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 1-6 of 6 results.

A152729 a(n) = (n-2)^4 - a(n-1) - a(n-2), with a(1) = a(2) = 0.

Original entry on oeis.org

0, 0, 1, 15, 65, 176, 384, 736, 1281, 2079, 3201, 4720, 6720, 9296, 12545, 16575, 21505, 27456, 34560, 42960, 52801, 64239, 77441, 92576, 109824, 129376, 151425, 176175, 203841, 234640, 268800, 306560, 348161, 393855, 443905, 498576, 558144
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Keywords

Comments

a(n+2) - a(n-1) = n^4 - (n-1)^4 = A005917(n) for all n in Z. - Michael Somos, Sep 02 2018

Examples

			0 + 0 + 1 = 1^4; 0 + 1 + 15 = 2^4; 1 + 15 + 65 = 3^4; ...
G.f. = x^3 + 15*x^4 + 65*x^5 + 176*x^6 + 384*x^7 + 736*x^8 + 1281*x^9 + ... - _Michael Somos_, Sep 02 2018

Links

Crossrefs

Cf. A005917, A152728, A152725, A152726, A000212.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!(x^3*(x+1)*(x^2+10*x+1)/((1-x)^5*(x^2+x+1)))); // G. C. Greubel, Sep 01 2018
  • Mathematica
    k0=k1=0;lst={k0,k1};Do[kt=k1;k1=n^4-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,4!}];lst
LinearRecurrence[{4,-6,5,-5,6,-4,1}, {0,0,1,15,65,176,384}, 50] (* G. C. Greubel, Sep 01 2018 *)
a[ n_] := With[ {m = Max[n, 2 - n]}, SeriesCoefficient[ x^3 (1 + x) (1 + 10 x + x^2) / ((1 - x)^5 (1 + x + x^2)), {x , 0, m}]]; (* Michael Somos, Sep 02 2018 *)
  • PARI
    concat([0,0], Vec(-x^3*(x+1)*(x^2+10*x+1)/((x-1)^5*(x^2+x+1)) + O(x^100))) \\ Colin Barker, Oct 28 2014
  • PARI
    {a(n) = my(m = max(n, 2 - n)); polcoeff( x^3 * (1 + x) * (1 + 10*x + x^2) / ((1 - x)^5 * (1 + x + x^2)) + x * O(x^m), m)}; /* Michael Somos, Sep 02 2018 */

Formula

G.f.: -x^3*(x+1)*(x^2+10*x+1) / ((x-1)^5*(x^2+x+1)). - Colin Barker, Oct 28 2014
a(n) = a(2 - n) for all n in Z. - Michael Somos, Sep 02 2018

Extensions

Definition adapted to offset by Georg Fischer, Jun 18 2021

A152730 a(n) + a(n+1) + a(n+2) = n^5, with a(1) = a(2) = 0.

Original entry on oeis.org

0, 0, 1, 31, 211, 782, 2132, 4862, 9813, 18093, 31143, 50764, 79144, 118924, 173225, 245675, 340475, 462426, 616956, 810186, 1048957, 1340857, 1694287, 2118488, 2623568, 3220568, 3921489, 4739319, 5688099, 6782950, 8040100, 9476950
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Keywords

Examples

			0 + 0 + 1 = 1^5; 0 + 1 + 31 = 2^5; 1 + 31 + 211 = 3^5; ...

Links

Crossrefs

Cf. A152728, A152729, A152725, A152726, A000212.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!(x^3*(x^4+26*x^3+66*x^2+26*x+1)/((x-1)^6*(x^2+x+1)))); // G. C. Greubel, Sep 01 2018
  • Mathematica
    k0=k1=0;lst={k0,k1};Do[kt=k1;k1=n^5-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,5!}];lst
LinearRecurrence[{5,-10,11,-10,11,-10,5,-1}, {0,0,1,31,211,782,2132, 4862}, 50] (* G. C. Greubel, Sep 01 2018 *)
CoefficientList[Series[x^2*(x^4 + 26*x^3 + 66*x^2 + 26*x + 1) / ((x - 1)^6*(x^2 + x + 1)),{x, 0, 50}], x] (* Stefano Spezia, Sep 02 2018 *)
  • PARI
    concat([0,0], Vec(x^3*(x^4+26*x^3+66*x^2+26*x+1)/((x-1)^6*(x^2+x+1)) + O(x^100))) \\ Colin Barker, Oct 28 2014

Formula

G.f.: x^3*(x^4 + 26*x^3 + 66*x^2 + 26*x + 1) / ((x-1)^6*(x^2 + x + 1)). - Colin Barker, Oct 28 2014

A152731 a(n) + a(n+1) + a(n+2) = n^6, a(1)=a(2)=0.

