A156035 -id:A156035 - cp's OEIS frontend

A157119 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+103)^2 = y^2.

0, 84, 105, 309, 765, 884, 2060, 4712, 5405, 12257, 27713, 31752, 71688, 161772, 185313, 418077, 943125, 1080332, 2436980, 5497184, 6296885, 14204009, 32040185, 36701184, 82787280, 186744132, 213910425, 482519877, 1088424813, 1246761572

Offset: 1

Klaus Brockhaus, Feb 25 2009

Corresponding values y of solutions (x, y) are in A157120.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*(11-3*sqrt(2))^2/(11+3*sqrt(2))^2 for n mod 3 = 0.

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

Cf. A157120, A001652, 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))).

{forstep(n=0, 1300000000, [1, 3], if(issquare(2*n^2+206*n+10609), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+206 for n > 6; a(1) = 0, a(2) = 84, a(3) = 105, a(4) = 309, a(5) = 765, a(6) = 884.
G.f.: x*(84+21*x+204*x^2-48*x^3-7*x^4-48*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 103*A001652(k) for k >= 0.

A118576 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+16807)^2 = y^2.

Original entry on oeis.org

0, 2145, 3773, 6468, 8540, 12005, 19208, 24521, 28665, 35672, 41148, 50421, 61388, 69972, 84525, 95921, 115248, 156065, 186480, 210308, 250733, 282405, 336140, 399797, 449673, 534296, 600600, 713097, 950796, 1127973, 1266797, 1502340

Offset: 1

Mohamed Bouhamida, May 16 2006

Also values x of Pythagorean triples (x, x+16807, y); 16807 = 7^5.
Corresponding values y of solutions (x, y) are in A156713.
Limit_{n -> oo} a(n)/a(n-11) = 3+2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 11 = {1, 2, 4, 6, 8, 10}.
Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))^3 / ((9+4*sqrt(2))/7)^7 for n mod 11 = {0, 3, 5, 9}.
Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2)) / ((9+4*sqrt(2))/7)^2 for n mod 11 = 7.

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

Cf. A156713, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).

{forstep(n=0, 1600000, [1, 3], if(issquare(2*n^2 + 33614*n + 282475249), print1(n, ",")))}

a(n) = 6*a(n-11)-a(n-22)+33614 for n > 22; a(1) = 0, a(2) = 2145, a(3) = 3773, a(4) = 6468, a(5) = 8540, a(6) = 12005, a(7) = 19208, a(8) = 24521, a(9) = 28665, a(10) = 35672, a(11) = 41148, a(12) = 50421, a(13) = 61388, a(14) = 69972, a(15) = 84525, a(16) = 95921, a(17) = 115248, a(18) = 156065, a(19) = 186480, a(20) = 210308, a(21) = 250733, a(22) = 282405.

G.f.: x*(2145+1628*x+2695*x^2+2072*x^3+3465*x^4+7203*x^5+5313*x^6+4144*x^7+7007*x^8+5476*x^9+9273*x^10-1903*x^11-1184*x^12-1617*x^13-1036*x^14-1463*x^15-2401*x^16-1463*x^17 -1036*x^18-1617*x^19-1184*x^20-1903*x^21 )/((1-x)*(1-6*x^11+x^22)).

Edited by Klaus Brockhaus, Feb 14 2009

A118630 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+2401)^2 = y^2.

Original entry on oeis.org

0, 539, 924, 1220, 1715, 2744, 3503, 4095, 5096, 7203, 9996, 12075, 13703, 16464, 22295, 26640, 30044, 35819, 48020, 64239, 76328, 85800, 101871, 135828, 161139, 180971, 214620, 285719, 380240, 450695, 505899, 599564, 797475, 944996, 1060584

Offset: 1

Mohamed Bouhamida, May 09 2006

Also values x of Pythagorean triples (x, x+2401, y); 2401=7^4.
Corresponding values y of solutions (x, y) are in A157247.
Limit_{n -> oo} a(n)/a(n-9) = 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 9 = {1, 2, 6}.
Limit_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 9 = {0, 3, 5, 7}.
Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))^3 / ((9+4*sqrt(2))/7)^7 for n mod 9 = {4, 8}.

			924^2+(924+2401)^2 = 853776+11055625 = 11909401 = 3451^2.

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. A157247, A001652, A118576, A118554, A118611, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).

