A156163 -id:A156163 - cp's OEIS frontend

A156164 Decimal expansion of 17 + 12*sqrt(2).

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

Offset: 2

Klaus Brockhaus, Feb 09 2009

Lim_{n -> infinity} b(n)/b(n-1) = 17+12*sqrt(2) for b = A156160, A156161, A156162, A278310.

Conjecturally, the fractional part 0.97056 27484 ... of this constant equals ( (1 + 2 * Sum_{n >= 1} (-1)^n*exp(-2*Pi*n^2))/(1 + 2 * Sum_{n >= 1} exp(-2*Pi*n^2)) )^4. The series are rapidly converging. For example, summing both series from n = 1 to n = 2 approximates the fractional part of the constant as ( (1 - 2*exp(-2*Pi) + 2*exp(-8*Pi))/(1 + 2*exp(-2*Pi) + 2*exp(-8*Pi)) )^4 = 0.97056 27484 77140 58562 026(89) ..., correct to 23 decimal places. - Peter Bala, Jun 05 2019

			17 + 12*sqrt(2) = 33.97056274847714058562026469051637694283606250452337687...

G. C. Greubel, Table of n, a(n) for n = 2..10000
M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22. Published in B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001, pp. 771-808, section 2.4. Example 4.
Index entries for algebraic numbers, degree 2

Cf. A002193: decimal expansion of sqrt(2); A156035: decimal expansion of 3+2*sqrt(2); A156163: decimal expansion of (19+6*sqrt(2))/17.

Cf. A005259, A059415/A059416.

RealDigits[17 + 12 Sqrt[2], 10, 120][[1]] (* Harvey P. Dale, Nov 30 2014 *)

17+sqrt(288) \\ Charles R Greathouse IV, May 07 2015

17+12*sqrt(2) = (3+2*sqrt(2))^2 = (1+sqrt(2))^4. - Klaus Brockhaus, Feb 14 2009. (corrected by Bruno Berselli, Feb 19 2013)

A156649 Decimal expansion of (9+4*sqrt(2))/7.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Feb 13 2009

lim_{n -> infinity} b(n)/b(n-1) = ((9+4*sqrt(2))/7)/((19+6*sqrt(2))/17) for n mod 9 = {1, 2}, b = A129837, A156650.

			(9+4*sqrt(2))/7 = 2.09383632135605431360...

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)). A156163 (decimal expansion of (19+6*sqrt(2))/17), A129837, A156650.

RealDigits[(9 + 4*Sqrt[2])/7, 10, 100][[1]] (* G. C. Greubel, Jul 05 2017 *)

(9+4*sqrt(2))/7 \\ G. C. Greubel, Jul 05 2017

A155923 Positive numbers y such that y^2 is of the form x^2+(x+17)^2 with integer x.

Original entry on oeis.org

13, 17, 25, 53, 85, 137, 305, 493, 797, 1777, 2873, 4645, 10357, 16745, 27073, 60365, 97597, 157793, 351833, 568837, 919685, 2050633, 3315425, 5360317, 11951965, 19323713, 31242217, 69661157, 112626853, 182092985, 406014977, 656437405

Offset: 1

Klaus Brockhaus, Feb 09 2009

(-5,a(1)) and (A118120(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+17)^2 = y^2. (Offset 1 is assumed for A118120.)
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (19+6*sqrt(2))/17 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (387+182*sqrt(2))/17^2 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)=2*m^2+2m+1, b(3)=10m^2-14m+5, b(4)=5p, b(5)=10m^2+14m+5, b(6)=58m^2-82m+29. [From Mohamed Bouhamida, Sep 09 2009]

			(-5,a(1)) = (-5,13) is a solution: (-5)^2+(-5+17)^2 = 25+144 = 169 = 13^2;
(A118120(1), a(2)) = (0, 17) is a solution: 0^2+(0+17)^2 = 289 = 17^2;
(A118120(2), a(3)) = (7, 25) is a solution: 7^2+(7+17)^2 = 49+576 = 625 = 25^2.

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

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

Cf. A156156 (first trisection), A156157 (second trisection), A156158 (third trisection).

