A156035 -id:A156035 - cp's OEIS frontend

A130645 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+439)^2 = y^2.

0, 44, 1121, 1317, 1541, 7644, 8780, 10080, 45621, 52241, 59817, 266960, 305544, 349700, 1557017, 1781901, 2039261, 9076020, 10386740, 11886744, 52899981, 60539417, 69282081, 308324744, 352850640, 403806620, 1797049361, 2056565301, 2353558517, 10473972300

Offset: 1

Mohamed Bouhamida, Jun 20 2007

Also values x of Pythagorean triples (x, x+439, y).
Corresponding values y of solutions (x, y) are in A159890.
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) = (443+42*sqrt(2))/439 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (450483+287918*sqrt(2))/439^2 for n mod 3 = 0.

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

Cf. A159890, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159891 (decimal expansion of (443+42*sqrt(2))/439), A159892 (decimal expansion of (450483+287918*sqrt(2))/439^2).

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 44, 1121, 1317, 1541, 7644, 8780}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2012 *)

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+878*n+192721), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+878 for n > 6; a(1)=0, a(2)=44, a(3)=1121, a(4)=1317, a(5)=1541, a(6)=7644.
G.f.: x*(44+1077*x+196*x^2-40*x^3-359*x^4-40*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 439*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 30 2009

A130646 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+727)^2 = y^2.

Original entry on oeis.org

0, 56, 1925, 2181, 2465, 13056, 14540, 16188, 77865, 86513, 96117, 455588, 505992, 561968, 2657117, 2950893, 3277145, 15488568, 17200820, 19102356, 90275745, 100255481, 111338445, 526167356, 584333520, 648929768, 3066729845

Offset: 1

Mohamed Bouhamida, Jun 20 2007

Also values x of Pythagorean triples (x, x+727, y).
Corresponding values y of solutions (x, y) are in A159893.
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) = (731+54*sqrt(2))/727 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1304787+843542*sqrt(2))/727^2 for n mod 3 = 0.

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

Cf. A159893, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159894 (decimal expansion of (731+54*sqrt(2))/727), A159895 (decimal expansion of (1304787+843542*sqrt(2))/727^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,56,1925,2181,2465,13056,14540},40] (* or *) RecurrenceTable[{a[1]==0,a[2]==56,a[3]==1925,a[4]==2181,a[5] == 2465, a[6] == 13056, a[n] ==6a[n-3]-a[n-6]+1454},a,{n,40}] (* Harvey P. Dale, Jan 16 2013 *)

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+1454*n+528529), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+1454 for n > 6; a(1)=0, a(2)=56, a(3)=1925, a(4)=2181, a(5)=2465, a(6)=13056.
G.f.: x*(56+1869*x+256*x^2-52*x^3-623*x^4-52*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 727*A001652(k) for k >= 0.

Edited and one term added by Klaus Brockhaus, Apr 30 2009

A130647 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+839)^2 = y^2.

Original entry on oeis.org

0, 60, 2241, 2517, 2821, 15180, 16780, 18544, 90517, 99841, 110121, 529600, 583944, 643860, 3088761, 3405501, 3754717, 18004644, 19850740, 21886120, 104940781, 115700617, 127563681, 611641720, 674354640, 743497644, 3564911217

Offset: 1

Mohamed Bouhamida, Jun 20 2007

Also values x of Pythagorean triples (x, x+839, y).
Corresponding values y of solutions (x, y) are in A159896.
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) = (843+58*sqrt(2))/839 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1760979+1141390*sqrt(2))/839^2 for n mod 3 = 0.

G. C. Greubel, Table of n, a(n) for n = 1..3895
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A159896, A028871, A118337, A130645, A130646, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159897 (decimal expansion of (843+58*sqrt(2))/839), A159898 (decimal expansion of (1760979+1141390*sqrt(2))/839^2).

I:=[0,60,2241,2517,2821,15180,16780]; [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 17 2018

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,60,2241,2517,2821,15180,16780},30] (* Harvey P. Dale, Jun 19 2014 *)

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+1678*n+703921), print1(n, ",")))}

a(n) = 6*a(n-3) -a(n-6) +1678 for n > 6; a(1)=0, a(2)=60, a(3)=2241, a(4)=2517, a(5)=2821, a(6)=15180.
G.f.: x*(60+2181*x+276*x^2-56*x^3-727*x^4-56*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 839*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 30 2009

A157257 Positive numbers y such that y^2 is of the form x^2+(x+41)^2 with integer x.

