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.

A032769 Numbers that are congruent to {0, 1, 2, 4} mod 5.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 72, 74, 75, 76, 77, 79, 80, 81, 82, 84, 85
Offset: 1

Views

Author

Patrick De Geest, May 15 1998

Keywords

Comments

Also, numbers m such that m*(m+1)*(m+2)*(m+3)*(m+4)/(m+(m+1)+(m+2)+(m+3)+(m+4)) is an integer.

Links

Crossrefs

Cf. A001622, A032768, A032770, A047209, A047215.

Programs

Formula

a(n) = (1/8)*(10*n-11+(-1)^n+2*(-1)^floor(n/2)). - Ralf Stephan, Jun 09 2005
a(n) = floor((5*n-4)/4). - Gary Detlefs, Mar 06 2010
G.f.: x^2*(1+x+2*x^2+x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 30 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (10*n-11+i^(2*n)+(1+i)*I^(-n)+(1-i)*i^n)/8 where i=sqrt(-1).
a(2k) = A047209(k), a(2k-1) = A047215(k). (End)
E.g.f.: (4 + sin(x) + cos(x) + (5*x - 6)*sinh(x) + 5*(x - 1)*cosh(x))/4. - Ilya Gutkovskiy, May 31 2016
Sum_{n>=2} (-1)^n/a(n) = log(5)/4 + 3*sqrt(5)*log(phi)/10 - sqrt(1-2/sqrt(5))*Pi/10, where phi is the golden ratio (A001622). - Amiram Eldar, Dec 10 2021

Extensions

Better description from Michael Somos, Jun 08 2000

A032774 a(n) = floor( n*(n+1)*(n+2)*...*(n+6) / (n+(n+1)+(n+2)+...+(n+6)) ).

Original entry on oeis.org

0, 180, 1152, 4320, 12342, 29700, 63360, 123552, 224640, 386100, 633600, 1000182, 1527552, 2267460, 3283200, 4651200, 6462720, 8825652, 11866422, 15732000, 20592000, 26640900, 34100352, 43221600, 54288000, 67617642, 83566080, 102529152, 124945920, 151301700, 182131200
Offset: 0

Views

Author

Patrick De Geest, May 15 1998

Keywords

Comments

In general, such sequences a(n) = floor((Product_{m=0..k} n+i) / (Sum_{m=0..k} n+i)) have rational generating functions. - Georg Fischer, Feb 23 2021

Links

Crossrefs

Cf. A032775, A032776.
Cf. A004526 (k=2), A032765 (k=3), A032768 (k=4), A032771 (k=5), A032774 (k=6), A032777 (k=7), A032780 (k=8), A032790 (k=9).

Programs

  • Maple
    seq(coeff(series( -(6*x^10-36*x^9 + 90*x^8 - 120*x^7 - 90*x^6 - 108*x^5 - 102*x^4 - 108*x^3 - 72*x^2 - 180*x) / (-x^13+6*x^12 - 15*x^11+20*x^10 - 15*x^9+6*x^8 - x^7+x^6 - 6*x^5+15*x^4 - 20*x^3+15*x^2 - 6*x + 1) , x, n+1), x, n), n = 0..40); # Georg Fischer, Feb 23 2021
  • Mathematica
    Table[Floor[(Times @@ Range[n, n + 6])/(7 n + 21)], {n, 0, 30}] (* Harvey P. Dale, May 16 2020 *)

Extensions

More terms from Georg Fischer, Feb 23 2021

A032770 Integer quotients of n(n + 1)(n + 2)(n + 3)(n + 4) / (n+(n+1)+(n+2)+(n+3)+(n+4)).

Original entry on oeis.org

0, 8, 36, 224, 432, 756, 1232, 2808, 4004, 5544, 7488, 12852, 16416, 20672, 25704, 38456, 46368, 55440, 65780, 90720, 105560, 122148, 140616, 183744, 208692, 236096, 266112, 334628, 373464, 415584, 461168, 563472, 620576, 681912, 747684
Offset: 0

Views

Author

Patrick De Geest, May 15 1998

Keywords

Comments

Sequence contains all numbers of form i(5i+c)(5i+d)(5i+e), with from {<1,3,4>, <-1,-3,-4>, <-1,2,3>, <1,-2,-3>}. - Ralf Stephan, May 16 2005

Crossrefs

Cf. A032768, A032769.

Programs

  • Mathematica
    Select[Times@@(Range[0,4]+#)/Total[Range[0,4]+#]&/@ Range[0,60],IntegerQ] (* Harvey P. Dale, Mar 17 2011 *)
Showing 1-3 of 3 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.