A191903 -id:A191903 - cp's OEIS frontend

A171714 a(n) = ceiling((n+1)^4/2).

1, 8, 41, 128, 313, 648, 1201, 2048, 3281, 5000, 7321, 10368, 14281, 19208, 25313, 32768, 41761, 52488, 65161, 80000, 97241, 117128, 139921, 165888, 195313, 228488, 265721, 307328, 353641, 405000, 461761, 524288, 592961, 668168, 750313, 839808

Offset: 0

Adi Dani, May 29 2011

Number of compositions of even natural numbers into 4 parts <=n.

Number of ways of placings of an even number of indistinguishable objects into 4 distinguishable boxes with the condition that in each box there can be at most n objects.

			a(1)=8: there are 8 compositions of even natural numbers into 4 parts <=1
(0,0,0,0);
(0,0,1,1), (0,1,0,1), (0,1,1,0), (1,0,0,1), (1,0,1,0), (1,1,0,0);
(1,1,1,1).
a(2)=41: there are 41 compositions of even natural numbers into 4 parts <=2
for 0: (0,0,0,0);
for 2: (0,0,0,2), (0,0,2,0), (0,2,0,0), (2,0,0,0), (0,0,1,1), (0,1,0,1), (0,1,1,0), (1,0,0,1), (1,0,1,0), (1,1,0,0);
for 4: (0,0,2,2), (0,2,0,2), (0,2,2,0), (2,0,0,2), (2,0,2,0), (2,2,0,0), (0,1,1,2), (0,1,2,1), (0,2,1,1), (1,0,1,2), (1,0,2,1), (1,1,0,2), (1,1,2,0), (1,2,0,1), (1,2,1,0), (2,0,1,1), (2,1,0,1), (2,1,1,0), (1,1,1,1);
for 6: (0,2,2,2), (2,0,2,2), (2,2,0,2), (2,2,2,0), (1,1,2,2), (1,2,1,2), (1,2,2,1), (2,1,1,2), (2,1,2,1), (2,2,1,1);
for 8: (2,2,2,2).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A175110, A191903.

[1/2*((n+1)^4+((1+(-1)^n)*1/2)^4): n in [0..40]]; // Vincenzo Librandi, Jun 16 2011

Table[1/2((n + 1)^4 + ((1 + (-1)^n)*1/2)^4), {n, 0, 25}]
Ceiling[Range[40]^4/2] (* Bruno Berselli, Jan 18 2017 *)

a(n) = ceil(n^4/2); \\ Michel Marcus, Dec 14 2013

a(n) = 1/2*((n + 1)^4 + ((1 + (-1)^n)*1/2)^4).
a(n) = +4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +1*a(n-6).
G.f.: (1 + 4*x + 14*x^2 + 4*x^3 + x^4)/((1 + x)*(1 - x)^5).
a(n) = (n+1)^4 - floor((n+1)^4/2). - Bruno Berselli, Jan 18 2017

Better name from Enrique Pérez Herrero, Dec 14 2013

A192396 Square array T(n, k) = floor(((k+1)^n - (1+(-1)^k)/2)/2) read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 4, 4, 2, 0, 0, 8, 13, 8, 2, 0, 0, 16, 40, 32, 12, 3, 0, 0, 32, 121, 128, 62, 18, 3, 0, 0, 64, 364, 512, 312, 108, 24, 4, 0, 0, 128, 1093, 2048, 1562, 648, 171, 32, 4, 0, 0, 256, 3280, 8192, 7812, 3888, 1200, 256, 40, 5, 0

Offset: 0

Adi Dani, Jun 29 2011

T(n,k) is the number of compositions of odd natural numbers into n parts <=k.

			T(2,4)=12: there are 12 compositions of odd natural numbers into 2 parts <=4
  1: (0,1), (1,0);
  3: (1,2), (2,1), (0,3), (3,0);
  5: (1,4), (4,1), (2,3), (3,2);
  7: (3,4), (4,3).
The table starts
    0,  0,   0,   0,    0,    0, ... A000004;
    0,  1,   1,   2,    2,    3, ... A004526;
    0,  2,   4,   8,   12,   18, ... A007590;
    0,  4,  13,  32,   62,  108, ... A036487;
    0,  8,  40, 128,  312,  648, ... A191903;
    0, 16, 121, 512, 1562, 3888, ... A191902;
    .        .      .       .    ...
with columns: A000004, A000079, A003462, A004171, A128531, A081341, ... .
Antidiagonal triangle begins:
  0;
  0,  0;
  0,  1,   0;
  0,  2,   1,   0;
  0,  4,   4,   2,   0;
  0,  8,  13,   8,   2,   0;
  0, 16,  40,  32,  12,   3,  0;
  0, 32, 121, 128,  62,  18,  3,  0;
  0, 64, 364, 512, 312, 108, 24,  4,  0;

G. C. Greubel, Antidiagonals n = 0..50, flattened
Adi Dani, Restricted compositions of natural numbers

Rows: A000004, A004526, A007590, A036487, A191902, A191902.

Columns: A000004, A000079, A003462, A004171, A081341, A128531.

A192396:= func< n,k | Floor(((k+1)^n - (1+(-1)^k)/2)/2) >;
[A192396(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 11 2023

A192396 := proc(n,k) (k+1)^n-(1+(-1)^k)/2 ; floor(%/2) ; end proc:
seq(seq( A192396(d-k,k),k=0..d),d=0..10) ; # R. J. Mathar, Jun 30 2011

T[n_, k_]:= Floor[((k+1)^n - (1+(-1)^k)/2)/2];
Table[T[n-k,k], {n,0,12}, {k,0,n}]//Flatten

def A192396(n,k): return ((k+1)^n - ((k+1)%2))//2
flatten([[A192396(n-k,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 11 2023

cp's OEIS Frontend

A171714 a(n) = ceiling((n+1)^4/2).

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