Original entry on oeis.org

29, 41, 85, 89, 205, 481, 505, 1189, 2801, 2941, 6929, 16325, 17141, 40385, 95149, 99905, 235381, 554569, 582289, 1371901, 3232265, 3393829, 7996025, 18839021, 19780685, 46604249, 109801861, 115290281, 271629469, 639972145, 671961001

Offset: 1

Klaus Brockhaus, Feb 26 2009

(-20, a(1)) and (A129288(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+41)^2 = y^2.
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 = {0, 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 = 1.

			(-20, a(1)) = (-20, 29) is a solution: (-20)^2+(-20+41)^2 = 400+441 = 841 = 29^2.
(A129288(1), a(2)) = (0, 41) is a solution: 0^2+(0+41)^2 = 1681 = 41^2.
(A129288(3), a(4)) = (39, 89) is a solution: 39^2+(39+41)^2 = 1521+6400 = 7921 = 89^2.

G. C. Greubel, Table of n, a(n) for n = 1..1001
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A129288, A001653, 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))).

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6))) // G. C. Greubel, Feb 04 2018

CoefficientList[Series[(1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6), {x,0,50}], x] (* G. C. Greubel, Feb 04 2018 *)

{forstep(n=-20, 500000000, [3 ,1], if(issquare(n^2+(n+41)^2, &k), print1(k, ",")))}

x='x+O('x^30); Vec((1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6)) \\ G. C. Greubel, Feb 04 2018

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=29, a(2)=41, a(3)=85, a(4)=89, a(5)=205, a(6)=481.
G.f.: (1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 41*A001653(k) for k >= 1.

A378352 Decimal expansion of the volume of a (small) triakis octahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 23 2024

The (small) triakis octahedron is the dual polyhedron of the truncated cube.

			2.9142135623730950488016887242096980785696718753769...

Eric Weisstein's World of Mathematics, Small Triakis Octahedron.
Wikipedia, Triakis octahedron.

Cf. A378351 (surface area), A378353 (inradius), A201488 (midradius), A378354 (dihedral angle).
Cf. A377299 (volume of a truncated cube with unit edge).
Cf. A156035.
Essentially the same as A002193 and A188582.

First[RealDigits[Sqrt[2] + 3/2, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisOctahedron", "Volume"], 10, 100]]

Equals sqrt(2) + 3/2 = A002193 + 3/2.

Equals A156035/2. - Hugo Pfoertner, Nov 24 2024

A072280 Product representation of the Pell numbers A000129 and A002203.

Original entry on oeis.org

2, 1, 7, 6, 41, 5, 239, 34, 199, 29, 8119, 33, 47321, 169, 961, 1154, 1607521, 197, 9369319, 1121, 32641, 5741, 318281039, 1153, 45245801, 33461, 7761799, 38081, 63018038201, 1345, 367296043199, 1331714, 37667521, 1136689, 1273319041, 39201, 72722761475561

Offset: 1

Miklos Kristof, Jul 10 2002

nonn

Define the silver mean constants h=1+sqrt(2) = A014176, h^2=1+2h = A156035, and 1/h=h-2.
Let Phi(n,x) be the n-th cyclotomic polynomial A013595, so that x^n-1 = Product_{d | n} Phi(d, x). Let g(n) be the order of Phi(n, x), A000010. Then a(n)=(h-2)^g(n)*Phi(n, h^2) if n <> 2.
The Binet representations of the Pell numbers yields:
For even n, A000129(n) = Product_{d|n} a(d).
For odd n, A000129(n)=Product_{ d|n} a(2d).
For odd prime p, a(p)=A002203(p)/2, a(2p)=A000129(p).
a(2^(k+1))=A002203(2^k).
For odd n, A002203(n)=Product_{ d|n} a(d).
For k>0 and odd n, A002203(n*2^k)=Product_{ d | n} a(d*2^(k+1)).