Original entry on oeis.org

0, 0, 1, 63, 665, 3368, 11592, 31696, 74361, 156087, 300993, 542920, 927648, 1515416, 2383745, 3630375, 5376505, 7770336, 10990728, 15251160, 20803993, 27944847, 37017281, 48417776, 62600832, 80084368, 101455425, 127375983
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Keywords

Comments

0 + 0 + 1 = 1^6; 0 + 1 + 63 = 2^6; ...

Links

Crossrefs

Cf. A152728, A152729, A152730, A152725, A152726, A000212.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!(x^3*(1+x)*(x^4+56*x^3+246*x^2+56*x+1)/((1-x)^7*(1 +x+ x^2)))); // G. C. Greubel, Sep 01 2018
  • Mathematica
    k0=k1=0;lst={k0,k1};Do[kt=k1;k1=n^6-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,5!}];lst
LinearRecurrence[{6, -15, 21, -21, 21, -21, 15, -6, 1}, {0, 0, 1, 63, 665, 3368, 11592, 31696, 74361}, 5000]
CoefficientList[Series[x^2*(1+x)*(x^4 + 56*x^3 + 246*x^2 + 56*x + 1)/((1-x)^7*(1+x+x^2)),{x, 0, 5000}], x] (* Stefano Spezia, Sep 02 2018 *)
  • PARI
    x='x+O('x^30); concat([0,0], Vec(x^3*(1+x)*(x^4+56*x^3 +246*x^2 +56*x+1)/((1-x)^7*(1+x+x^2)))) \\ G. C. Greubel, Sep 01 2018

Formula

From R. J. Mathar, Dec 12 2008: (Start)
a(n) = -26*n/3 + 20*n^3/3 - 5*n^2 + 7/3 - 2*n^5 + n^6/3 + 5*n^4/3 - 7*A131713(n)/3.
G.f.: x^3*(1+x)*(x^4 + 56*x^3 + 246*x^2 + 56*x + 1)/((1-x)^7*(1+x+x^2)). (End)

A152732 a(n) + a(n+1) + a(n+2) = 2^n.

Original entry on oeis.org

0, 0, 2, 2, 4, 10, 18, 36, 74, 146, 292, 586, 1170, 2340, 4682, 9362, 18724, 37450, 74898, 149796, 299594, 599186, 1198372, 2396746, 4793490, 9586980, 19173962, 38347922, 76695844, 153391690, 306783378, 613566756, 1227133514, 2454267026, 4908534052
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Keywords

Comments

0 + 0 + 2 = 2^1; 0 + 2 + 2 = 2^2; 2 + 2 + 4 = 2^3; 2 + 4 + 10 = 2^4; ...
With a(0)=1, a(n) is the number of length n strings in the language over alphabet {0,1} generated by the regular expression: ((0+1)(0*(11)*)*10)*. - Geoffrey Critzer, Jan 25 2014

Links

Crossrefs

Cf. A152728, A152729, A152730, A152731, A152725, A152726, A000212.

Programs

  • Magma
    I:=[0,0,2]; [n le 3 select I[n] else Self(n-1) +Self(n-2) +2*Self(n-3): n in [1..30]]; // G. C. Greubel, Sep 01 2018
  • Mathematica
    k0=k1=0;lst={k0,k1};Do[kt=k1;k1=2^n-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,5!}];lst
LinearRecurrence[{1, 1, 2}, {0, 0, 2}, 70] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)
  • PARI
    concat([0,0],Vec(2/(1-2*x)/(1+x+x^2)+O(x^99))) \\ Charles R Greathouse IV, Feb 24 2012

Formula

From R. J. Mathar, Dec 12 2008: (Start)
a(n) = 2*A077947(n-3).
G.f.: 2*x^3/((1-2*x)*(1+x+x^2)). (End)
a(n) = (1/21)*(3*2^n + 18*cos((2*n*Pi)/3) + 2*sqrt(3)*sin((2*n*Pi)/3)). - Zak Seidov, Dec 12 2008

A152733 a(n) + a(n+1) + a(n+2) = 3^n.