{forstep(n=0, 1100000, [3 ,1], if(issquare(n^2+(n+2401)^2), print1(n, ",")))}

a(n) = 6*a(n-9)-a(n-18)+4802 for n > 18; a(1)=0, a(2)=539, a(3)=924, a(4)=1220, a(5)=1715, a(6)=2744, a(7)=3503, a(8)=4095, a(9)=5096, a(10)=7203, a(11)=9996, a(12)=12075, a(13)=13703, a(14)=16464,a (15)=22295, a(16)=26640, a(17)=30044, a(18)=35819.
G.f.: x*(539+385*x+296*x^2+495*x^3+1029*x^4+759*x^5+592*x^6 +1001*x^7+2107*x^8-441*x^9-231*x^10-148*x^11-209*x^12-343*x^13 -209*x^14-148*x^15-231*x^16-441*x^17) / ((1-x)*(1-6*x^9+x^18)).
a(9*k+1) = 2401*A001652(k) for k >= 0.

Edited by Klaus Brockhaus, Feb 25 2009

A129626 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+281)^2 = y^2.

Original entry on oeis.org

0, 76, 559, 843, 1239, 3976, 5620, 7920, 23859, 33439, 46843, 139740, 195576, 273700, 815143, 1140579, 1595919, 4751680, 6648460, 9302376, 27695499, 38750743, 54218899, 161421876, 225856560, 316011580, 940836319, 1316389179, 1841851143, 5483596600

Offset: 1

Mohamed Bouhamida, May 30 2007

Also values x of Pythagorean triples (x, x+281, y).
Corresponding values y of solutions (x, y) are in A157348.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (297+68*sqrt(2))/281 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (130803+73738*sqrt(2))/281^2 for n mod 3 = 0.

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

Cf. A157348, A001652, A129288, A129289, A129298, A156035 (decimal expansion of 3+2*sqrt(2)), A157349 (decimal expansion of (297+68*sqrt(2))/281), A157350 (decimal expansion of (130803+73738*sqrt(2))/281^2).

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 76, 559, 843, 1239, 3976, 5620}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

{forstep(n=0, 1000000000, [3, 1], if(issquare(2*n^2+562*n+78961), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+562 for n > 6; a(1)=0, a(2)=76, a(3)=559, a(4)=843, a(5)=1239, a(6)=3976.
G.f.: x*(76+483*x+284*x^2-60*x^3-161*x^4-60*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 281*A001652(k) for k >= 0.

Edited by Klaus Brockhaus, Apr 12 2009

A129725 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+521)^2 = y^2.

Original entry on oeis.org

0, 100, 1159, 1563, 2079, 8080, 10420, 13416, 48363, 61999, 79459, 283140, 362616, 464380, 1651519, 2114739, 2707863, 9627016, 12326860, 15783840, 56111619, 71847463, 91996219, 327043740, 418758960, 536194516, 1906151863, 2440707339, 3125171919, 11109868480

Offset: 1

Mohamed Bouhamida, Jun 02 2007

Also values x of Pythagorean triples (x, x+521, y).
Corresponding values y of solutions (x, y) are in A160583.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (537+92*sqrt(2))/521 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (520659+314170*sqrt(2))/521^2 for n mod 3 = 0.

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

Cf. A160583, A001652, A129642, A156035 (decimal expansion of 3+2*sqrt(2)), A160584 (decimal expansion of (537+92*sqrt(2))/521), A160585 (decimal expansion of (520659+314170*sqrt(2))/521^2).

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 100, 1159, 1563, 2079, 8080, 10420}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1042*n+271441), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+1042 for n > 6; a(1)=0, a(2)=100, a(3)=1159, a(4)=1563, a(5)=2079, a(6)=8080.
G.f.: x*(100+1059*x+404*x^2-84*x^3-353*x^4-84*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 521*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Jun 08 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

A129975 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+953)^2 = y^2.

Original entry on oeis.org

0, 132, 2295, 2859, 3535, 15792, 19060, 22984, 94363, 113407, 136275, 552292, 663288, 796572, 3221295, 3868227, 4645063, 18777384, 22547980, 27075712, 109444915, 131421559, 157811115, 637894012, 765983280, 919792884, 3717921063

Offset: 1

Mohamed Bouhamida, Jun 13 2007

Also values x of Pythagorean triples (x, x+953, y).
Corresponding values y of solutions (x, y) are in A160212.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (969+124*sqrt(2))/953 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1947891+1218490*sqrt(2))/953^2 for n mod 3 = 0.