			For even n=12, A000129(12) = a(1)*a(2)*a(3)*a(4)*a(6)*a(12) = 2*1*7*6*5*33 = 13860.
For odd n=9, A000129(9) = a(2)*a(6)*a(18)= 1*5*197 = 985.
For even n=8, A002203(12) = a(8)*a(24)=34*1153 = 39202.
For odd n=21, A002203(21) = a(1)*a(3)*a(7)*a(21) = 2*7*239*32641 = 109216786.

Dan Kalman and Robert Mena, The Fibonacci Numbers: Exposed, Math. Mag. 76 (3) (2003) 167-181.
Index to sequences related to cyclotomic polynomials.

Cf. A000129, A002203.

A072280 := proc(n) if n <= 2 then 3-n ; else g := numtheory[phi](n) ; h := 1+sqrt(2) ; (h-2)^g*numtheory[cyclotomic](n,h^2) ; simplify(expand(%)) ; end if; end proc:
seq(A072280(n),n=1..80) ; # R. J. Mathar, Nov 27 2009

a[n_] := If[n <= 2, 3-n, g = EulerPhi[n]; h = 1 + Sqrt[2]; (h - 2)^g*Cyclotomic[n, h^2] // Expand];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, May 08 2023, after R. J. Mathar *)

Edited and extended by R. J. Mathar, Nov 27 2009

A118611 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+343)^2 = y^2.

Original entry on oeis.org

0, 77, 132, 245, 392, 585, 728, 1029, 1428, 1725, 2352, 3185, 4292, 5117, 6860, 9177, 10904, 14553, 19404, 25853, 30660, 40817, 54320, 64385, 85652, 113925, 151512, 179529, 238728, 317429, 376092, 500045, 664832, 883905, 1047200, 1392237, 1850940, 2192853

Offset: 1

Mohamed Bouhamida, May 08 2006

Also values x of Pythagorean triples (x, x+343, y); 343=7^3.
Corresponding values y of solutions (x, y) are in A157246.
Limit_{n -> oo} a(n)/a(n-7) = 3+2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2)) / ((9+4*sqrt(2))/7)^2 for n mod 7 = {1, 2, 4, 5, 6}.
Limit_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 7 = {0, 3}.

			132^2+(132+343)^2 = 17424+225625 = 243049 = 493^2.

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

Cf. A157246, A001652, A118576, A118554, A118611, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).

LinearRecurrence[{1, 0, 0, 0, 0, 0, 6, -6, 0, 0, 0, 0, 0, -1, 1}, {0, 77, 132, 245, 392, 585, 728, 1029, 1428, 1725, 2352, 3185, 4292, 5117, 6860}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

{forstep(n=0, 1400000, [1, 3], if(issquare(n^2+(n+343)^2), print1(n, ",")))}

a(n) = 6*a(n-7)-a(n-14)+686 for n > 14; a(1)=0, a(2)=77, a(3)=132, a(4)=245, a(5)=392, a(6)=585, a(7)=728, a(8)=1029, a(9)=1428, a(10)=1725, a(11)=2352, a(12)=3185, a(13)=4292, a(14)=5117.
G.f.: x*(77+55*x+113*x^2+147*x^3+193*x^4+143*x^5+301*x^6-63*x^7 -33*x^8-51*x^9-49*x^10-51*x^11-33*x^12-63*x^13)/((1-x)*(1-6*x^7+x^14)).
a(7*k+1) = 343*A001652(k) for k >= 0.

Edited by Klaus Brockhaus, Feb 25 2009

A122694 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+761)^2 = y^2.

Original entry on oeis.org

0, 583, 820, 2283, 5440, 6783, 15220, 33579, 41400, 90559, 197556, 243139, 529656, 1153279, 1418956, 3088899, 6723640, 8272119, 18005260, 39190083, 48215280, 104944183, 228418380, 281021083, 611661360, 1331321719, 1637912740

Offset: 1

Mohamed Bouhamida, Jun 03 2007

Also values x of Pythagorean triples (x, x+761, y).
Corresponding values y of solutions (x, y) are in A160200.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2) = A156035.
lim_{n -> infinity} a(n)/a(n-1) = (1003+462*sqrt(2))/761 = A160201 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (591603+85478*sqrt(2))/761^2 = A160202 for n mod 3 = 0.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A160200, A001652, A115135.

