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 1-3 of 3 results.

A280321 Number of 2 X 2 matrices with all elements in {0,..,n} having determinant = n*permanent.

Original entry on oeis.org

1, 12, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6561, 6889, 7225, 7569, 7921, 8281, 8649, 9025, 9409, 9801, 10201
Offset: 0

Views

Author

Indranil Ghosh, Jan 01 2017

Keywords

Comments

Same as A016754, except for n=1. Here a(1)=12 but A016754(1)=9.

Links

Crossrefs

Cf. A016754.

Programs

  • Python
    def t(n):
    s=0
    for a in range(n+1):
        for b in range(n+1):
            for c in range(n+1):
                for d in range(n+1):
                    if (a*d-b*c)==n*(a*d+b*c):
                        s+=1
    return s
for i in range(41):
    print(str(i)+" "+str(t(i)))

Formula

a(n+1) = (((n-2)*a(n))/(n-1)) + ((12*(n)^2-12*(n)+1)/(n-1)) for n>=1.
Conjectures from Colin Barker, Jan 01 2017: (Start)
a(n) = (1 + 2*n)^2 = A273789(n) = A273743(n) for n>1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>4.
G.f.: (1 + 9*x - 8*x^2 + 9*x^3 - 3*x^4) / (1 - x)^3.
(End)

A273790 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 931", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 6, 31, 80, 161, 282, 451, 676, 965, 1326, 1767, 2296, 2921, 3650, 4491, 5452, 6541, 7766, 9135, 10656, 12337, 14186, 16211, 18420, 20821, 23422, 26231, 29256, 32505, 35986, 39707, 43676, 47901, 52390, 57151, 62192, 67521, 73146, 79075, 85316, 91877, 98766
Offset: 0

Views

Author

Robert Price, May 30 2016

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Crossrefs

Cf. A273789.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=931; stages=128;
rule=IntegerDigits[code,2,10];
g=2*stages+1; (* Maximum size of grid *)
a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
ca=a;
ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
PrependTo[ca,a];
(* Trim full grid to reflect growth by one cell at each stage *)
k=(Length[ca[[1]]]+1)/2;
ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}];
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)

Formula

Conjectures from Colin Barker, May 31 2016: (Start)
a(n) = (4*n^3+12*n^2+11*n-9)/3 for n>0.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.
G.f.: (1+2*x+13*x^2-12*x^3+4*x^4) / (1-x)^4. (End)
Conjectured e.g.f.: 4 + exp(x)*(4*x^3/3 + 8*x^2 + 9*x - 3). - Stefano Spezia, Dec 30 2024

A273791 First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 931", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

4, 20, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 408, 416, 424, 432
Offset: 0

Views

Author

Robert Price, May 30 2016

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Crossrefs

Cf. A273789.

Programs

  • Mathematica
    CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=931; stages=128;
rule=IntegerDigits[code,2,10];
g=2*stages+1; (* Maximum size of grid *)
a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
ca=a;
ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
PrependTo[ca,a];
(* Trim full grid to reflect growth by one cell at each stage *)
k=(Length[ca[[1]]]+1)/2;
ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}];
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Table[on[[i+1]]-on[[i]],{i,1,Length[on]-1}] (* Difference at each stage *)

Formula

Conjectures from Colin Barker, May 31 2016: (Start)
a(n) = 8*(1+n) = A273745(n) = A273315(n) for n>1.
a(n) = 2*a(n-1)-a(n-2) for n>3.
G.f.: 4*(1+3*x-3*x^2+x^3) / (1-x)^2.
(End)
Showing 1-3 of 3 results.

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