A156035 -id:A156035 - cp's OEIS frontend

A129641 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+409)^2 = y^2.

0, 200, 611, 1227, 2291, 4620, 8180, 14364, 27927, 48671, 84711, 163760, 284664, 494720, 955451, 1660131, 2884427, 5569764, 9676940, 16812660, 32463951, 56402327, 97992351, 189214760, 328737840, 571142264, 1102825427, 1916025531, 3328862051, 6427738620

Offset: 1

Mohamed Bouhamida, May 31 2007

Also values x of Pythagorean triples (x, x+409, y).
Corresponding values y of solutions (x, y) are in A160577.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (473+168*sqrt(2))/409 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (204819+83570*sqrt(2))/409^2 for n mod 3 = 0.

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

Cf. A160577, A001652, A129640, A156035 (decimal expansion of 3+2*sqrt(2)), A160578 (decimal expansion of (473+168*sqrt(2))/409), A160579 (decimal expansion of (204819+83570*sqrt(2))/409^2).

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 200, 611, 1227, 2291, 4620, 8180}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+818*n+167281), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+818 for n > 6; a(1)=0, a(2)=200, a(3)=611, a(4)=1227, a(5)=2291, a(6)=4620.
G.f.: x*(200+411*x+616*x^2-136*x^3-137*x^4-136*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 409*A001652(k) for k >= 0.

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

A129974 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+937)^2 = y^2.

Original entry on oeis.org

0, 627, 1128, 2811, 6188, 9027, 18740, 38375, 54908, 111503, 225936, 322295, 652152, 1319115, 1880736, 3803283, 7690628, 10963995, 22169420, 44826527, 63905108, 129215111, 261270408, 372468527, 753123120, 1522797795, 2170907928

Offset: 1

Mohamed Bouhamida, Jun 13 2007

Also values x of Pythagorean triples (x, x+937, y).
Corresponding values y of solutions (x, y) are in A160209.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (1179+506*sqrt(2))/937 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (933747+224782*sqrt(2))/937^2 for n mod 3 = 0.

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

Cf. A160209, A001652, A129857, A156035 (decimal expansion of 3+2*sqrt(2)), A160210 (decimal expansion of (1179+506*sqrt(2))/937), A160211 (decimal expansion of (933747+224782*sqrt(2))/937^2).

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1874*n+877969), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+1874 for n > 6; a(1)=0, a(2)=627, a(3)=1128, a(4)=2811, a(5)=6188, a(6)=9027.
G.f.: x*(627+501*x+1683*x^2-385*x^3-167*x^4-385*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 937*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, May 18 2009

A129999 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+337)^2 = y^2.

Original entry on oeis.org

0, 27, 888, 1011, 1148, 6027, 6740, 7535, 35948, 40103, 44736, 210335, 234552, 261555, 1226736, 1367883, 1525268, 7150755, 7973420, 8890727, 41678468, 46473311, 51819768, 242920727, 270867120, 302028555, 1415846568, 1578730083

Offset: 1

Mohamed Bouhamida, Jun 15 2007

Also values x of Pythagorean triples (x, x+337, y).
Corresponding values y of solutions (x, y) are in A159574.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (339+26*sqrt(2))/337 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (278307+179662*sqrt(2))/337^2 for n mod 3 = 0.

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

Cf. A159574, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159575 (decimal expansion of (339+26*sqrt(2))/337), A159576 (decimal expansion of (278307+179662*sqrt(2))/337^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,27,888,1011,1148,6027,6740},40] (* Harvey P. Dale, Feb 26 2015 *)

{forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+674*n+113569), print1(n, ",")))}

a(n)=6*a(n-3)-a(n-6)+674 for n > 6; a(1)=0, a(2)=27, a(3)=888, a(4)=1011, a(5)=1148, a(6)=6027.
G.f.: x*(27+861*x+123*x^2-25*x^3-287*x^4-25*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 337*A001652(k) for k >= 0.
a(0)=0, a(1)=27, a(2)=888, a(3)=1011, a(4)=1148, a(5)=6027, a(6)=6740, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Feb 26 2015

Edited and two terms added by Klaus Brockhaus, Apr 16 2009

A130005 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+577)^2 = y^2.

Original entry on oeis.org

0, 35, 1568, 1731, 1908, 10595, 11540, 12567, 63156, 68663, 74648, 369495, 401592, 436475, 2154968, 2342043, 2545356, 12561467, 13651820, 14836815, 73214988, 79570031, 86476688, 426729615, 463769520, 504024467, 2487163856, 2703048243

Offset: 1

Mohamed Bouhamida, Jun 15 2007

Also values x of Pythagorean triples (x, x+577, y).
Corresponding values y of solutions (x, y) are in A159626.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (579+34*sqrt(2))/577 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (855171+556990*sqrt(2))/577^2 for n mod 3 = 0.

