A156035 -id:A156035 - cp's OEIS frontend

A129836 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 97)^2 = y^2.

0, 15, 228, 291, 368, 1575, 1940, 2387, 9416, 11543, 14148, 55115, 67512, 82695, 321468, 393723, 482216, 1873887, 2295020, 2810795, 10922048, 13376591, 16382748, 63658595, 77964720, 95485887, 371029716, 454411923, 556532768

Offset: 1

Mohamed Bouhamida, May 21 2007

Also values x of Pythagorean triples (x, x + 97, y).
Corresponding values y of solutions (x, y) are in A157469.
For the generic case x^2 + (x + p)^2 = y^2 with p = 2*m^2 - 1 a (prime) number in A066436, 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 (cf. A118673).
Pairs (p, m) are (7, 2), (17, 3), (31, 4), (71, 6), (97, 7), (127, 8), (199, 10), (241, 11), (337, 13), (449, 15), (577, 17), (647, 18), (881, 21), (967, 22), ...
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 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (19491 + 12070*sqrt(2))/97^2 for n mod 3 = 0.
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(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

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. A157469, A066436 (primes of the form 2*n^2 - 1), A001652, A118673, A118674, 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).

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(15+213*x+63*x^2-13*x^3-71*x^4-13*x^5)/((1-x)*(1-6*x^3 + x^6)))); // G. C. Greubel, May 07 2018

ClearAll[a]; Evaluate[Array[a, 6]] = {0, 15, 228, 291, 368, 1575}; a[n_] := a[n] = 6*a[n-3] - a[n-6] + 194; Table[a[n], {n, 1, 29}] (* Jean-François Alcover, Dec 27 2011, after given formula *)
LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,15,228,291,368,1575,1940}, 50] (* G. C. Greubel, May 07 2018 *)

forstep(n=0, 600000000, [3, 1], if(issquare(2*n^2+194*n+9409), print1(n, ",")))

a(n) = 6*a(n-3) - a(n-6) + 194 for n > 6; a(1)=0, a(2)=15, a(3)=228, a(4)=291, a(5)=368, a(6)=1575.
G.f.: x*(15 + 213*x + 63*x^2 - 13*x^3 - 71*x^4 - 13*x^5)/((1-x)*(1 - 6*x^3 + x^6)).
a(3*k + 1) = 97*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Mar 12 2009

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

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

Original entry on oeis.org

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

A244920 Decimal expansion of 2*log(1+sqrt(2)), the integral over the square [0,1]x[0,1] of 1/sqrt(x^2+y^2) dx dy.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 08 2014

Number field regulator of the cyclotomic number field Q(zeta_8), where zeta_8 = sqrt(i), an eighth root of 1. - Alonso del Arte, Mar 11 2017

			1.7627471740390860504652186499595846180563206565232708215065912173...

G. C. Greubel, Table of n, a(n) for n = 1..10000
D. H. Bailey, J. M. Borwein and R. E. Crandall, Advances in the theory of box integrals, Math. Comp., Vol. 79, No. 271 (2010), pp. 1839-1866. See p. 1860.
LMFDB, Global Number Field 4.0.256.1: Q(zeta_8).
Index entries for transcendental numbers

Equals twice A091648. - Michel Marcus, Mar 18 2017

Cf. A156035.

RealDigits[2 * Log[1 + Sqrt[2]], 10, 101] // First
RealDigits[NumberFieldRegulator[Sqrt[I]], 10, 100][[1]] (* Alonso del Arte, Mar 11 2017 *)

2*asinh(1) \\ Michel Marcus, Mar 18 2017

Equals 2*arcsinh(1).
Equals Integral_{x>=1} 1/(x*(1+x)^(1/2)) dx. - Pointed out by Robert FERREOL.
Equals arccosh(3). - Vaclav Kotesovec, Dec 11 2016
Equals Integral_{x>=1} arcsinh(x)/x^2 dx. - Amiram Eldar, Jun 26 2021
Equals Integral_{x = 0..Pi/2} x/cos(x/2) dx. - Peter Bala, Aug 13 2024
Equals log(A156035). - Hugo Pfoertner, Aug 17 2024
Equals arcsinh(2*sqrt(2)). - Akiva Weinberger, Dec 03 2024
Equals Integral_{x=0..oo} erf(sqrt(x))/(x*e^x) dx. - Kritsada Moomuang, May 25 2025

A129288 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 41)^2 = y^2.

