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.

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A111258 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+601)^2 = y^2.

Original entry on oeis.org

0, 539, 560, 1803, 4740, 4859, 12020, 29103, 29796, 71519, 171080, 175119, 418296, 998579, 1022120, 2439459, 5821596, 5958803, 14219660, 33932199, 34731900, 82879703, 197772800, 202433799, 483059760, 1152705803, 1179872096
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 03 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+601, y).
Corresponding values y of solutions (x, y) are in A160098.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (843+418*sqrt(2))/601 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (361299+5950*sqrt(2))/601^2 for n mod 3 = 0.

Crossrefs

Cf. A160098, A001652, A101152, A156035 (decimal expansion of 3+2*sqrt(2)), A160099 (decimal expansion of (843+418*sqrt(2))/601), A160100 (decimal expansion of (361299+5950*sqrt(2))/601^2).

Programs

  • Magma
    I:=[0,539,560,1803,4740,4859,12020]; [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 22 2018
  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,539,560,1803,4740,4859,12020}, 50] (* G. C. Greubel, Apr 22 2018 *)
  • PARI
    {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1202*n+361201), print1(n, ",")))}
    
  • PARI
    x='x+O('x^30); concat([0], Vec(x*(539 +21*x +1243*x^2 -297*x^3 -7*x^4 -297*x^5)/((1-x)*(1 -6*x^3 +x^6)))) \\ G. C. Greubel, Apr 22 2018
    

Formula

a(n) = 6*a(n-3) - a(n-6) + 1202 for n > 6; a(1)=0, a(2)=539, a(3)=560, a(4)=1803, a(5)=4740, a(6)=4859.
G.f.: x*(539 +21*x +1243*x^2 -297*x^3 -7*x^4 -297*x^5)/((1-x)*(1 -6*x^3 +x^6)).
a(3*k+1) = 601*A001652(k) for k >= 0.

Extensions

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

A123654 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+809)^2 = y^2.

Original entry on oeis.org

0, 264, 1491, 2427, 3811, 10764, 16180, 24220, 64711, 96271, 143127, 379120, 563064, 836160, 2211627, 3283731, 4875451, 12892260, 19140940, 28418164, 75143551, 111563527, 165635151, 437970664, 650241840, 965394360, 2552682051
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 03 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+809, y).
Corresponding values y of solutions (x, y) are in A160203.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (873+232*sqrt(2))/809 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (989043+524338*sqrt(2))/809^2 for n mod 3 = 0.

Crossrefs

Cf. A160203, A001652, A115135, A156035 (decimal expansion of 3+2*sqrt(2)), A160204 (decimal expansion of (873+232*sqrt(2))/809), A160205 (decimal expansion of (989043+524338*sqrt(2))/809^2).

Programs

  • Magma
    I:=[0,264,1491,2427,3811,10764,16180]; [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
  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,264,1491,2427,3811,10764,16180}, 50] (* G. C. Greubel, May 04 2018 *)
  • PARI
    {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1618*n+654481), print1(n, ",")))}
    
  • PARI
    x='x+O('x^30); concat([0], Vec(x*(264+1227*x+936*x^2-200*x^3 -409*x^4 -200*x^5)/((1-x)*(1-6*x^3 +x^6)))) \\ G. C. Greubel, May 04 2018
    

Formula

a(n) = 6*a(n-3)-a(n-6)+1618 for n > 6; a(1)=0, a(2)=264, a(3)=1491, a(4)=2427, a(5)=3811, a(6)=10764.
G.f.: x*(264+1227*x+936*x^2-200*x^3-409*x^4-200*x^5) / ((1-x)*(1-6*x^3 +x^6)).
a(3*k+1) = 809*A001652(k) for k >= 0.

Extensions

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

A129642 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+457)^2 = y^2.

Original entry on oeis.org

0, 348, 495, 1371, 3255, 4088, 9140, 20096, 24947, 54383, 118235, 146508, 318072, 690228, 855015, 1854963, 4024047, 4984496, 10812620, 23454968, 29052875, 63021671, 136706675, 169333668, 367318320, 796785996, 986950047, 2140889163, 4644010215, 5752367528
Offset: 1

Views

Author

Mohamed Bouhamida, May 31 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+457, y).
Corresponding values y of solutions (x, y) are in A160580.
Limit_{n->oo} a(n)/a(n-3) = 3+2*sqrt(2).
Limit_{n->oo} a(n)/a(n-1) = (601+276*sqrt(2))/457 for n mod 3 = {1, 2}.
Limit_{n->oo} a(n)/a(n-1) = (213651+31850*sqrt(2))/457^2 for n mod 3 = 0.

Crossrefs

Cf. A160580, A001652, A129641, A156035 (decimal expansion of 3+2*sqrt(2)), A160581 (decimal expansion of (601+276*sqrt(2))/457), A160582 (decimal expansion of (213651+31850*sqrt(2))/457^2).