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

Cf. A159626, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159627 (decimal expansion of (579+34*sqrt(2))/577), A159628 (decimal expansion of (855171+556990*sqrt(2))/577^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,35,1568,1731,1908,10595,11540},30] (* Harvey P. Dale, May 27 2018 *)

{forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+1154*n+332929), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+1154 for n > 6; a(1)=0, a(2)=35, a(3)=1568, a(4)=1731, a(5)=1908, a(6)=10595.
G.f.: x*(35+1533*x+163*x^2-33*x^3-511*x^4-33*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 577*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 21 2009

A130014 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+881)^2 = y^2.

Original entry on oeis.org

0, 43, 2440, 2643, 2860, 16443, 17620, 18879, 97980, 104839, 112176, 573199, 613176, 655939, 3342976, 3575979, 3825220, 19486419, 20844460, 22297143, 113577300, 121492543, 129959400, 661979143, 708112560, 757461019, 3858299320

Offset: 1

Mohamed Bouhamida, Jun 15 2007

Also values x of Pythagorean triples (x, x+881, y).
Corresponding values y of solutions (x, y) are in A159690.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (883+42*sqrt(2))/881 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (2052963+1343918*sqrt(2))/881^2 for n mod 3 = 0.

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

Cf. A159690, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159691 (decimal expansion of (883+42*sqrt(2))/881), A159692 (decimal expansion of (2052963+1343918*sqrt(2))/881^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,43,2440,2643,2860,16443,17620},30] (* Harvey P. Dale, Aug 13 2015 *)

{forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+1762*n+776161), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+1762 for n > 6; a(1)=0, a(2)=43, a(3)=2440, a(4)=2643, a(5)=2860, a(6)=16443.
G.f.: x*(43+2397*x+203*x^2-41*x^3-799*x^4-41*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 881*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 21 2009

A130608 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+167)^2 = y^2.

Original entry on oeis.org

0, 28, 385, 501, 645, 2668, 3340, 4176, 15957, 19873, 24745, 93408, 116232, 144628, 544825, 677853, 843357, 3175876, 3951220, 4915848, 18510765, 23029801, 28652065, 107889048, 134227920, 166996876, 628823857, 782338053, 973329525, 3665054428, 4559800732

Offset: 1

Mohamed Bouhamida, Jun 17 2007

Also values x of Pythagorean triples (x, x+167, y).
Corresponding values y of solutions (x, y) are in A159777.
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) = (171+26*sqrt(2))/167 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (56211+34510*sqrt(2))/167^2 for n mod 3 = 0.

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

Cf. A159777, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159778 (decimal expansion of (171+26*sqrt(2))/167), A159779 (decimal expansion of (56211+34510*sqrt(2))/167^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,28,385,501,645,2668,3340},80] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+334*n+27889), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+334 for n > 6; a(1)=0, a(2)=28, a(3)=385, a(4)=501, a(5)=645, a(6)=2668.
G.f.: x*(28+357*x+116*x^2-24*x^3-119*x^4-24*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 167*A001652(k) for k >= 0.

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

A155464 a(n) = 7a(n-1) - 7a(n-2) + a(n-3) for n > 2; a(0) = 0, a(1) = 51, a(2) = 340.

Original entry on oeis.org

0, 51, 340, 2023, 11832, 69003, 402220, 2344351, 13663920, 79639203, 464171332, 2705388823, 15768161640, 91903581051, 535653324700, 3122016367183, 18196444878432, 106056652903443, 618143472542260, 3602804182350151

Offset: 0

Klaus Brockhaus, Jan 30 2009

lim_{n -> infinity} a(n+1)/a(n) = 3+2*sqrt(2).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

First trisection of A118120. Equals 17*A001652.

Cf. A155465, A155466, A156035 (decimal expansion of 3+2*sqrt(2)).

I:=[0,51,340]; [n le 3 select I[n] else 7*Self(n-1) - 7*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Aug 21 2018

LinearRecurrence[{7,-7,1},{0,51,340},30] (* Harvey P. Dale, Jun 10 2013 *)
Table[17*(LucasL[2*n+1,2] - 2)/4, {n, 0, 50}] (* G. C. Greubel, Aug 21 2018 *)

{m=20; v=concat([0, 51, 340], vector(m-3)); for(n=4, m, v[n]=7*v[n-1]-7*v[n-2]+v[n-3]); v}

a(n) = 6*a(n-1) - a(n-2) + 34 for n > 1; a(0) = 0, a(1) = 51.
a(n) = ((1+sqrt(2))*(3+2*sqrt(2))^n + (1-sqrt(2))*(3-2*sqrt(2))^n -2)*(17/4).
G.f.: 17*x*(3-x)/((1-x)*(1-6*x+x^2)).
a(n) = 17*(A002203(2*n+1) - 2)/4. - G. C. Greubel, Aug 21 2018

Comment and recursion formula added, cross-references edited by Klaus Brockhaus, Sep 23 2009

A155465 a(n) = 7a(n-1) - 7a(n-2) + a(n-3) for n > 2; a(0) = 7, a(1) = 88, a(2) = 555.