LinearRecurrence[{0,0,6,0,0,-1},{13,17,25,53,85,137},50] (* Harvey P. Dale, Feb 11 2015 *)

{forstep(n=-5, 660000000, [1,3], if(issquare(2*n*(n+17)+289, &k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1) = 13, a(2) = 17, a(3) = 25, a(4) = 53, a(5) = 85, a(6) = 137.

G.f.: x*(1-x)*(13+30*x+55*x^2+30*x^3+13*x^4)/(1-6*x^3+x^6).

G.f. corrected, first and fourth comment and examples edited, cross-reference added by Klaus Brockhaus, Sep 22 2009

A129837 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+119)^2 = y^2.

Original entry on oeis.org

0, 24, 49, 57, 85, 136, 180, 196, 261, 357, 481, 616, 660, 816, 1105, 1357, 1449, 1824, 2380, 3100, 3885, 4141, 5049, 6732, 8200, 8736, 10921, 14161, 18357, 22932, 24424, 29716, 39525, 48081, 51205, 63940, 82824, 107280, 133945, 142641, 173485, 230656

Offset: 1

Mohamed Bouhamida, May 21 2007

nonn

Also values x of Pythagorean triples (x, x+119, y).
Corresponding values y of solutions (x, y) are in A156650.
lim_{n -> infinity} a(n)/a(n-9) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)/((19+6*sqrt(2))/17) for n mod 9 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = ((19+6*sqrt(2))/17)^2/((9+4*sqrt(2))/7) for n mod 9 = {0, 3}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/(((9+4*sqrt(2))/7)*((19+6*sqrt(2))/17)^2) for n mod 9 = {4, 8}.
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^2*((19+6*sqrt(2))/17)/(3+2*sqrt(2)) for n mod 9 = {5, 7}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/((9+4*sqrt(2))/7)^2 for n mod 9 = 6.

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

Cf. A156650, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7), A156163 (decimal expansion of (19+6*sqrt(2))/17), A118630.

LinearRecurrence[{1,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,-1,1}, {0,24,49,57,85,136,180,196,261,357,481,616,660,816,1105,1357,1449,1824,2380}, 140] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)

{forstep(n=0, 240000, [1, 3], if(issquare(n^2+(n+119)^2), print1(n, ",")))}

a(n) = 6*a(n-9)-a(n-18)+238 for n > 18; a(1)=0, a(2)=24, a(3)=49, a(4)=57, a(5)=85, a(6)=136, a(7)=180, a(8)=196, a(9)=261, a(10)=357, a(11)=481, a(12)=616, a(13)=660, a(14)=816, a(15)=1105, a(16)=1357, a(17)=1449, a(18)=1824.

G.f.: x*(24+25*x+8*x^2+28*x^3+51*x^4+44*x^5+16*x^6+65*x^7+96*x^8-20*x^9-15*x^10-4*x^11-12*x^12-17*x^13-12*x^14-4*x^15-15*x^16-20*x^17 )/((1-x)*(1-6*x^9+x^18))

Edited and extended by Klaus Brockhaus, Feb 13 2009

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.

A129010 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+833)^2 = y^2.

Original entry on oeis.org

0, 124, 168, 187, 343, 399, 595, 624, 915, 952, 1260, 1372, 1768, 1827, 1975, 2499, 3135, 3367, 3468, 4312, 4620, 5712, 5875, 7524, 7735, 9499, 10143, 12427, 12768, 13624, 16660, 20352, 21700, 22287, 27195, 28987, 35343, 36292, 45895, 47124

Offset: 1

Mohamed Bouhamida, May 27 2007

nonn

Also values x of Pythagorean triples (x, x+833, y); 833=7^2*17.
Corresponding values y of solutions (x, y) are in A156835.
lim_{n -> infinity} a(n)/a(n-15) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^2*((19+6*sqrt(2))/17)/(3+2*sqrt(2)) for n mod 15 = {1, 2, 5, 7, 11, 13}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/(((9+4*sqrt(2))/7)*((19+6*sqrt(2))/17)^2) for n mod 15 = {0, 3, 6, 12}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*((19+6*sqrt(2))/17)/((9+4*sqrt(2))/7)^3 for n mod 15 = {4, 8, 10, 14}.
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^4/((3+2*sqrt(2))*((19+6*sqrt(2))/17)^2) for n mod 15 = 9.

