cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 15 results. Next

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

Original entry on oeis.org

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

Views

Author

Mohamed Bouhamida, May 21 2007

Keywords

Comments

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

Links

Crossrefs

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).

Programs

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

Formula

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.

Extensions

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

A118673 Positive solutions x to the equation x^2 + (x+71)^2 = y^2.

Original entry on oeis.org

0, 13, 160, 213, 280, 1113, 1420, 1809, 6660, 8449, 10716, 38989, 49416, 62629, 227416, 288189, 365200, 1325649, 1679860, 2128713, 7726620, 9791113, 12407220, 45034213, 57066960, 72314749, 262478800, 332610789, 421481416, 1529838729, 1938597916, 2456573889
Offset: 0

Views

Author

Mohamed Bouhamida, May 19 2006

Keywords

Comments

Consider all Pythagorean triples (x,x+71,y) ordered by increasing y; sequence gives x values.
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 associated value in A066049, the x values are given by the sequence defined by: a(n) = 6*a(n-3) -a(n-6) + 2*p with a(0)=0, a(1)=2m+1, a(2)=6m^2-10m+4, a(3)=3p, a(4)=6m^2+10m+4, a(5)=40m^2-58m+21.
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(0)=p, b(1)=2m^2+2m+1, b(2)=10m^2-14m+5, b(3)=5p, b(4)=10m^2+14m+5, b(5)=58m^2-82m+29. - Mohamed Bouhamida, Sep 09 2009

Links

Crossrefs

Cf. A076296 (p=7), A118120 (p=17), A118674 (p=31), A129836 (p=97), A129992 (p=127), A129993 (p=199), A129991 (p=241), A129999 (p=337), A130004 (p=449), A130005 (p=577), A130013 (p=647), A130014 (p=881), A130017 (p=967).

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(13+147*x+53*x^2-11*x^3-49*x^4-11*x^5)/((1-x)*(1 - 6*x^3 +x^6)))); // G. C. Greubel, May 07 2018
  • Mathematica
    Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+71)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,13,160,213,280,1113,1420},100] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)
  • PARI
    a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 1,-1,0,-6,6,0,1]^n*[0;13;160;213;280;1113;1420])[1,1] \\ Charles R Greathouse IV, Apr 22 2016
  • PARI
    x='x+O('x^30); concat([0], Vec(x*(13+147*x+53*x^2-11*x^3 -49*x^4 -11*x^5)/((1-x)*(1-6*x^3+x^6)))) \\ G. C. Greubel, May 07 2018

Formula

a(n) = 6*a(n-3) -a(n-6) +142 with a(0)=0, a(1)=13, a(2)=160, a(3)=213, a(4)=280, a(5)=1113.
O.g.f.: x*(13+147*x+53*x^2-11*x^3-49*x^4-11*x^5)/((1-x)*(1-6*x^3+x^6)). - R. J. Mathar, Jun 10 2008

Extensions

Edited by R. J. Mathar, Jun 10 2008

A129993 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+199)^2 = y^2.

Original entry on oeis.org

0, 21, 504, 597, 704, 3441, 3980, 4601, 20540, 23681, 27300, 120197, 138504, 159597, 701040, 807741, 930680, 4086441, 4708340, 5424881, 23818004, 27442697, 31619004, 138821981, 159948240, 184289541, 809114280, 932247141, 1074118640
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 14 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+199, y).
Corresponding values y of solutions (x, y) are in A159548.
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) = (201+20*sqrt(2))/199 for n mod 3 = {1, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (91443+58282*sqrt(2))/199^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159548, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159549 (decimal expansion of (201+20*sqrt(2))/199), A159550 (decimal expansion of (91443+58282*sqrt(2))/199^2).

