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 1-2 of 2 results.

A344418 a(n) = n*a(n-1) + n^(1+n mod 2), a(0) = 0.

Original entry on oeis.org

0, 1, 4, 13, 56, 281, 1692, 11845, 94768, 852913, 8529140, 93820541, 1125846504, 14636004553, 204904063756, 3073560956341, 49176975301472, 836008580125025, 15048154442250468, 285914934402758893, 5718298688055177880, 120084272449158735481, 2641853993881492180604
Offset: 0

Views

Author

Alois P. Heinz, May 17 2021

Keywords

Crossrefs

Programs

  • Maple
    a:= proc(n) a(n):= n*a(n-1) + n^(1+n mod 2) end: a(0):= 0:
    seq(a(n), n=0..23);

Formula

E.g.f.: (x+1)*sinh(x)/(1-x).
a(n) = A344317(n) - n! = A344317(n) - A000142(n).
a(n) = A155521(n-1) + A344419(n) for n > 0.
Lim_{n-> infinity} a(n)/n! = 2*sinh(1) = 2*A073742 = e-1/e = A174548. - Amrit Awasthi, May 20 2021

A344935 a(0)=1; for n > 0, a(n) = n*(a(n-1) + i^(n-1)) if n is odd, n*a(n-1) + i^n otherwise, where i = sqrt(-1).

Original entry on oeis.org

1, 2, 3, 6, 25, 130, 779, 5446, 43569, 392130, 3921299, 43134278, 517611337, 6728947394, 94205263515, 1413078952710, 22609263243361, 384357475137154, 6918434552468771, 131450256496906630, 2629005129938132601, 55209107728700784642, 1214600370031417262123
Offset: 0

Views

Author

Amrit Awasthi, Jun 03 2021

Keywords

Examples

			a(0) = 1;
a(1) = 1*(a(0) + i^(1-1)) =  2;
a(2) = 2*a(1)  + i^2      =  3;
a(3) = 3*(a(2) + i^2)     =  6;
a(4) = 4*a(3)  + i^4      = 25.
		

Crossrefs

Programs

  • Maple
    A344935 := proc(n)
        option remember ;
        if n = 0 then
            1;
        elif type(n,'odd') then
            n*(procname(n-1)+I^(n-1)) ;
        else
            n*procname(n-1)+I^n ;
        end if;
        simplify(%) ;
    end proc:
    seq(A344935(n),n=0..40) ; # R. J. Mathar, Aug 19 2022
  • Mathematica
    a[0] = 1; a[n_] := a[n] = If[OddQ[n], n*(a[n - 1] + I^(n - 1)), n*a[n - 1] + I^n]; Array[a, 30, 0] (* Amiram Eldar, Jun 03 2021 *)
  • PARI
    a(n) = if (n==0, 1, if (n%2, n*(a(n-1) + I^(n-1)), n*a(n-1) + I^n)); \\ Michel Marcus, Jun 05 2021

Formula

E.g.f.: (1+x)*cos(x)/(1-x).
Lim_{n->infinity} a(n)/n! = 2*cos(1) = 2*A049470.
D-finite with recurrence a(n) -n*a(n-1) +2*a(n-2) +2*(-n+2)*a(n-3) +a(n-4) +(-n+4)*a(n-5)=0. - R. J. Mathar, Aug 19 2022
Showing 1-2 of 2 results.