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.

A035598 Number of points of L1 norm 4 in cubic lattice Z^n.

Original entry on oeis.org

0, 2, 16, 66, 192, 450, 912, 1666, 2816, 4482, 6800, 9922, 14016, 19266, 25872, 34050, 44032, 56066, 70416, 87362, 107200, 130242, 156816, 187266, 221952, 261250, 305552, 355266, 410816, 472642, 541200, 616962, 700416, 792066
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A035596, A035597, A035599, A035600, A035601, A035602, A035603, A035604, A035605, A035606.
Cf. A001845, A001846, A008412, A014820.
Column 4 of A035607, A266213, A343599.
Row 4 of A113413, A119800, A122542.

Programs

  • Magma
    [( 2*n^4 +4*n^2 )/3: n in [0..40]]; // Vincenzo Librandi, Apr 22 2012
  • Maple
    f := proc(d,m) local i; sum( 2^i*binomial(d,i)*binomial(m-1,i-1),i=1..min(d,m)); end; # n=dimension, m=norm
  • Mathematica
    CoefficientList[Series[2*x*(1+x)^3/(1-x)^5,{x,0,40}],x] (* Vincenzo Librandi, Apr 22 2012 *)
LinearRecurrence[{5,-10,10,-5,1},{0,2,16,66,192},50] (* Harvey P. Dale, Dec 11 2019 *)
  • PARI
    a(n)=2*n^2*(n^2+2)/3 \\ Charles R Greathouse IV, Dec 07 2011

Formula

a(n) = 2*n^2*(n^2 + 2)/3. - Frank Ellermann, Mar 16 2002
G.f.: 2*x*(1+x)^3/(1-x)^5. - Colin Barker, Apr 15 2012
a(n) = 2*A014820(n-1). - R. J. Mathar, Dec 10 2013
a(n) = a(n-1) + A035597(n) + A035597(n-1). - Bruce J. Nicholson, Mar 11 2018
From Shel Kaphan, Feb 28 2023: (Start)
a(n) = 2*n*Hypergeometric2F1(1-n,1-k,2,2), where k=4.
a(n) = A001846(n) - A001845(n).
a(n) = A008412(n)*n/4. (End)
From Amiram Eldar, Mar 12 2023: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/8 - 3*Pi*coth(sqrt(2)*Pi)/(8*sqrt(2)) + 3/16.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/16 + 3*Pi*cosech(sqrt(2)*Pi)/(8*sqrt(2)) - 3/16. (End)
E.g.f.: 2*exp(x)*x*(3 + 9*x + 6*x^2 + x^3)/3. - Stefano Spezia, Mar 14 2024

A035599 Number of points of L1 norm 5 in cubic lattice Z^n.

Original entry on oeis.org

0, 2, 20, 102, 360, 1002, 2364, 4942, 9424, 16722, 28004, 44726, 68664, 101946, 147084, 207006, 285088, 385186, 511668, 669446, 864008, 1101450, 1388508, 1732590, 2141808, 2625010, 3191812, 3852630, 4618712, 5502170, 6516012
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A035596-A035606.
Column 5 of A035607, A266213. Row 5 of A113413, A119800, A122542.

Programs

  • Magma
    [(4*n^4+20*n^2+6)*n/15: n in [0..30]]; // Vincenzo Librandi, Apr 23 2012
  • Maple
    f := proc(d,m) local i; sum( 2^i*binomial(d,i)*binomial(m-1,i-1),i=1..min(d,m)); end; # n=dimension, m=norm
  • Mathematica
    CoefficientList[Series[2*x*(1+x)^4/(1-x)^6,{x,0,33}],x] (* Vincenzo Librandi, Apr 23 2012 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,2,20,102,360,1002},40] (* Harvey P. Dale, Dec 30 2023 *)
  • PARI
    a(n)=(4*n^5+20*n^3+6*n)/15 \\ Charles R Greathouse IV, Dec 07 2011

Formula

a(n) = (4*n^4+20*n^2+6)*n/15. - Frank Ellermann, Mar 16 2002
G.f.: 2*x*(1+x)^4/(1-x)^6. - Colin Barker, Mar 19 2012
a(n) = 2*A069038(n). - R. J. Mathar, Dec 10 2013
From Shel Kaphan, Mar 01 2023: (Start)
a(n) = 2*n*Hypergeometric2F1(1-n,1-k,2,2), where k=5.
a(n) = A001847(n) - A001846(n).
a(n) = A008413(n)*n/5. (End)

A035600 Number of points of L1 norm 6 in cubic lattice Z^n.

Original entry on oeis.org

0, 2, 24, 146, 608, 1970, 5336, 12642, 27008, 53154, 97880, 170610, 284000, 454610, 703640, 1057730, 1549824, 2220098, 3116952, 4298066, 5831520, 7796978, 10286936, 13408034, 17282432, 22049250, 27866072, 34910514, 43381856, 53502738, 65520920, 79711106
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A035596-A035607.