Original entry on oeis.org

7, 88, 555, 3276, 19135, 111568, 650307, 3790308, 22091575, 128759176, 750463515, 4374021948, 25493668207, 148587987328, 866034255795, 5047617547476, 29419671029095, 171470408627128, 999402780733707, 5824946275775148

Offset: 0

Klaus Brockhaus, Jan 30 2009

lim_{n -> infinity} a(n+1)/a(n) = 3+2*sqrt(2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Second trisection of A118120. Cf. A001652.

Cf. A155464, A155466, A156035 (decimal expansion of 3+2*sqrt(2)).

I:=[7,88,555]; [n le 3 select I[n] else 7*Self(n-1) - 7*Self(n-2) + Self(n-3): n in [1..50]]; // G. C. Greubel, Aug 21 2018

LinearRecurrence[{7,-7,1},{7,88,555},30] (* Harvey P. Dale, Apr 29 2012 *)
Table[(3*LucasL[2*n+3,2] + 10*LucasL[2*n+1,2] - 34)/4, {n, 0, 50}] (* G. C. Greubel, Aug 21 2018 *)

{m=20; v=concat([7, 88, 555], vector(m-3)); for(n=4, m, v[n]=7*v[n-1]-7*v[n-2]+v[n-3]); v}

a(n) = 6*a(n-1) - a(n-2) + 34 for n > 1; a(0) = 7, a(1) = 88.
a(n) = ((31+25*sqrt(2))*(3+2*sqrt(2))^n + (31-25*sqrt(2))*(3-2*sqrt(2))^n - 34)/4.
G.f.: (7+39*x-12*x^2)/((1-x)*(1-6*x+x^2)).
a(n) = (3*A002203(2*n+3) + 10*A002203(2*n+1) - 34)/4. - G. C. Greubel, Aug 21 2018

Comment and recursion formula added, cross-references edited by Klaus Brockhaus, Sep 23 2009

A155466 a(n) = 7a(n-1) - 7a(n-2) + a(n-3) for n > 2; a(0) = 28, a(1) = 207, a(2) = 1248.

Original entry on oeis.org

28, 207, 1248, 7315, 42676, 248775, 1450008, 8451307, 49257868, 287095935, 1673317776, 9752810755, 56843546788, 331308470007, 1931007273288, 11254735169755, 65597403745276, 382329687301935, 2228380720066368

Offset: 0

Klaus Brockhaus, Jan 30 2009

lim_{n -> infinity} a(n+1)/a(n) = 3+2*sqrt(2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Third trisection of A118120. Cf. A001652.

Cf. A155464, A155465, A156035 (decimal expansion of 3+2*sqrt(2)).

I:=[28, 207, 1248]; [n le 3 select I[n] else 7*Self(n-1) - 7*Self(n-2) + Self(n-3): n in [1..50]]; // G. C. Greubel, Aug 21 2018

Table[(10*LucasL[2*n+3,2] + 3*LucasL[2*n+1, 2] -34)/4, {n, 0, 50}] (* or *) LinearRecurrence[{7,-7,1}, {28, 207, 1248}, 50] (* G. C. Greubel, Aug 21 2018 *)

{m=19; v=concat([28, 207, 1248], vector(m-3)); for(n=4, m, v[n]=7*v[n-1]-7*v[n-2]+v[n-3]); v}

a(n) = 6*a(n-1) - a(n-2) + 34 for n > 1; a(0) = 28, a(1) = 207.
a(n) = ((73+53*sqrt(2))*(3+2*sqrt(2))^n + (73-53*sqrt(2))*(3-2*sqrt(2))^n - 34)/4.
G.f.: (28+11*x-5*x^2)/((1-x)*(1-6*x+x^2)).
a(n) = (10*A002203(2*n+3) + 3*A002203(2*n+1) - 34)/4. - G. C. Greubel, Aug 21 2018

Comment and recursion formula added, cross-references edited by Klaus Brockhaus, Sep 23 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

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A129641 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+409)^2 = y^2.

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A129974 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+937)^2 = y^2.

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A129999 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+337)^2 = y^2.

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A130005 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+577)^2 = y^2.

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A130014 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+881)^2 = y^2.

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A130608 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+167)^2 = y^2.

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A155464 a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3) for n > 2; a(0) = 0, a(1) = 51, a(2) = 340.

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A155464 a(n) = 7a(n-1) - 7a(n-2) + a(n-3) for n > 2; a(0) = 0, a(1) = 51, a(2) = 340.

A155465 a(n) = 7a(n-1) - 7a(n-2) + a(n-3) for n > 2; a(0) = 7, a(1) = 88, a(2) = 555.

A155466 a(n) = 7a(n-1) - 7a(n-2) + a(n-3) for n > 2; a(0) = 28, a(1) = 207, a(2) = 1248.