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.

Previous Showing 31-33 of 33 results.

A008535 Coordination sequence for {A_7}* lattice.

Original entry on oeis.org

1, 16, 128, 688, 2746, 8752, 23536, 55568, 118498, 232976, 428752, 747056, 1243258, 1989808, 3079456, 4628752, 6781826, 9714448, 13638368, 18805936, 25515002, 34114096, 45007888, 58662928, 75613666, 96468752, 121917616, 152737328, 189799738, 234078896
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Programs

  • GAP
    Concatenation([1], List([1..45], n-> (36+175*n^2+70*n^4+7*n^6)/18 )); # G. C. Greubel, Nov 10 2019
  • Magma
    [1] cat [(36+175*n^2+70*n^4+7*n^6)/18: n in [1..45]]; // G. C. Greubel, Nov 10 2019
  • Maple
    1, seq( (7*k^6+70*k^4+175*k^2+36)/18, k=1..40);
  • Mathematica
    Table[If[n==0,1,(36+175*n^2+70*n^4+7*n^6)/18], {n,0,40}] (* G. C. Greubel, Nov 10 2019 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,16,128,688,2746,8752,23536,55568},40] (* Harvey P. Dale, Jun 04 2023 *)
  • PARI
    Vec(-(x+1)*(x^6+8*x^5+29*x^4+64*x^3+29*x^2+8*x+1) / (x-1)^7 + O(x^40)) \\ Colin Barker, Mar 03 2015
  • PARI
    vector(46, n, if(n==1,1, (36+175*(n-1)^2+70*(n-1)^4+7*(n-1)^6)/18 ) ) \\ G. C. Greubel, Nov 10 2019
  • Sage
    [1]+[(36+175*n^2+70*n^4+7*n^6)/18 for n in (1..45)]; # G. C. Greubel, Nov 10 2019

Formula

G.f.: (1+x)*(1+8*x+29*x^2+64*x^3+29*x^4+8*x^5+x^6)/(1-x)^7. - Colin Barker, Mar 03 2015
E.g.f.: -1 + (36 + 252*x + 882*x^2 + 1050*x^3 + 525*x^4 + 105*x^5 + 7*x^6)*exp(x)/18. - G. C. Greubel, Nov 10 2019

A010079 Coordination sequence for net formed by holes in D_4 lattice.

Original entry on oeis.org

1, 16, 104, 344, 792, 1528, 2632, 4152, 6200, 8792, 12072, 16024, 20824, 26424, 33032, 40568, 49272, 59032, 70120, 82392, 96152, 111224, 127944, 146104, 166072, 187608, 211112, 236312, 263640, 292792, 324232, 357624, 393464, 431384, 471912, 514648, 560152
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.

Links

Programs

  • Maple
    f := n-> if n mod 2 = 0 then 12*n^3+8*n-8 else 12*n^3+4*n+8; fi; #(for n>1).
  • Mathematica
    CoefficientList[Series[-(8 x^7 - 25 x^6 + 2 x^5 - 63 x^4 - 124 x^3 - 71 x^2 - 14 x-1)/((x - 1)^4 (x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 15 2013 *)
LinearRecurrence[{2,1,-4,1,2,-1},{1,16,104,344,792,1528,2632,4152},40] (* Harvey P. Dale, Nov 08 2017 *)

Formula

a(n) = 2*(-4*(-1)^n+(3+(-1)^n)*n+6*n^3) for n>1. G.f.: -(8*x^7 -25*x^6 +2*x^5 -63*x^4 -124*x^3 -71*x^2 -14*x -1) / ((x-1)^4*(x+1)^2). - Colin Barker, Jul 07 2013

Extensions

More terms from Colin Barker, Jul 07 2013

A088307 Triangle, read by rows, T(n,k) = n^2 + k^2 if gcd(n,k)=1, otherwise 0.

Original entry on oeis.org

2, 5, 0, 10, 13, 0, 17, 0, 25, 0, 26, 29, 34, 41, 0, 37, 0, 0, 0, 61, 0, 50, 53, 58, 65, 74, 85, 0, 65, 0, 73, 0, 89, 0, 113, 0, 82, 85, 0, 97, 106, 0, 130, 145, 0, 101, 0, 109, 0, 0, 0, 149, 0, 181, 0, 122, 125, 130, 137, 146, 157, 170, 185, 202, 221, 0
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 05 2003

Keywords

Comments

(n^2-k^2, 2*k*n, T(n,k)) is a primitive Pythagorean triple iff T(n,k) > 0.

Examples

			Triangle begins:
   2;
   5,  0;
  10, 13,  0;
  17,  0, 25,  0;
  26, 29, 34, 41,  0;
  37,  0,  0,  0, 61, 0;
  ...

Links

Crossrefs

Cf. A000007, A000010, A001844, A002522, A008527, A053818.
Cf. A057427, A070216, A078370, A079273, A190816.

Programs

  • Magma
    function A088307(n,k)
  if GCD(k,n) eq 1 then return n^2+k^2;
  else return 0;
  end if; return A088307;
end function;
[A088307(n,k): k in [1..n], n in [1..13]]; // G. C. Greubel, Dec 16 2022
  • Mathematica
    Table[If[CoprimeQ[n,k],n^2+k^2,0],{n,20},{k,n}]//Flatten (* Harvey P. Dale, Jul 13 2018 *)
  • SageMath
    def A088307(n,k):
    if (gcd(n,k)==1): return n^2 + k^2
    else: return 0
flatten([[A088307(n,k) for k in range(1,n+1)] for n in range(1,14)]) # G. C. Greubel, Dec 16 2022

Formula

T(n, n) = 2*A000007(n-1).
T(n, 1) = A002522(n).
T(2*n+1, 2) = A078370(n).
Sum_{k=1..n} A057427(T(n,m)) = A000010(n).
From G. C. Greubel, Dec 15 2022: (Start)
T(n, n-1) = A001844(n).
T(n, n-2) = ((1-(-1)^n)/2) * A008527((n+1)/2).
T(2*n, n) = 5*A000007(n-1).
T(2*n+1, n) = A079273(n+1).
T(2*n-1, n) = A190816(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A053818(n+1) + [n=1]. (End)
Previous Showing 31-33 of 33 results.

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

Content is available under The OEIS End-User License Agreement.