A036486 -id:A036486 - cp's OEIS frontend

A191484 Number of compositions of even natural numbers into 5 parts <= n.

1, 16, 122, 512, 1563, 3888, 8404, 16384, 29525, 50000, 80526, 124416, 185647, 268912, 379688, 524288, 709929, 944784, 1238050, 1600000, 2042051, 2576816, 3218172, 3981312, 4882813, 5940688

Offset: 0

Adi Dani, Jun 03 2011

nonn

Number of ways of placing an even number of indistinguishable objects in 5 distinguishable boxes with the condition that in each box can be at most n objects.

			a(1)=16 as there are 16 compositions of even natural numbers into 5 parts <= 1:
(0,0,0,0,0);
(0,0,0,1,1), (0,0,1,0,1), (0,0,1,1,0), (0,1,1,0,0), (0,1,0,1,0), (0,1,0,0,1), (1,1,0,0,0), (1,0,1,0,0), (1,0,0,1,0), (1,0,0,0,1);
(0,1,1,1,1), (1,0,1,1,1), (1,1,0,1,1), (1,1,1,0,1), (1,1,1,1,0).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Adi Dani, Restricted compositions of natural numbers
Index entries for linear recurrences with constant coefficients, signature (5,-9,5,5,-9,5,-1).

Cf. A036486 (3 parts), A171714 (4 parts), A191489 (6 parts), A191494 (7 parts), A191495 (8 parts).

[((n + 1)^5 + (1 + (-1)^n)/2 )/2: n in [0..40]]; // Vincenzo Librandi, Jun 16 2011

Table[1/2*((n + 1)^5 + (1 + (-1)^n)*1/2), {n, 0, 25}]
LinearRecurrence[{5,-9,5,5,-9,5,-1},{1,16,122,512,1563,3888,8404},50] (* Harvey P. Dale, Nov 09 2011 *)

a(n) = ((n + 1)^5 + (1 + (-1)^n)/2 )/2.
a(n) = 5*a(n-1) - 9*a(n-2) + 5*a(n-3) + 5*a(n-4) - 9*a(n-5) + 5*a(n-6) - a(n-7).
G.f.: (16*x^4 + 41*x^3 + 51*x^2 + 11*x + 1)/((1-x)^6*(1+x)).

A191494 Number of compositions of even natural numbers in 7 parts <= n.

Original entry on oeis.org

1, 64, 1094, 8192, 39063, 139968, 411772, 1048576, 2391485, 5000000, 9743586, 17915904, 31374259, 52706752, 85429688, 134217728, 205169337, 306110016, 446935870, 640000000, 900544271, 1247178944, 1702412724, 2293235712, 3051757813, 4015905088

Offset: 0

Adi Dani, Jun 03 2011

Number of ways of placing an even number of indistinguishable objects in 7 distinguishable boxes with the condition that in each box can be at most n objects.

			a(1)=64 and compositions of even natural numbers into 7 parts no greater than 1 are
:(0,0,0,0,0,0,0) --> 7!/7!0! =  1
:(0,0,0,0,0,1,1) --> 7!/5!2! = 21
:(0,0,0,1,1,1,1) --> 7!/3!4! = 35
:(0,1,1,1,1,1,1) --> 7!/1!6! =  7

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Adi Dani, Restricted compositions of natural numbers
Index entries for linear recurrences with constant coefficients, signature (7,-20,28,-14,-14,28,-20,7,-1).

Cf. A036486 (3 parts), A171714 (4 parts), A191484 (5 parts), A191489 (6 parts), A191495 (8 parts).

[((n + 1)^7 + (1+(-1)^n)/2 )/2: n in [0..40]]; // Vincenzo Librandi, Jun 16 2011

Table[1/2*((n + 1)^7 + (1 + (-1)^n)*1/2), {n, 0, 25}]

a(n) = ((n + 1)^7 + (1 + (-1)^n)/2 )/2.

G.f.: ( 1 + 57*x + 666*x^2 + 1786*x^3 + 1821*x^4 + 645*x^5 + 64*x^6 ) / ( (1+x)*(x-1)^8 ). - R. J. Mathar, Jun 08 2011

A191489 Number of compositions of even natural numbers into 6 parts <= n.