Original entry on oeis.org

0, 36, 39, 123, 319, 336, 820, 1960, 2059, 4879, 11523, 12100, 28536, 67260, 70623, 166419, 392119, 411720, 970060, 2285536, 2399779, 5654023, 13321179, 13987036, 32954160, 77641620, 81522519, 192071019, 452528623, 475148160, 1119472036

Offset: 1

Mohamed Bouhamida, May 26 2007

Also values x of Pythagorean triples (x, x+41, y).
Corresponding values y of solutions (x, y) are in A157257.
lim_{n -> infinity} a(n)/a(n-3) = 3 + 2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (7 + 2*sqrt(2))/(7 - 2*sqrt(2)) for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3 + 2*sqrt(2))*(7 - 2*sqrt(2))^2/(7 + 2*sqrt(2))^2 for n mod 3 = 0.

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. A157257, A001652, A156035 (decimal expansion of 3 + 2*sqrt(2)), A157258 (decimal expansion of 7 + 2*sqrt(2)), A157259 (decimal expansion of 7 - 2*sqrt(2)), A157260 (decimal expansion of (7 + 2*sqrt(2))/(7 - 2*sqrt(2))).

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(36+3*x+84*x^2-20*x^3-x^4-20*x^5)/((1-x)*(1-6*x^3+ x^6)))); // G. C. Greubel, May 07 2018

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,36,39,123,319,336,820},40] (* Harvey P. Dale, Jan 18 2015 *)

forstep(n=0, 1200000000, [3 ,1], if(issquare(2*n^2+82*n+1681), print1(n, ",")))

a(n) = 6*a(n-3) - a(n-6) + 82 for n > 6; a(1)=0, a(2)=36, a(3)=39, a(4)=123, a(5)=319, a(6)=336.
G.f.: x*(36 + 3*x + 84*x^2 - 20*x^3 - x^4 - 20*x^5)/((1-x)*(1 - 6*x^3 + x^6)).
a(3*k + 1) = 41*A001652(k) for k >= 0.

Edited and extended by Klaus Brockhaus, Feb 26 2009

A118676 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+79)^2 = y^2.

Original entry on oeis.org

0, 20, 161, 237, 341, 1140, 1580, 2184, 6837, 9401, 12921, 40040, 54984, 75500, 233561, 320661, 440237, 1361484, 1869140, 2566080, 7935501, 10894337, 14956401, 46251680, 63497040, 87172484, 269574737, 370088061, 508078661, 1571196900, 2157031484, 2961299640

Offset: 1

Mohamed Bouhamida, May 19 2006

Also values x of Pythagorean triples (x, x+79, y).
Corresponding values y of solutions (x, y) are in A159758.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (83+18*sqrt(2))/79 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (10659+6110*sqrt(2))/79^2 for n mod 3 = 0.

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. A159758, A028871, A118337, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159759 (decimal expansion of (83+18*sqrt(2))/79), A159760 (decimal expansion of (10659+6110*sqrt(2))/79^2).

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(20+141*x+76*x^2-16*x^3-47*x^4-16*x^5)/((1-x)*(1- 6*x^3+x^6)))); // G. C. Greubel, May 07 2018

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,20,161,237,341,1140,1580},75] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)

forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+158*n+6241), print1(n, ",")))

a(n) = 6*a(n-3) -a(n-6) +158 for n > 6; a(1)=0, a(2)=20, a(3)=161, a(4)=237, a(5)=341, a(6)=1140.
G.f.: x*(20+141*x+76*x^2-16*x^3-47*x^4-16*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 79*A001652(k) for k >= 0.

Edited by Klaus Brockhaus, Apr 30 2009

A129298 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+89)^2 = y^2.

Original entry on oeis.org

0, 51, 120, 267, 540, 931, 1780, 3367, 5644, 10591, 19840, 33111, 61944, 115851, 193200, 361251, 675444, 1126267, 2105740, 3936991, 6564580, 12273367, 22946680, 38261391, 71534640, 133743267, 223003944, 416934651, 779513100, 1299762451

Offset: 1

Mohamed Bouhamida, May 26 2007

Also values x of Pythagorean triples (x, x+89, y).
Corresponding values y of solutions (x, y) are in A160055.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (107+42*sqrt(2))/89 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (8979+2990*sqrt(2))/89^2 for n mod 3 = 0.

