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.
Table[Round[Cosh[(2 n - 1) ArcSinh[n]]^2], {n, 0, 10}] (* Artur Jasinski *)
Table[Round[Sinh[(2 n - 1) ArcSinh[n]]], {n, 0, 20}] (* Artur Jasinski *)
Round[Table[1/2 (n - Sqrt[1 + n^2])^(2 n - 1) + 1/2 (n + Sqrt[1 + n^2])^(2 n - 1), {n, 0, 10}]] (* Artur Jasinski, Feb 14 2010 *)
Primitive square with s = 10:
a = 1, b = 2, s = 10, m = 10, z = 20, and
Non-primitive square with s = 20:
a = 1, b = 2, s = 20, m = 40, z = 80.
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with respectively m = 10 (and m = 40) elementary 2 X 5 rectangles as below:
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There are these three possibilities:
- 10 is a primitive term because the square 10 X 10 can be tiled with 20 squares of size 1 X 1 and 20 squares of size 2 X 2, and no smaller square can be tiled with a same number of squares of size 1 X 1 and of squares of size 2 X 2.
- 20 is a non-primitive term because the square 20 X 20 can be tiled with 80 squares of size 1 X 1 and 80 squares of size 2 X 2.
- 30 is a primitive term because the square 30 X 30 can be tiled with 90 squares of size 1 X 1 and 90 squares of size 3 X 3, and no smaller square can be tiled with a same number of squares of size 1 X 1 and of squares of size 3 X 3,
but also, 30 is a non-primitive term because the square 30 X 30 can be tiled with 180 squares of size 1 X 1 and 180 squares of size 2 X 2.
isokp1(s) = {if (!(s % 2) && ispower(s/2, 4), return (0)); my(m = sqrtnint(s, 3)); vecsearch(setbinop((x, y)->x*y*(x^2+y^2), [1..m]), s); }
isok(s) = {if (isokp1(s), return (1)); fordiv(s, d, if ((d>1) || (dMichel Marcus, Dec 22 2021
Table[Round[Sinh[(2 n - 1) ArcCosh[n]]^2], {n, 0, 20}]
(x,y) = (5,1), (515,101), (52525,10301), ... give the positive integer solutions to x^2 - 26*y^2 = -1.
Table[(1/5) Round[N[Sinh[(2 n - 1) ArcSinh[5]], 100]], {n, 1, 50}] (* Artur Jasinski, Feb 10 2010 *)
CoefficientList[Series[(1 + x)/(1 - 102 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 13 2014 *)
LinearRecurrence[{102,-1},{1,103},20] (* Harvey P. Dale, Aug 20 2017 *)
x='x+O('x^99); Vec((1+x)/(1-102*x+x^2)) \\ Altug Alkan, Apr 05 2018
Square 10 x 10 with a = 1, b = 2, s = 10, z = 20.
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core[n_] := Times @@ Apply[ Power, {#[[1]], Mod[#[[2]], 2]}& /@ FactorInteger[n], {1}]; Select[Range[500000], core[#] == Ceiling[Sqrt[#]]&] (* Jean-François Alcover, Jul 26 2011 *)
The primary square with side A345285(1) = 10 can be tiled with a(1) = 20 small squares of side a = 1 and 20 large squares of side b = 2.
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The primary square with side A345285(6) = 160 can be tiled with a(6) = 1280 small squares of side a = 2 and 1280 large squares of side b = 4.
[n^4+n^3+n: n in [0..50]]; // Vincenzo Librandi, Jun 09 2011
Table[n^4+n^3+n,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,3,26,111,324},40] (* Harvey P. Dale, Apr 25 2015 *)
a(n)=n^4+n^3+n \\ Charles R Greathouse IV, Oct 21 2022
a(2) = 89 is in the sequence because 3^4 + 3^2 - 1 = 89 is prime.
select(isprime, [seq(n^4+n^2-1, n=1..1000)]); # Robert Israel, Mar 24 2017
a={}; Do[p=n^4+n^2-1; If[PrimeQ[p], AppendTo[a, p]], {n, 10^2}]; Print[a];
for(n=1, 1e3, if(isprime(p=n^4+n^2-1), print1(p ", "))) \\ Altug Alkan, Mar 24 2017
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