1

Klaus Brockhaus, Apr 11 2009

(-57, a(1)) and (A129625(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+233)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (251+66*sqrt(2))/233 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (82611+44030*sqrt(2))/233^2 for n mod 3 = 1.

			(-57, a(1)) = (-57, 185) is a solution: (-57)^2+(-57+233)^2 = 3249+30976 = 34225 = 185^2.
(A129625(1), a(2)) = (0, 233) is a solution: 0^2+(0+233)^2 = 54289 = 233^2.
(A129625(3), a(4)) = (432, 793) is a solution: 432^2+(432+233)^2 = 186624+442225 = 628849 = 793^2.

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. A129625, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157298 (decimal expansion of (251+66*sqrt(2))/233), A157299 (decimal expansion of (82611+44030*sqrt(2))/233^2).

I:=[185,233,317,793,1165,1717]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Mar 29 2018

LinearRecurrence[{0,0,6,0,0,-1}, {185,233,317,793,1165,1717}, 50] (* G. C. Greubel, Mar 29 2018 *)

{forstep(n=-60, 1100000000, [3,1], if(issquare(2*n^2+466*n+54289, &k),print1(k, ",")))};

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=185, a(2)=233, a(3)=317, a(4)=793, a(5)=1165, a(6)=1717.
G.f.: (1-x)*(185 +418*x +735*x^2 +418*x^3 +185*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 233*A001653(k) for k >= 1.

A157299 Decimal expansion of (82611+44030*sqrt(2))/233^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 11 2009

lim_{n -> infinity} b(n)/b(n-1) = (82611+44030*sqrt(2))/233^2 for n mod 3 = 0, b = A129625.

lim_{n -> infinity} b(n)/b(n-1) = (82611+44030*sqrt(2))/233^2 for n mod 3 = 1, b = A157297.

			(82611+44030*sqrt(2))/233^2 = 2.66865890237962340434...

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

Cf. A129625, A157297, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A157298 (decimal expansion of (251+66*sqrt(2))/233).

(82611+44030*Sqrt(2))/233^2; // G. C. Greubel, Mar 29 2018

RealDigits[(82611+44030Sqrt[2])/233^2,10,120][[1]] (* Harvey P. Dale, Feb 25 2014 *)

(82611+44030*sqrt(2))/233^2 \\ G. C. Greubel, Mar 29 2018

(82611+44030*sqrt(2))/233^2 = (370+119*sqrt(2))/(370-119*sqrt(2))

= (3+2*sqrt(2))*(22-3*sqrt(2))^2/(22+3*sqrt(2))^2.

A157350 Decimal expansion of (130803 + 73738*sqrt(2))/281^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 12 2009

lim_{n -> infinity} b(n)/b(n-1) = (130803+73738*sqrt(2))/281^2 for n mod 3 = 0, b = A129626.

lim_{n -> infinity} b(n)/b(n-1) = (130803+73738*sqrt(2))/281^2 for n mod 3 = 1, b = A157348.

			(130803 + 73738*sqrt(2))/281^2 = 2.97722014237746840476...

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

Cf. A129626, A157348, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A157349 (decimal expansion of (297+68*sqrt(2))/281).

(130803+73738*Sqrt(2))/281^2 // G. C. Greubel, Feb 01 2018

RealDigits[(130803 + 73738*Sqrt[2])/281^2, 10, 100][[1]] (* G. C. Greubel, Feb 01 2018 *)

(130803+73738*sqrt(2))/281^2 \\ G. C. Greubel, Feb 01 2018

(130803 + 73738*sqrt(2))/281^2 = (458 + 161*sqrt(2))/(458 - 161*sqrt(2)) = (3 + 2*sqrt(2))*(17 - 2*sqrt(2))^2/(17 + 2*sqrt(2))^2.

A157469 Positive numbers y such that y^2 is of the form x^2 + (x+97)^2 with integer x.

Original entry on oeis.org

85, 97, 113, 397, 485, 593, 2297, 2813, 3445, 13385, 16393, 20077, 78013, 95545, 117017, 454693, 556877, 682025, 2650145, 3245717, 3975133, 15446177, 18917425, 23168773, 90026917, 110258833, 135037505, 524715325, 642635573, 787056257

Offset: 1

Klaus Brockhaus, Mar 12 2009

(-13,a(1)) and (A129836(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2 + (x+97)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (99+14*sqrt(2))/97 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (19491+12070*sqrt(2))/97^2 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

			(-13, a(1)) = (-13, 85) is a solution: (-13)^2+(-13+97)^2 = 169+7056 = 7225 = 85^2.
(A129836(1), a(2)) = (0, 97) is a solution: 0^2+(0+97)^2 = 9409 = 97^2.
(A129836(3), a(4)) = (228, 397) is a solution: 228^2+(228+97)^2 = 51984+105625 = 157609 = 397^2.

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. A129836, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157470 (decimal expansion of (99+14*sqrt(2))/97), A157471 (decimal expansion of (19491+12070*sqrt(2))/97^2).

I:=[85,97,113,397,485,593]; [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},{85,97,113,397,485,593},30] (* Harvey P. Dale, Apr 04 2013 *)

{forstep(n=-20, 800000000, [3, 1], if(issquare(2*n^2+194*n+9409, &k), print1(k, ",")))};

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=85, a(2)=97, a(3)=113, a(4)=397, a(5)=485, a(6)=593.
G.f.: (1-x)*(85 + 182*x + 295*x^2 + 182*x^3 + 85*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 97*A001653(k) for k >= 1.

A157471 Decimal expansion of (19491+12070*sqrt(2))/97^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Mar 12 2009

lim_{n -> infinity} b(n)/b(n-1) = (19491+12070*sqrt(2))/97^2 for n mod 3 = 0, b = A129836.

lim_{n -> infinity} b(n)/b(n-1) = (19491+12070*sqrt(2))/97^2 for n mod 3 = 1, b = A157469.

			(19491+12070*sqrt(2))/97^2 = 3.88570067997058744170...

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

Cf. A129836, A157469, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A157470 (decimal expansion of (99+14*sqrt(2))/97).

(19491+12070*Sqrt(2))/97^2; // G. C. Greubel, Mar 30 2018

with(MmaTranslator[Mma]): Digits:=100:
RealDigits(evalf((19491+12070*sqrt(2))/97^2))[1]; # Muniru A Asiru, Mar 31 2018

RealDigits[(19491+12070*Sqrt[2])/97^2, 10, 100][[1]] (* G. C. Greubel, Mar 30 2018 *)

(19491+12070*sqrt(2))/97^2 \\ G. C. Greubel, Mar 30 2018

(19491+12070*sqrt(2))/97^2 = (170+71*sqrt(2))/(170-71*sqrt(2))

= (3+2*sqrt(2))*(14-sqrt(2))^2/(14+sqrt(2))^2.

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.

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.

cp's OEIS Frontend

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

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

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A157120 Positive numbers y such that y^2 is of the form x^2+(x+103)^2 with integer x.

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A157297 Positive numbers y such that y^2 is of the form x^2+(x+233)^2 with integer x.

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A157299 Decimal expansion of (82611+44030*sqrt(2))/233^2.

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A157350 Decimal expansion of (130803 + 73738*sqrt(2))/281^2.

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