Programs

  • Magma
    [( 4*n^6 +40*n^4 +46*n^2 )/45: n in [0..30]]; // Vincenzo Librandi, Apr 23 2012
  • Maple
    f := proc(d,m) local i; sum( 2^i*binomial(d,i)*binomial(m-1,i-1),i=1..min(d,m)); end; # n=dimension, m=norm
  • Mathematica
    CoefficientList[Series[2*x*(1+x)^5/(1-x)^7,{x,0,33}],x] (* Vincenzo Librandi, Apr 23 2012 *)
  • PARI
    a(n)=(4*n^6+40*n^4+46*n^2)/45 \\ Charles R Greathouse IV, Dec 07 2011

Formula

a(n) = (4*n^6 + 40*n^4 + 46*n^2)/45. - Frank Ellermann, Mar 16 2002
G.f.: 2*x*(1+x)^5/(1-x)^7. - Colin Barker, Apr 15 2012
a(n) = 2*A069039(n). - R. J. Mathar, Dec 10 2013

A035602 Number of points of L1 norm 8 in cubic lattice Z^n.

Original entry on oeis.org

0, 2, 32, 258, 1408, 5890, 20256, 59906, 157184, 374274, 822560, 1690370, 3281280, 6065410, 10746400, 18347010, 30316544, 48663554, 76117536, 116323586, 174074240, 255582978, 368804128, 523804162, 733189632
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A035596-A035607.

Programs

  • Magma
    [(2*n^8+8*7*n^6+4*7*11*n^4+8*3*11*n^2)/315: n in [0..30]]; // Vincenzo Librandi, Apr 24 2012
  • Maple
    f := proc(d,m) local i; sum( 2^i*binomial(d,i)*binomial(m-1,i-1),i=1..min(d,m)); end; # n=dimension, m=norm
  • Mathematica
    CoefficientList[Series[2*x*(1+x)^7/(1-x)^9,{x,0,30}],x] (* Vincenzo Librandi, Apr 24 2012 *)
  • PARI
    a(n)=2*n^2*(n^6+28*n^4+154*n^2+132)/315 \\ Charles R Greathouse IV, Dec 07 2011

Formula

a(n) = (2*n^8 + 8*7*n^6 + 4*7*11*n^4 + 8*3*11*n^2)/(5*7*9). - Frank Ellermann, Mar 16 2002
G.f.: 2*x*(1+x)^7/(1-x)^9. - Colin Barker, Apr 15 2012
a(n) = 2*A099195(n). - R. J. Mathar, Dec 10 2013

A035603 Number of points of L1 norm 9 in cubic lattice Z^n.

Original entry on oeis.org

0, 2, 36, 326, 1992, 9290, 35436, 115598, 332688, 864146, 2060980, 4573910, 9545560, 18892250, 35704060, 64797470, 113461024, 192441122, 317222212, 509663334, 800061160, 1229718378, 1854105484, 2746713774, 4003707568
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A035596-A035607.

Programs

  • Magma
    [(4*n^9+168*n^7+1596*n^5+3272*n^3+630*n)/2835: n in [0..30]]; // Vincenzo Librandi, Apr 24 2012
  • Maple
    f := proc(d,m) local i; sum( 2^i*binomial(d,i)*binomial(m-1,i-1),i=1..min(d,m)); end; # n=dimension, m=norm
  • Mathematica
    CoefficientList[Series[2*x*(1+x)^8/(1-x)^10,{x,0,30}],x] (* Vincenzo Librandi, Apr 24 2012 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,2,36,326,1992,9290,35436,115598,332688,864146},30] (* Harvey P. Dale, Jan 17 2021 *)
  • PARI
    a(n)=(4*n^9+168*n^7+1596*n^5+3272*n^3+630*n)/2835 \\ Charles R Greathouse IV, Dec 07 2011

Formula

a(n) = (4*n^9 + 168*n^7 + 1596*n^5 + 3272*n^3 + 630*n)/(5*7*9*9). - Frank Ellermann, Mar 16 2002
G.f.: 2*x*(1+x)^8/(1-x)^10. - Colin Barker, Apr 15 2012
a(n) = 2*A099196(n). - R. J. Mathar, Dec 10 2013

A035601 Number of points of L1 norm 7 in cubic lattice Z^n.

Original entry on oeis.org

0, 2, 28, 198, 952, 3530, 10836, 28814, 68464, 148626, 299660, 568150, 1022760, 1761370, 2919620, 4680990, 7288544, 11058466, 16395516, 23810534, 33940120, 47568618, 65652532, 89347502, 120037968, 159369650, 209284972
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A035596-A035607.

Programs

  • Magma
    [( 8*n^6 +4*5*7*n^4 +8*7*7*n^2 +2*5*9 )*n/(5*7*9): n in [0..30]]; // Vincenzo Librandi, Apr 23 2012
  • Maple
    f := proc(d,m) local i; sum( 2^i*binomial(d,i)*binomial(m-1,i-1),i=1..min(d,m)); end; # n=dimension, m=norm
  • Mathematica
    CoefficientList[Series[2*x*(1+x)^6/(1-x)^8,{x,0,30}],x] (* Vincenzo Librandi, Apr 23 2012 *)
  • PARI
    (8*n^7+140*n^5+392*n^3+90*n)/315 \\ Charles R Greathouse IV, Dec 07 2011

Formula

a(n) = (8*n^6 + 4*5*7*n^4 + 8*7*7*n^2 + 2*5*9)*n/(5*7*9). - Frank Ellermann, Mar 16 2002
G.f.: 2*x*(1+x)^6/(1-x)^8. - Colin Barker, Apr 15 2012
a(n) = 2*A099193(n). - R. J. Mathar, Dec 10 2013
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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