A032769 -id:A032769 - cp's OEIS frontend

A032768 a(n) = floor( n(n+1)(n+2)(n+3)(n+4) / (n+(n+1)+(n+2)+(n+3)+(n+4)) ).

0, 8, 36, 100, 224, 432, 756, 1232, 1900, 2808, 4004, 5544, 7488, 9900, 12852, 16416, 20672, 25704, 31600, 38456, 46368, 55440, 65780, 77500, 90720, 105560, 122148, 140616, 161100, 183744, 208692, 236096, 266112, 298900, 334628, 373464, 415584

Offset: 0

Patrick De Geest, May 15 1998

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,1,-4,6,-4,1).

Cf. A032769, A032770.

[Floor(n*(n+1)*(n+3)*(n+4)/5): n in [0..36]]; // Bruno Berselli, Jun 20 2012

CoefficientList[Series[4*x*(1+x)*(2-x+2*x^2)/((1-x)^5*(1+x+x^2+x^3+x^4)),{x,0,40}],x] (* Vincenzo Librandi, Jun 20 2012 *)

G.f.: 4*x*(1+x)*(2-x+2*x^2)/((1-x)^5*(1+x+x^2+x^3+x^4)).

a(n) = floor( n(n+1)(n+3)(n+4)/5 ). [Bruno Berselli, Jun 20 2012]

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

Patrick De Geest, May 15 1998

nonn

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

Cf. A032768, A032769.

Select[Times@@(Range[0,4]+#)/Total[Range[0,4]+#]&/@ Range[0,60],IntegerQ] (* Harvey P. Dale, Mar 17 2011 *)

A271324 a(n) = n + floor(n/4) + (n mod 4).

Original entry on oeis.org

0, 2, 4, 6, 5, 7, 9, 11, 10, 12, 14, 16, 15, 17, 19, 21, 20, 22, 24, 26, 25, 27, 29, 31, 30, 32, 34, 36, 35, 37, 39, 41, 40, 42, 44, 46, 45, 47, 49, 51, 50, 52, 54, 56, 55, 57, 59, 61, 60, 62, 64, 66, 65, 67, 69, 71, 70, 72, 74, 76, 75, 77, 79, 81, 80, 82, 84, 86, 85, 87, 89

Offset: 0

Bruno Berselli, Apr 04 2016

Sort the terms in increasing order and add 1 to get sequence A032769.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A032769.

Cf. numbers of the form m + floor(m/k) + (m mod k): A028242 (k=-2), A000004 (k=-1), A005843 (k=1), A007494 (k=2), A063224 (k=3).

[n + Floor(n/4) + (n mod 4): n in [0..80]];

Table[n + Floor[n/4] + Mod[n, 4], {n, 0, 80}]

makelist(n + floor(n/4) + mod(n, 4), n, 0, 80);

vector(80, n, n--; n + floor(n/4) + n%4)

def A271324(n): return n+(n>>2)+(n&3) # Chai Wah Wu, Jan 29 2023

[n + floor(n/4) + n%4 for n in (0..80)]

O.g.f.: x*(2 + 2*x + 2*x^2 - x^3)/((1 - x)^2*(1 + x + x^2 + x^3)).
E.g.f.: ((6 + 5*x)*sinh(x) + (3 + 5*x)*cosh(x) - 3*(sin(x) + cos(x)))/4.
a(n) = 1 + (10*n - 6*(-1)^((n-1)*n/2) - 3*(-1)^n + 1)/8.
a(4*k + r) = 5*k + 2*r, with r = 0, 1, 2 or 3.
a(n + 4*k) = a(n) + 5*k.

cp's OEIS Frontend

A032768 a(n) = floor( n(n+1)(n+2)(n+3)(n+4) / (n+(n+1)+(n+2)+(n+3)+(n+4)) ).

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

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A271324 a(n) = n + floor(n/4) + (n mod 4).

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