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. A160055, A129289, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A160056 (decimal expansion of (107+42*sqrt(2))/89), A160057 (decimal expansion of (8979+2990*sqrt(2))/89^2).

I:=[0,51,120,267,540,931,1780]; [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, Apr 19 2018

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,51,120,267,540,931,1780},30] (* Harvey P. Dale, Sep 21 2013 *)

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+178*n+7921), print1(n, ",")))};

x='x+O('x^30); concat(0, Vec(x*(51+69*x+147*x^2-33*x^3-23*x^4-33*x^5)/((1-x)*(1-6*x^3+x^6)))) \\ G. C. Greubel, Apr 19 2018

a(n) = 6*a(n-3) - a(n-6) + 178 with for n > 6; a(1)=0, a(2)=51, a(3)=120, a(4)=267, a(5)=540, a(6)=931.
G.f.: x*(51+69*x+147*x^2-33*x^3-23*x^4-33*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 89*A001652(k), k >= 0. (Zak Seidov, May 28 2007)
a(1)=0, a(2)=51, a(3)=120, a(4)=267, a(5)=540, a(6)=931, a(7)=1780, a(n) = a(n-1) + 6*a(n-3) - 6*a(n-4) - a(n-6) + a(n-7). - Harvey P. Dale, Sep 21 2013

Edited and three terms added by Klaus Brockhaus, May 04 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

A118675 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+47)^2 = y^2.

Original entry on oeis.org

0, 16, 85, 141, 225, 616, 940, 1428, 3705, 5593, 8437, 21708, 32712, 49288, 126637, 190773, 287385, 738208, 1112020, 1675116, 4302705, 6481441, 9763405, 25078116, 37776720, 56905408, 146166085, 220178973, 331669137, 851918488, 1283297212, 1933109508

Offset: 1

Mohamed Bouhamida, May 19 2006

nonn

Also values x of Pythagorean triples (x, x+47, y).
Corresponding values y of solutions (x, y) are in A159750.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (51+14*sqrt(2))/47 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3267+1702*sqrt(2))/47^2 for n mod 3 = 0.

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

Cf. A159750, A028871, A118337, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159751 (decimal expansion of (51+14*sqrt(2))/47), A159752 (decimal expansion of (3267+1702*sqrt(2))/47^2).

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(16+69*x+56*x^2-12*x^3-23*x^4-12*x^5)/((1-x)*(1-6*x^3 +x^6)))); // G. C. Greubel, May 07 2018

Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+47)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,16,85,141,225,616,940},50] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+94+2209), print1(n, ",")))}

a(n) = 6*a(n-3) -a(n-6) +94 for n > 6; a(1)=0, a(2)=16, a(3)=85, a(4)=141, a(5)=225, a(6)=616.
G.f.: x*(16+69*x+56*x^2-12*x^3-23*x^4-12*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 47*A001652(k) for k >= 0.

Edited by Klaus Brockhaus, Apr 30 2009

A129289 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+73)^2 = y^2.

Original entry on oeis.org

0, 44, 95, 219, 455, 744, 1460, 2832, 4515, 8687, 16683, 26492, 50808, 97412, 154583, 296307, 567935, 901152, 1727180, 3310344, 5252475, 10066919, 19294275, 30613844, 58674480, 112455452, 178430735, 341980107, 655438583, 1039970712, 1993206308, 3820176192

Offset: 1

Mohamed Bouhamida, May 26 2007

nonn

Also values x of Pythagorean triples (x, x+73, y).
Corresponding values y of solutions (x, y) are in A160041.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (89+36*sqrt(2))/73 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (5907+1802*sqrt(2))/73^2 for n mod 3 = 0.

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. A160041, A129288, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A160042 (decimal expansion of (89+36*sqrt(2))/73), A160043 (decimal expansion of (5907+1802*sqrt(2))/73^2).

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(44+51*x+124*x^2-28*x^3-17*x^4-28*x^5)/((1-x)*(1-6*x^3+x^6)))); // G. C. Greubel, May 07 2018

Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+73)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,44,95,219,455,744,1460},70] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

{forstep(n=0, 100000000, [3 ,1], if(issquare(2*n^2+146*n+5329), print1(n, ",")))}

a(n) = 6*a(n-3) -a(n-6) +146 for n > 6; a(1)=0, a(2)=44, a(3)=95, a(4)=219, a(5)=455, a(6)=744.
G.f.: x*(44+51*x+124*x^2-28*x^3-17*x^4-28*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 73*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, May 04 2009

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