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

Cf. A160212, A001652, A129974, A156035 (decimal expansion of 3+2*sqrt(2)), A160213 (decimal expansion of (969+124*sqrt(2))/953), A160214 (decimal expansion of (1947891+1218490*sqrt(2))/953^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,132,2295,2859,3535,15792,19060},30] (* Harvey P. Dale, Apr 12 2013 *)

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1906*n+908209), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+1906 for n > 6; a(1)=0, a(2)=132, a(3)=2295, a(4)=2859, a(5)=3535, a(6)=15792.
G.f.: x*(132+2163*x+564*x^2-116*x^3-721*x^4-116*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 953*A001652(k) for k >= 0.
a(1)=0, a(2)=132, a(3)=2295, a(4)=2859, a(5)=3535, a(6)=15792, a(7)=19060, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Apr 12 2013

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

A156156 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1) = 13, a(2) = 53.

Original entry on oeis.org

13, 53, 305, 1777, 10357, 60365, 351833, 2050633, 11951965, 69661157, 406014977, 2366428705, 13792557253, 80388914813, 468540931625, 2730856674937, 15916599117997, 92768738033045, 540695829080273, 3151406236448593

Offset: 1

Klaus Brockhaus, Feb 09 2009

nonn

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

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

First trisection of A155923.

Cf. A156035 (decimal expansion of 3+2*sqrt(2)), A156157, A156158.

{m=20; v=concat([13, 53], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-v[n-2]); v}

a(n) = ((50+31*sqrt(2))*(3-2*sqrt(2))^n+(50-31*sqrt(2))*(3+2*sqrt(2))^n)/4.

G.f.: x*(13-25*x)/(1-6*x+x^2).

Replaced abbreviation by sqrt(2) Klaus Brockhaus, Feb 12 2009

G.f. corrected by Klaus Brockhaus, Sep 23 2009

A156157 a(n) = 6*a(n-1)-a(n-2) for n > 2; a(1) = 17, a(2) = 85.

Original entry on oeis.org

17, 85, 493, 2873, 16745, 97597, 568837, 3315425, 19323713, 112626853, 656437405, 3825997577, 22299548057, 129971290765, 757528196533, 4415197888433, 25733659134065, 149986756915957, 874186882361677, 5095134537254105

Offset: 1

Klaus Brockhaus, Feb 09 2009

nonn

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

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

Second trisection of A155923. Equals 17*A001653.

Cf. A156035 (decimal expansion of 3+2*sqrt(2)), A156156, A156158.

{m=20; v=concat([17, 85], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-v[n-2]); v}

a(n) = ((2+sqrt(2))*(3-2*sqrt(2))^n+(2-sqrt(2))*(3+2*sqrt(2))^n)*17/4.

G.f.: 17*x*(1-x)/(1-6*x+x^2).

Replaced abbreviation by sqrt(2) Klaus Brockhaus, Feb 12 2009

G.f. corrected by Klaus Brockhaus, Sep 23 2009

A156158 a(n) = 6*a(n-1) - a(n-2) for n > 2; a(1) = 25, a(2) = 137.

Original entry on oeis.org

25, 137, 797, 4645, 27073, 157793, 919685, 5360317, 31242217, 182092985, 1061315693, 6185801173, 36053491345, 210135146897, 1224757390037, 7138409193325, 41605697769913, 242495777426153, 1413368966787005

Offset: 1

Klaus Brockhaus, Feb 09 2009

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

Third trisection of A155923.

Cf. A156035 (decimal expansion of 3+2*sqrt(2)), A156156, A156157.

LinearRecurrence[{6,-1},{25,137},30] (* Harvey P. Dale, Jan 02 2019 *)

{m=19; v=concat([25,137], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-v[n-2]); v}

a(n) = ((26+7*sqrt(2))*(3-2*sqrt(2))^n+(26-7*sqrt(2))*(3+2*sqrt(2))^n)/4.
G.f.: x*(25-13*x)/(1-6*x+x^2).
Limit_{n -> oo} a(n)/a(n-1) = 3+2*sqrt(2).

cp's OEIS Frontend

A157119 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+103)^2 = y^2.

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A118576 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+16807)^2 = y^2.

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A118630 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+2401)^2 = y^2.

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A129626 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+281)^2 = y^2.

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A129725 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+521)^2 = y^2.

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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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A129975 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+953)^2 = y^2.

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