Programs

  • Magma
    I:=[0,21,504,597,704,3441,3980]; [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..50]]; // G. C. Greubel, Mar 31 2018
  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,21,504,597,704,3441,3980},30] (* Harvey P. Dale, Jun 03 2012 *)
  • PARI
    {forstep(n=0, 500000000, [1, 3], if(issquare(2*n^2+398*n+39601), print1(n, ",")))};

Formula

a(n) = 6*a(n-3) - a(n-6) + 398 for n > 6; a(1)=0, a(2)=21, a(3)=504, a(4)=597, a(5)=704, a(6)=3441.
G.f.: x*(21+483*x+93*x^2-19*x^3-161*x^4-19*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 199*A001652(k) for k >= 0.
a(1)=0, a(2)=21, a(3)=504, a(4)=597, a(5)=704, a(6)=3441, a(7)=3980, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Jun 03 2012

Extensions

Edited and two terms added by Klaus Brockhaus, Apr 14 2009

A129992 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+127)^2 = y^2.

Original entry on oeis.org

0, 17, 308, 381, 468, 2117, 2540, 3045, 12648, 15113, 18056, 74025, 88392, 105545, 431756, 515493, 615468, 2516765, 3004820, 3587517, 14669088, 17513681, 20909888, 85498017, 102077520, 121872065, 498319268, 594951693, 710322756, 2904417845, 3467632892
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 14 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+127, y).
Corresponding values y of solutions (x, y) are in A159466.
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) = (129+16*sqrt(2))/127 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (34947+21922*sqrt(2))/127^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159466, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159467 (decimal expansion of (129+16*sqrt(2))/127), A159468 (decimal expansion of (34947+21922*sqrt(2))/127^2).

Programs

  • Magma
    I:=[0,17,308,381,468,2117,2540]; [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..50]]; // G. C. Greubel, Mar 31 2018
  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,17,308,381,468,2117,2540},80] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)
  • PARI
    {forstep(n=0, 500000000, [1, 3], if(issquare(2*n^2+254*n+16129), print1(n, ",")))};

Formula

a(n) = 6*a(n-3) - a(n-6) + 254 for n > 6; a(1)=0, a(2)=17, a(3)=308, a(4)=381, a(5)=468, a(6)=2117.
G.f.: x*(17+291*x+73*x^2-15*x^3-97*x^4-15*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 127*A001652(k) for k >= 0.

Extensions

Edited and two more terms added by Klaus Brockhaus, Apr 13 2009

A129991 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+241)^2 = y^2.

Original entry on oeis.org

0, 23, 620, 723, 840, 4223, 4820, 5499, 25200, 28679, 32636, 147459, 167736, 190799, 860036, 978219, 1112640, 5013239, 5702060, 6485523, 29219880, 33234623, 37800980, 170306523, 193706160, 220320839, 992619740, 1129002819, 1284124536, 5785412399, 6580311236
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 14 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+241, y).
Corresponding values y of solutions (x, y) are in A159565.
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) = (243+22*sqrt(2))/241 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (137283+87958*sqrt(2))/241^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159565, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159566 (decimal expansion of (243+22*sqrt(2))/241), A159567 (decimal expansion of (137283+87958*sqrt(2))/241^2).

Programs

  • Mathematica
    LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 23, 620, 723, 840, 4223, 4820}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2012 *)
  • PARI
    {forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+482*n+58081), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+482 for n > 6; a(1)=0, a(2)=23, a(3)=620, a(4)=723, a(5)=840, a(6)=4223.
G.f.: x*(23+597*x+103*x^2-21*x^3-199*x^4-21*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 241*A001652(k) for k >= 0.

Extensions

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

A130004 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+449)^2 = y^2.

Original entry on oeis.org

0, 31, 1204, 1347, 1504, 8151, 8980, 9891, 48600, 53431, 58740, 284347, 312504, 343447, 1658380, 1822491, 2002840, 9666831, 10623340, 11674491, 56343504, 61918447, 68045004, 328395091, 360888240, 396596431, 1914027940, 2103411891, 2311534480, 11155773447
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 15 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+449, y).
Corresponding values y of solutions (x, y) are in A159589.
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) = (451+30*sqrt(2))/449 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (507363+329222*sqrt(2))/449^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159589, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159590 (decimal expansion of (451+30*sqrt(2))/449), A159591 (decimal expansion of (507363+329222*sqrt(2))/449^2).