Original entry on oeis.org

1, 32, 365, 2048, 7813, 23328, 58825, 131072, 265721, 500000, 885781, 1492992, 2413405, 3764768, 5695313, 8388608, 12068785, 17006112, 23522941, 32000000, 42883061, 56689952, 74017945, 95551488, 122070313, 154457888

Offset: 0

Adi Dani, Jun 03 2011

Number of ways of placing of an even number of indistinguishable objects in 6 distinguishable boxes with condition that in each box can be at most n objects.

			a(1)=32 compositions of even natural numbers in 6 parts <= 1 are
:(0,0,0,0,0,0)--> 6!/(6!0!) =  1
:(0,0,0,0,1,1)--> 6!/(4!2!) = 15
:(0,0,1,1,1,1)--> 6!/(2!4!) = 15
:(1,1,1,1,1,1)--> 6!/(0!6!) =  1
a(2)=365 compositions of even natural numbers in 6 parts <= 2 are
:(0,0,0,0,0,0)--> 6!/(6!0!0!) =  1
:(0,0,0,0,1,1)--> 6!/(4!2!0!) = 15
:(0,0,0,0,0,2)--> 6!/(5!0!1!) =  6
:(0,0,1,1,1,1)--> 6!/(2!4!0!) = 15
:(0,0,0,1,1,2)--> 6!/(3!2!1!) = 60
:(0,0,0,0,2,2)--> 6!/(4!0!2!) = 15
:(0,1,1,1,1,2)--> 6!/(1!4!1!) = 30
:(0,0,0,2,2,2)--> 6!/(3!0!3!) = 20
:(0,0,1,1,2,2)--> 6!/(2!2!2!) = 90
:(1,1,1,1,1,1)--> 6!/(0!6!0!) =  1
:(0,1,1,2,2,2)--> 6!/(1!2!3!) = 60
:(0,0,2,2,2,2)--> 6!/(2!0!4!) = 15
:(1,1,1,1,2,2)--> 6!/(0!4!2!) = 15
:(0,2,2,2,2,2)--> 6!/(1!0!5!) =  6
:(1,1,2,2,2,2)--> 6!/(0!2!4!) = 15
:(2,2,2,2,2,2)--> 6!/(0!0!6!) =  1

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Adi Dani, Restricted compositions of natural numbers
Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).

Cf. A036486 (3 parts), A171714 (4 parts), A191484 (5 parts), A191494 (7 parts), A191495 (8 parts).

[((n + 1)^6 + (1+(-1)^n)/2 )/2: n in [0..40]]; // Vincenzo Librandi, Jun 16 2011

Table[1/2*((n + 1)^6 + (1 + (-1)^n)*1/2), {n, 0, 25}]

a(n) = ((n + 1)^6 + (1+(-1)^n)/2 )/2.
G.f.: (x^2 + 10*x + 1)*(x^4 + 16*x^3 + 26*x^2 + 16*x + 1) / ( (1+x)*(1-x)^7 ). - R. J. Mathar, Jun 06 2011
a(2n) = A175113(n). - R. J. Mathar, Jun 07 2011

A191495 Number of compositions of even natural numbers into 8 parts <= n.

Original entry on oeis.org

1, 128, 3281, 32768, 195313, 839808, 2882401, 8388608, 21523361, 50000000, 107179441, 214990848, 407865361, 737894528, 1281445313, 2147483648, 3487878721, 5509980288, 8491781521, 12800000000, 18911429681, 27437936768, 39155492641, 55037657088, 76293945313, 104413532288

Offset: 0

Adi Dani, Jun 03 2011

Number of ways of placing an even number of indistinguishable objects in 8 distinguishable boxes with the condition that in each box can be at most n objects.

			a(1)=128 compositions of even natural numbers into 8 parts no greater than 1 are
  (0,0,0,0,0,0,0,0) --> 8!/(8!0!) =  1
  (0,0,0,0,0,0,1,1) --> 8!/(6!2!) = 28
  (0,0,0,0,1,1,1,1) --> 8!/(4!4!) = 70
  (0,0,1,1,1,1,1,1) --> 8!/(2!6!) = 28
  (1,1,1,1,1,1,1,1) --> 8!/(0!8!) =  1

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Adi Dani, Restricted compositions of natural numbers
Index entries for linear recurrences with constant coefficients, signature (8,-27,48,-42,0,42,-48,27,-8,1).

Cf. A036486 (3 parts), A171714 (4 parts), A191484 (5 parts), A191489 (6 parts), A191494 (7 parts).

[ 1/2*((n + 1)^8 + (1 + (-1)^n)*1/2): n in [0..35]]; // Vincenzo Librandi, Jun 06 2011

Table[1/2*((n + 1)^8 + (1 + (-1)^n)*1/2), {n, 0, 25}]
LinearRecurrence[{8,-27,48,-42,0,42,-48,27,-8,1},{1,128,3281,32768,195313,839808,2882401,8388608,21523361,50000000},30] (* Harvey P. Dale, Mar 13 2013 *)

a(n)=((n+1)^8+1)>>1 \\ Charles R Greathouse IV, Jun 06 2011

a(n) = ( (n+1)^8 + (1 + (-1)^n)/2 )/2.
G.f.: ( -1 - 120*x - 2284*x^2 - 9928*x^3 - 15654*x^4 - 9928*x^5 - 2284*x^6 - 120*x^7 - x^8 ) / ( (1+x)*(x-1)^9 ). - R. J. Mathar, Jun 06 2011
a(n) = 8*a(n-1) - 27*a(n-2) + 48*a(n-3) - 42*a(n-4) + 42*a(n-6) - 48*a(n-7) + 27*a(n-8) - 8*a(n-9) + a(n-10); a(0)=1, a(1)=128, a(2)=3281, a(3)=32768, a(4)=195313, a(5)=839808, a(6)=2882401, a(7)=8388608, a(8)=21523361, a(9)=50000000. - Harvey P. Dale, Mar 13 2013

A245996 Number of length 1+2 0..n arrays with no pair in any consecutive three terms totaling exactly n.