Original entry on oeis.org

0, 0, 3, 6, 18, 57, 168, 504, 1515, 4542, 13626, 40881, 122640, 367920, 1103763, 3311286, 9933858, 29801577, 89404728, 268214184, 804642555, 2413927662, 7241782986, 21725348961, 65176046880, 195528140640, 586584421923, 1759753265766, 5279259797298
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Keywords

Examples

			0 + 0 + 3 = 3^1; 0 + 3 + 6 = 3^2; 3 + 6 + 18 = 3^3; ...

Links

Crossrefs

Cf. A152728, A152729, A152730, A152731, A152732, A152725, A152726, A000212.

Programs

  • Magma
    [n le 2 select 0 else 3^(n-2) -Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 31 2014
  • Mathematica
    k0=k1=0;lst={k0,k1};Do[kt=k1;k1=3^n-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,5!}];lst
Rest[CoefficientList[Series[3x^3/((1-3x)(1+x+x^2)),{x,0,30}],x]] (* Harvey P. Dale, Aug 31 2014 *)
  • PARI
    x='x+O('x^30); concat([0,0], Vec(3*x^3/((1-3*x)*(1+x+x^2)))) \\ G. C. Greubel, Sep 01 2018

Formula

From R. J. Mathar, Dec 12 2008: (Start)
a(n) = 3*A077834(n-3).
G.f.: 3*x^3/((1-3*x)*(1+x+x^2)). (End)
a(n) = (1/13)*(3^n + 12*cos((2*n*Pi)/3) + 2*sqrt(3)*sin((2*n*Pi)/3)), n=1,2,... - Zak Seidov, Dec 12 2008

A361134 a(1) = 1, a(2) = 2; for n >= 3, a(n) = (n-1)^3 - a(n-1) - a(n-2).

Original entry on oeis.org

1, 2, 5, 20, 39, 66, 111, 166, 235, 328, 437, 566, 725, 906, 1113, 1356, 1627, 1930, 2275, 2654, 3071, 3536, 4041, 4590, 5193, 5842, 6541, 7300, 8111, 8978, 9911, 10902, 11955, 13080, 14269, 15526, 16861, 18266, 19745, 21308, 22947, 24666, 26475, 28366, 30343
Offset: 1

Views

Author

Tamas Sandor Nagy, Mar 02 2023

Keywords

Comments

The sum of every three consecutive terms is equal to the cube of the index of the middle one, i.e., a(n-1) + a(n) + a(n+1) = n^3.

Examples

			a(5) = (5-1)^3 - a(4) - a(3) = 4^3 - 20 - 5 = 64 - 20 - 5 = 39.

Links

Crossrefs

Cf. A000578, A152728, A242135.

Programs

  • Mathematica
    a[1] = 1; a[2] = 2; a[n_] := a[n] = (n - 1)^3 - a[n - 1] - a[n - 2]; Array[a, 45] (* Amiram Eldar, Mar 03 2023 *)
  • PARI
    lista(nn) = my(va = vector(nn)); va[1] = 1; va[2] = 2; for (n=3, nn, va[n] = (n-1)^3 - va[n-1] - va[n-2];); va; \\ Michel Marcus, Mar 03 2023

Formula

G.f.: x*(2*x^5 - 7*x^4 + 9*x^3 + 2*x^2 - x + 1)/((x^2 + x + 1)*(x - 1)^4).
a(n) = (A242135(n) - 6*cos(2*n*Pi/3) + 2*sin(2*n*Pi/3)/sqrt(3))/3. - Stefano Spezia, Mar 04 2023
Showing 1-6 of 6 results.

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