			124^2+(124+833)^2 = 15376+915849 = 931225 = 965^2.

Cf. A156835, A076296, A118120, A118554, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7), A156163 (decimal expansion of (19+6*sqrt(2))/17).

{forstep(n=0, 50000, [3, 1], if(issquare(2*n^2+1666*n+693889), print1(n, ",")))}

a(n) = 6*a(n-15)-a(n-30)+1666 for n > 30; a(1) = 0, a(2) = 124, a(3) = 168, a(4) = 187, a(5) = 343, a(6) = 399, a(7) = 595, a(8) = 624, a(9) = 915, a(10) = 952, a(11) = 1260, a(12) = 1372, a(13) = 1768, a(14) = 1827, a(15) = 1975, a(16) = 2499, a(17) = 3135, a(18) = 3367, a(19) = 3468, a(20) = 4312, a(21) = 4620, a(22) = 5712, a(23) = 5875, a(24) = 7524, a(25) = 7735, a(26) = 9499, a(27) = 10143, a(28) = 12427, a(29) = 12768, a(30) = 13624.

G.f.: x*(124+44*x+19*x^2+156*x^3+56*x^4+196*x^5+29*x^6+291*x^7+37*x^8+308*x^9+112*x^10+396*x^11+59*x^12+148*x^13+524*x^14-108*x^15-32*x^16-13*x^17-92*x^18-28*x^19-84*x^20-11*x^21-97*x^22-11*x^23-84*x^24-28*x^25-92*x^26-13*x^27-32*x^28-108*x^29)/((1-x)*(1-6*x^15+x^30)).

Edited by Klaus Brockhaus, Feb 16 2009

A156650 Positive numbers y such that y^2 is of the form x^2+(x+119)^2 with integer x.

Original entry on oeis.org

85, 89, 91, 101, 119, 145, 175, 185, 221, 289, 349, 371, 461, 595, 769, 959, 1021, 1241, 1649, 2005, 2135, 2665, 3451, 4469, 5579, 5941, 7225, 9605, 11681, 12439, 15529, 20111, 26045, 32515, 34625, 42109, 55981, 68081, 72499, 90509, 117215, 151801

Offset: 1

Klaus Brockhaus, Feb 17 2009

nonn

(-51, a(1)), (-39, a(2)), (-35, a(3)), (-20, a(4)) and (A129837(n), a(n+4)) are solutions (x, y) to the Diophantine equation x^2+(x+119)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-9) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/((9+4*sqrt(2))/7)^2 for n mod 9 = 1.
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^2*((19+6*sqrt(2))/17)/(3+2*sqrt(2)) for n mod 9 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/(((9+4*sqrt(2))/7)*((19+6*sqrt(2))/17)^2) for n mod 9 = {3, 8}.
lim_{n -> infinity} a(n)/a(n-1) = ((19+6*sqrt(2))/17)^2/((9+4*sqrt(2))/7) for n mod 9 = {4, 7}.
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)/((19+6*sqrt(2))/17) for n mod 9 = {5, 6}.

			(-51, a(1)) = (-51, 85) is a solution: (-51)^2+(-51+119)^2 = 2601+4624 = 7225 = 85^2.
(A129837(1), a(5)) = (0, 119) is a solution: 0^2+(0+119)^2 = 14161 = 119^2.
(A129837(3), a(7)) = (49, 175) is a solution: 49^2+(49+119)^2 = 2401+28224 = 30625 = 175^2.

Cf. A129837, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7), A156163 (decimal expansion of (19+6*sqrt(2))/17).

upto=200000; With[{max=Ceiling[(Sqrt[2*upto^2]-119)/2]},Union[ Sqrt[#]&/@ Select[Table[x^2+(x+119)^2,{x,-250,max}],IntegerQ[Sqrt[#]]&]]](* Harvey P. Dale, Aug 11 2011 *)

{forstep(n=-52, 120000, [1, 3], if(issquare(n^2+(n+119)^2, &k), print1(k, ",")))}

a(n) = 6*a(n-9)-a(n-18) for n > 18; a(1)=85, a(2)=89, a(3)=91, a(4)=101, a(5)=119, a(6)=145, a(7)=175, a(8)=185, a(9)=221, a(10)=289, a(11)=349, a(12)=371, a(13)=461, a(14)=595, a(15)=769, a(16)=959, a(17)=1021, a(18)=1241.