Original entry on oeis.org

2, 8, 28, 64, 126, 216, 344, 512, 730, 1000, 1332, 1728, 2198, 2744, 3376, 4096, 4914, 5832, 6860, 8000, 9262, 10648, 12168, 13824, 15626, 17576, 19684, 21952, 24390, 27000, 29792, 32768, 35938, 39304, 42876, 46656, 50654, 54872, 59320, 64000, 68922

Offset: 1

R. H. Hardin, Aug 09 2014

nonn

From Pontus von Brömssen, Jan 10 2022: (Start)
Proof of the empirical observations in the Formula section:
For k = 1, 2, 3, let N_k be the number of triples (x, y, z) with x, y, and z in 0..n, that satisfy x+y = n (if k=1), x+y = y+z = n (if k=2), or x+y = y+z = z+x = n (if k=3).
By inclusion-exclusion (and symmetry between x, y, and z), a(n) = (n+1)^3 - 3*N_1 + 3*N_2 - N_3. The unique solution to x+y = y+z = z+x = n is x = y = z = n/2, so N_3 = 1 if n is even, otherwise N_3 = 0. We write this as N_3 = [n even]. It is easily seen that N_1 = (n+1)^2 (x and z can be chosen freely and y = n-x) and that N_2 = n+1 (y can be chosen freely and x = z = n-y), so a(n) = (n+1)^3 - 3*(n+1)^2 + 3*(n+1) - [n even] = n^3 + [n odd] = 2*ceiling(n^3/2) = 2*A036486(n).
The recurrence and the generating function follow from this. (End)

			Some solutions for n=10:
  6   9   5   8   0   5   8   6   9   8   5   0   4   8   5   2
  3   8   3   0   0   7   9   5   0   4   7   5   2   4   7   6
  6   9   6   9   5   9   7   3   7   4   1   7  10   0   2   6

R. H. Hardin, Table of n, a(n) for n = 1..210
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Row 1 of A245995.

Cf. A036486.

Empirical: a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
From R. J. Mathar, Aug 10 2014: (Start)
Empirical: a(n) = 2*A036486(n).
G.f.: 2*x*(1+x+4*x^2) / ( (1+x)*(x-1)^4 ). (End)

A191687 Table T(n,k) = ceiling((1/2)*((k+1)^n+(1+(-1)^k)/2)) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 14, 8, 3, 1, 1, 16, 41, 32, 13, 3, 1, 1, 32, 122, 128, 63, 18, 4, 1, 1, 64, 365, 512, 313, 108, 25, 4, 1, 1, 128, 1094, 2048, 1563, 648, 172, 32, 5, 1

Offset: 1

Adi Dani, Jun 11 2011

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

			Top left corner:
  1, 1, 1,  1,  1,...
  1, 1, 2,  2,  3,...
  1, 2, 5,  8, 13,...
  1, 4,14, 32, 63,...
  1, 8,41,128,313,...
T(2,4)=13: there are 13 compositions of even natural numbers into 2 parts <=4
0: (0,0);
2: (0,2), (2,0), (1,1);
4: (0,4), (4,0), (1,3), (3,1), (2,2);
6: (2,4), (4,2), (3,3);
8: (4,4).

Adi Dani, Restricted compositions of natural numbers

Rows sums gives: A000012, A004526, A000982, A036486, A171714, A191484, A191489, A191494, A191495, A191496

Columns sums gives: A000012, A000079, A007051, A004171, A034478, A081341, A034494, A092811, A083884, A093143

Table[Table[Ceiling[1/2*((k+1)^n+(1+(-1)^k)/2)],{n,0,9},{k,0,9}]]

cp's OEIS Frontend

A191484 Number of compositions of even natural numbers into 5 parts <= n.

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A191494 Number of compositions of even natural numbers in 7 parts <= n.

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A191489 Number of compositions of even natural numbers into 6 parts <= n.

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A191495 Number of compositions of even natural numbers into 8 parts <= n.

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A245996 Number of length 1+2 0..n arrays with no pair in any consecutive three terms totaling exactly n.

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A191687 Table T(n,k) = ceiling((1/2)*((k+1)^n+(1+(-1)^k)/2)) read by antidiagonals.

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