I:=[0,583,820,2283,5440,6783,15220]; [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

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 583, 820, 2283, 5440, 6783, 15220}, 27] (* Jean-François Alcover, Nov 13 2017 *)

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1522*n+579121), print1(n, ",")))}

x='x+O('x^30); concat([0], Vec(x*(583 +237*x +1463*x^2 -341*x^3 -79*x^4 -341*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) +1522 for n > 6; a(1)=0, a(2)=583, a(3)=820, a(4)=2283, a(5)=5440, a(6)=6783.
G.f.: x*(583 +237*x +1463*x^2 -341*x^3 -79*x^4 -341*x^5)/((1-x)*(1 -6*x^3 +x^6)).
a(3*k+1) = 761*A001652(k) for k >= 0.

Edited and one term added by Klaus Brockhaus, May 18 2009

A129625 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+233)^2 = y^2.

Original entry on oeis.org

0, 75, 432, 699, 1092, 3115, 4660, 6943, 18724, 27727, 41032, 109695, 162168, 239715, 639912, 945747, 1397724, 3730243, 5512780, 8147095, 21742012, 32131399, 47485312, 126722295, 187276080, 276765243, 738592224, 1091525547, 1613106612

Offset: 1

Mohamed Bouhamida, May 30 2007

Also values x of Pythagorean triples (x, x+233, y).
Corresponding values y of solutions (x, y) are in A157297.
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 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (82611+44030*sqrt(2))/233^2 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. A157297, A001652, A129288, A129289, A129298, 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:=[0,75,432,699,1092,3115,4660]; [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, Mar 29 2018

LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,75,432,699,1092,3115,4660}, 50] (* G. C. Greubel, Mar 29 2018 *)

{forstep(n=0, 1700000000, [3, 1], if(issquare(2*n^2+466*n+54289), print1(n, ",")))};

a(n) = 6*a(n-3) -a(n-6) +466 for n > 6; a(1)=0, a(2)=75, a(3)=432, a(4)=699, a(5)=1092, a(6)=3115.
G.f.: x*(75 +357*x +267*x^2 -57*x^3 -119*x^4 -57*x^5)/((1-x)*(1 -6*x^3 +x^6)).
a(3*k+1) = 233*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 11 2009

A129640 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+313)^2 = y^2.

Original entry on oeis.org

0, 155, 464, 939, 1764, 3515, 6260, 11055, 21252, 37247, 65192, 124623, 217848, 380723, 727112, 1270467, 2219772, 4238675, 7405580, 12938535, 24705564, 43163639, 75412064, 143995335, 251576880, 439534475, 839267072, 1466298267, 2561795412, 4891607723

Offset: 1

Mohamed Bouhamida, May 31 2007

Also values x of Pythagorean triples (x, x+313, y).
Corresponding values y of solutions (x, y) are in A160574.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (363+130*sqrt(2))/313 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (119187+47998*sqrt(2))/313^2 for n mod 3 = 0.

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

Cf. A160574, A001652, A129298, A156035 (decimal expansion of 3+2*sqrt(2)), A160575 (decimal expansion of (363+130*sqrt(2))/313), A160576 (decimal expansion of (119187+47998*sqrt(2))/313^2).

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 155, 464, 939, 1764, 3515, 6260}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+626*n+97969), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+626 for n > 6; a(1)=0, a(2)=155, a(3)=464, a(4)=939, a(5)=1764, a(6)=3515.
G.f.: x*(155+309*x+475*x^2-105*x^3-103*x^4-105*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 313*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Jun 08 2009

cp's OEIS Frontend

A130645 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+439)^2 = y^2.

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A130646 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+727)^2 = y^2.

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Mathematica

PARI

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Extensions

A130647 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+839)^2 = y^2.

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Magma

Mathematica

PARI

Formula

Extensions

A157257 Positive numbers y such that y^2 is of the form x^2+(x+41)^2 with integer x.

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Magma

Mathematica

PARI

PARI

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A378352 Decimal expansion of the volume of a (small) triakis octahedron with unit shorter edge length.

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Mathematica

Formula

A072280 Product representation of the Pell numbers A000129 and A002203.

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Maple

Mathematica

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A118611 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+343)^2 = y^2.

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