G.f.: x * (1-x) * (85 +174*x +265*x^2 +366*x^3 +485*x^4 +630*x^5 +805*x^6 +990*x^7 +1211*x^8 +990*x^9 +805*x^10 +630*x^11 +485*x^12 +366*x^13 +265*x^14 +174*x^15 +85*x^16) / (1 -6*x^9 +x^18). [adapted to the offset by Bruno Berselli, Apr 01 2011]

A156835 Positive numbers y such that y^2 is of the form x^2+(x+833)^2 with integer x.

Original entry on oeis.org

593, 595, 623, 637, 697, 707, 733, 833, 965, 1015, 1037, 1225, 1295, 1547, 1585, 1973, 2023, 2443, 2597, 3145, 3227, 3433, 4165, 5057, 5383, 5525, 6713, 7147, 8687, 8917, 11245, 11543, 14035, 14945, 18173, 18655, 19865, 24157, 29377, 31283, 32113

Offset: 1

Klaus Brockhaus, Feb 17 2009

nonn

(-368, a(1)), (-357, a(2)), (-273, a(3)), (-245, a(4)), (-153, a(5)), (-140, a(6)), (-108, a(7)) and (A129010(n), a(n+7)) are solutions (x, y) to the Diophantine equation x^2+(x+833)^2 = y^2.
lim_{n -> oo} a(n)/a(n-15) = 3+2*sqrt(2).
lim_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^4/((3+2*sqrt(2))*((19+6*sqrt(2))/17)^2) for n mod 15 = 1.
lim_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))*((19+6*sqrt(2))/17)/((9+4*sqrt(2))/7)^3 for n mod 15 = {0, 2, 6, 11}.
lim_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^2*((19+6*sqrt(2))/17)/(3+2*sqrt(2)) for n mod 15 = {3, 5, 8, 9, 12, 14}.
lim_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))/(((9+4*sqrt(2))/7)*((19+6*sqrt(2))/17)^2) for n mod 15 = {4, 7, 10, 13}.

			(-368, a(1)) = (-368, 593) is a solution: (-368)^2+(-368+833)^2 = 135424+216225 = 351649 = 593^2.
(A129010(1), a(8)) = (0, 833) is a solution: 0^2+(0+833)^2 = 693889 = 833^2.
(A129010(3), a(10)) = (168, 1015) is a solution: (168)^2+(168+833)^2 = 28224+1002001 = 1030225 = 1015^2.

Cf. A129010, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7), A156163 (decimal expansion of (19+6*sqrt(2))/17).

{forstep(n=-400, 26000, [3, 1], if(issquare(2*n^2+1666*n+693889, &k), print1(k, ",")))}

a(n) = 6*a(n-15)-a(n-30) for n > 30.

G.f.: (1-x)*(593 +1188*x+1811*x^2+2448*x^3+3145*x^4+3852*x^5+4585*x^6+5418*x^7+6383*x^8+7398*x^9+8435*x^10+9660*x^11+10955*x^12+12502*x^13+14087*x^14+12502*x^15+10955*x^16+9660*x^17+8435*x^18+7398*x^19+6383*x^20+5418*x^21+4585*x^22+3852*x^23+3145*x^24+2448*x^25+1811*x^26+1188*x^27+593*x^28)/(1-6*x^15+x^30).

cp's OEIS Frontend

A156164 Decimal expansion of 17 + 12*sqrt(2).

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A156649 Decimal expansion of (9+4*sqrt(2))/7.

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A155923 Positive numbers y such that y^2 is of the form x^2+(x+17)^2 with integer x.

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A129837 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+119)^2 = y^2.

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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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Magma

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A129010 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+833)^2 = y^2.

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