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.

A193263 G.f.: A(x) = x + Sum_{n>=1} x^(2*n) / (1+x)^A193259(n).

Original entry on oeis.org

1, 1, -1, 2, -5, 12, -26, 52, -101, 201, -422, 927, -2070, 4579, -9894, 20789, -42517, 84937, -166570, 322700, -622500, 1207056, -2376168, 4787523, -9908610, 21021499, -45404102, 98952388, -215756156, 467541948, -1002478352, 2121546013, -4427208709, 9110572776, -18503242145, 37135048484, -73759839074
Offset: 1

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Author

Paul D. Hanna, Jul 19 2011

Keywords

Comments

Note: A193259(n) = n + floor(log_2(n)) + A011371(n), where A011371(n) = highest power of 2 dividing n!.
The g.f. A(x), as a power series in x, diverges at x=-1/2 and converges at x=+1/2 to A(1/2) = 0.6811907120229079095390167697...
Other values: A(x) = 1/2 at x = 0.385874434537804442263..., A(x) = 1 at x = 0.685568171776262105563..., A((sqrt(5)-1)/2) = 0.880771363850914609641...

Examples

			G.f.: A(x) = x + x^2 - x^3 + 2*x^4 - 5*x^5 + 12*x^6 - 26*x^7 + 52*x^8 +...
where
A(x) = x + x^2/(1+x)^1 + x^4/(1+x)^4 + x^6/(1+x)^5 + x^8/(1+x)^9 + x^10/(1+x)^10 + x^12/(1+x)^12 + x^14/(1+x)^13 + x^16/(1+x)^18 +...+ x^(2*n)/(1+x)^A193259(n) +...
Also,
A(x) = x/(1+x)^0 + x^2/(1+x)^2 + x^3/(1+x)^2 + x^4/(1+x)^5 + x^5/(1+x)^5 + x^6/(1+x)^6 + x^7/(1+x)^6 + x^8/(1+x)^10 +...+ x^n/(1+x)^(A193259(n)-n) +...

Crossrefs

Cf. A193259, A011371, A000523.

Programs

  • PARI
    {a(n)=polcoeff(sum(m=1,n,x^m/(1+x+x*O(x^n))^(floor(log(m+1/2)/log(2)) + valuation(m!, 2))),n)}

Formula

G.f.: A(x) = Sum_{n>=1} x^n / (1+x)^(A193259(n) - n).

A193260 G.f.: x+x^2 = Sum_{n>=1} x^n * ((1+x+x^2)^n - x^(2*n)) / (1+x+x^2)^a(n).

Original entry on oeis.org

1, 2, 5, 6, 7, 9, 10, 11, 15, 16, 17, 19, 20, 21, 23, 24, 25, 28, 29, 30, 32, 33, 34, 36, 37, 38, 43, 44, 45, 47, 48, 49, 51, 52, 53, 56, 57, 58, 60, 61, 62, 64, 65, 66, 69, 70, 71, 73, 74, 75, 77, 78, 79, 83, 84, 85, 87, 88, 89, 91, 92, 93, 96, 97, 98, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113, 114, 115, 117, 118, 119, 125, 126, 127, 129, 130, 131, 133, 134, 135
Offset: 1

Views

Author

Paul D. Hanna, Jul 19 2011

Keywords

Examples

			G.f.: x+x^2 = x*((1+x+x^2) - x^2)/(1+x+x^2)  + x^2*((1+x+x^2)^2 - x^4)/(1+x+x^2)^2  + x^3*((1+x+x^2)^3 - x^6)/(1+x+x^2)^5  + x^4*((1+x+x^2)^4 - x^8)/(1+x+x^2)^6  + x^5*((1+x+x^2)^5 - x^10)/(1+x+x^2)^7  + x^6*((1+x+x^2)^6 - x^12)/(1+x+x^2)^9  + x^7*((1+x+x^2)^7 - x^14)/(1+x+x^2)^10  + x^8*((1+x+x^2)^8 - x^16)/(1+x+x^2)^11  + x^9*((1+x+x^2)^9 - x^18)/(1+x+x^2)^15 +...+ x^n*((1+x+x^2)^n - x^(2*n))/(1+x+x^2)^a(n) +...

Links

Crossrefs

Cf. A193259, A054861.

Programs

  • Mathematica
    Table[n+Floor[Log[3,n]]+IntegerExponent[n!,3],{n,90}] (* Harvey P. Dale, Oct 10 2012 *)
  • PARI
    {a(n)=if(n<1,0,n + floor(log(n+1/2)/log(3)) + valuation(n!,3))}
  • PARI
    {a(n)=if(n<1,0,if(n==1,1,polcoeff(sum(m=1,n+1,x^m*((1+x+x^2)^m-x^(2*m))/(1+x+x^2 +x^2*O(x^n))^if(m>=n,1,a(m)))+x^(n+1),n+1)))}

Formula

a(n) = n + floor(log_3(n)) + A054861(n) for n>=1, where A054861(n) = highest power of 3 dividing n!.

A193532 G.f.: x = Sum_{n>=1} x^n * ((1+x)^(n+1) - x^(n+1)) / (1+x)^a(n).

Original entry on oeis.org

3, 4, 8, 9, 11, 12, 17, 18, 20, 21, 24, 25, 27, 28, 34, 35, 37, 38, 41, 42, 44, 45, 49, 50, 52, 53, 56, 57, 59, 60, 67, 68, 70, 71, 74, 75, 77, 78, 82, 83, 85, 86, 89, 90, 92, 93, 98, 99, 101, 102, 105, 106, 108, 109, 113, 114, 116, 117, 120, 121, 123, 124, 132, 133, 135, 136, 139, 140, 142, 143, 147, 148, 150, 151, 154, 155, 157, 158, 163, 164, 166, 167, 170
Offset: 1

Views

Author

Paul D. Hanna, Jul 29 2011

Keywords

Examples

			G.f.: x = x*((1+x)^2 - x^2)/(1+x)^3 + x^2*((1+x)^3 - x^3)/(1+x)^4 + x^3*((1+x)^4 - x^4)/(1+x)^8 + x^4*((1+x)^5 - x^5)/(1+x)^9 + x^5*((1+x)^6 - x^6)/(1+x)^11 + x^6*((1+x)^7 - x^7)/(1+x)^12 + x^7*((1+x)^8 - x^8)/(1+x)^17 + x^8*((1+x)^9 - x^9)/(1+x)^18 +...+ x^n*((1+x)^(n+1) - x^(n+1))/(1+x)^a(n) +...

Crossrefs

Cf. A193259, A193260.

Programs

  • PARI
    {a(n)=if(n<1,0,n+ floor(log(n+1+1/100)/log(2)) + valuation((n+1)!,2))}
  • PARI
    {a(n)=if(n<1,0,if(n==1,3,polcoeff(sum(m=1,n+1,x^m*((1+x)^(m+1)-x^(m+1))/(1+x +x^2*O(x^n))^if(m>=n,1,a(m)))+x^(n+1),n+1)))}

Formula

a(n) = n + floor(log_2(n+1)) + A011371(n+1) for n>=1, where A011371(n) = highest power of 2 dividing n!.
Showing 1-3 of 3 results.

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