Programs

  • Magma
    I:=[0, 31, 1204, 1347, 1504, 8151, 8980]; [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 08 2018
  • Mathematica
    LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 31, 1204, 1347, 1504, 8151, 8980}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2012 *)
  • PARI
    {forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+898*n+201601), print1(n, ",")))}
  • PARI
    x='x+O('x^30); concat([0], Vec(x*(31+1173*x+143*x^2-29*x^3-391*x^4 -29*x^5)/((1-x)*(1-6*x^3+x^6)))) \\ G. C. Greubel, May 08 2018

Formula

a(n) = 6*a(n-3) -a(n-6) +898 for n > 6; a(1)=0, a(2)=31, a(3)=1204, a(4)=1347, a(5)=1504, a(6)=8151.
G.f.: x*(31+1173*x+143*x^2-29*x^3-391*x^4-29*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 449*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, Apr 17 2009

A130013 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+647)^2 = y^2.

Original entry on oeis.org

0, 37, 1768, 1941, 2128, 11937, 12940, 14025, 71148, 76993, 83316, 416245, 450312, 487165, 2427616, 2626173, 2840968, 14150745, 15308020, 16559937, 82478148, 89223241, 96519948, 480719437, 520032720, 562561045, 2801839768, 3030974373
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 15 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+647, y).
Corresponding values y of solutions (x, y) are in A159641.
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) = (649+36*sqrt(2))/647 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1084467+707402*sqrt(2))/647^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159641, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159642 (decimal expansion of (649+36*sqrt(2))/647), A159643 (decimal expansion of (1084467+707402*sqrt(2))/647^2).

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,37,1768,1941,2128,11937,12940},40] (* Harvey P. Dale, Jan 27 2025 *)
  • PARI
    {forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+1294*n+418609), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+1294 for n > 6; a(1)=0, a(2)=37, a(3)=1768, a(4)=1941, a(5)=2128, a(6)=11937.
G.f.: x*(37+1731*x+173*x^2-35*x^3-577*x^4-35*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 647*A001652(k) for k >= 0.

Extensions

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

A130017 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+967)^2 = y^2.

Original entry on oeis.org

0, 45, 2688, 2901, 3128, 18105, 19340, 20657, 107876, 115073, 122748, 631085, 673032, 717765, 3680568, 3925053, 4185776, 21454257, 22879220, 24398825, 125046908, 133352201, 142209108, 728829125, 777235920, 828857757, 4247929776
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 15 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+967, y).
Corresponding values y of solutions (x, y) are in A159701.
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) = (969+44**sqrt(2))/967 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (2487411+1629850*sqrt(2))/967^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159701, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159702 (decimal expansion of (969+44**sqrt(2))/967), A159703 (decimal expansion of (2487411+1629850*sqrt(2))/967^2).

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,45,2688,2901,3128,18105,19340},40] (* Harvey P. Dale, Nov 03 2013 *)
  • PARI
    {forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+1934*n+935089), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+1934 for n > 6; a(1)=0, a(2)=45, a(3)=2688, a(4)=2901, a(5)=3128, a(6)=18105.
G.f.: x*(45+2643*x+213*x^2-43*x^3-881*x^4-43*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 967*A001652(k) for k >= 0.
a(1)=0, a(2)=45, a(3)=2688, a(4)=2901, a(5)=3128, a(6)=18105, a(7)=19340, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Nov 03 2013

Extensions

Edited and two terms added by Klaus Brockhaus, Apr 21 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

Views

Author

Mohamed Bouhamida, Jun 15 2007

Keywords

Comments

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.

Links

Crossrefs

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).

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,27,888,1011,1148,6027,6740},40] (* Harvey P. Dale, Feb 26 2015 *)
  • PARI
    {forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+674*n+113569), print1(n, ",")))}

Formula

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

Extensions

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

Views

Author

Mohamed Bouhamida, Jun 15 2007

Keywords

Comments

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.

Links

Crossrefs

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).

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,35,1568,1731,1908,10595,11540},30] (* Harvey P. Dale, May 27 2018 *)
  • PARI
    {forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+1154*n+332929), print1(n, ",")))}

Formula

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.

Extensions

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