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,348,495,1371,3255,4088,9140},30] (* Harvey P. Dale, May 13 2012 *)
  • PARI
    {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+914*n+208849), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+914 for n > 6; a(1)=0, a(2)=348, a(3)=495, a(4)=1371, a(5)=3255, a(6)=4088.
G.f.: x^2*(348+147*x+876*x^2-204*x^3-49*x^4-204*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 457*A001652(k) for k >= 0.
a(1)=0, a(2)=348, a(3)=495, a(4)=1371, a(5)=3255, a(6)=4088, a(7)=9140, a(n) = a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, May 13 2012

Extensions

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

A129857 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+857)^2 = y^2.

Original entry on oeis.org

0, 235, 1696, 2571, 3796, 12075, 17140, 24255, 72468, 101983, 143448, 424447, 596472, 838147, 2475928, 3478563, 4887148, 14432835, 20276620, 28486455, 84122796, 118182871, 166033296, 490305655, 688822320, 967715035, 2857712848
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 03 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+857, y).
Corresponding values y of solutions (x, y) are in A160206.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (907+210*sqrt(2))/857 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1208787+678878*sqrt(2))/857^2 for n mod 3 = 0.

Crossrefs

Cf. A160206, A001652, A123654, A156035 (decimal expansion of 3+2*sqrt(2)), A160207 (decimal expansion of (907+210*sqrt(2))/857), A160208 (decimal expansion of (1208787+678878*sqrt(2))/857^2).

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat  Coefficients(R!(x*(235+1461*x+875*x^2-185*x^3-487*x^4-185*x^5)/((1-x)*(1-6*x^3+x^6))) );  // G. C. Greubel, May 03 2018
  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,235,1696,2571,3796,12075, 17140}, 30] (* or *) CoefficientList[Series[x (235+1461x+875x^2-185x^3- 487x^4- 185x^5)/((1-x)(1-6x^3+x^6)),{x,0,30}],x] (* Harvey P. Dale, Apr 24 2011 *)
  • PARI
    {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1714*n+734449), print1(n, ",")))}
    

Formula

a(n) = 6*a(n-3)-a(n-6)+1714 for n > 6; a(1)=0, a(2)=235, a(3)=1696, a(4)=2571, a(5)=3796, a(6)=12075.
G.f.: x*(235+1461*x+875*x^2-185*x^3-487*x^4-185*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 857*A001652(k) for k >= 0.

Extensions

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

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.

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.

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.

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

A130609 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+223)^2 = y^2.

Original entry on oeis.org

0, 32, 533, 669, 833, 3672, 4460, 5412, 21945, 26537, 32085, 128444, 155208, 187544, 749165, 905157, 1093625, 4366992, 5276180, 6374652, 25453233, 30752369, 37154733, 148352852, 179238480, 216554192, 864664325, 1044678957, 1262170865, 5039633544, 6088835708
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 17 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+223, y).
Corresponding values y of solutions (x, y) are in A159809.
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) = (227+30*sqrt(2))/223 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (105507+65798*sqrt(2))/223^2 for n mod 3 = 0.

Crossrefs

Cf. A159809, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159810 (decimal expansion of (227+30*sqrt(2))/223), A159811 (decimal expansion of (105507+65798*sqrt(2))/223^2).

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,32,533,669,833,3672,4460}, 70]  (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)
  • PARI
    {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+446*n+49729), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+446 for n > 6; a(1)=0, a(2)=32, a(3)=533, a(4)=669, a(5)=833, a(6)=3672.
G.f.: x*(32+501*x+136*x^2-28*x^3-167*x^4-28*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 223*A001652(k) for k >= 0.

Extensions

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

A130610 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+359)^2 = y^2.

Original entry on oeis.org

0, 40, 901, 1077, 1281, 6160, 7180, 8364, 36777, 42721, 49621, 215220, 249864, 290080, 1255261, 1457181, 1691577, 7317064, 8493940, 9860100, 42647841, 49507177, 57469741, 248570700, 288549840, 334959064, 1448777077, 1681792581
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 17 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+359, y).
Corresponding values y of solutions (x, y) are in A159844.
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) = (363+38*sqrt(2))/359 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (293619+186550*sqrt(2))/359^2 for n mod 3 = 0.

Crossrefs

Cf. A159844, A028871, A118337, A130609, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159845 (decimal expansion of (363+38*sqrt(2))/359), A159846 (decimal expansion of (293619+186550*sqrt(2))/359^2).

Programs

  • PARI
    {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+718*n+128881), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+718 for n > 6; a(1)=0, a(2)=40, a(3)=901, a(4)=1077, a(5)=1281, a(6)=6160.
G.f.: x*(40+861*x+176*x^2-36*x^3-287*x^4-36*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 359*A001652(k) for k >= 0.

Extensions

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