A175113 -id:A175113 - cp's OEIS frontend

A175114 First differences of A175113.

1, 364, 7448, 51012, 206896, 620060, 1527624, 3281908, 6373472, 11454156, 19360120, 31134884, 48052368, 71639932, 103701416, 146340180, 201982144, 273398828, 363730392, 476508676, 615680240, 785629404, 991201288, 1237724852

Offset: 0

R. J. Mathar, Feb 13 2010

Convolution of the finite sequence 1,358,5279,11764,5279,358,1 with A000389. Number of points in the standard root system of the D_6 lattice having L_infinity norm n.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A110907, A117216, A175112.

I:=[1,364,7448,51012,206896,620060,1527624]; [n le 7 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..40]]; // Vincenzo Librandi, Dec 20 2012

CoefficientList[Series[(358 x + 5279 x^2 + 11764 x^3 + 5279 x^4 + 358 x^5 + 1+x^6)/(x - 1)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 20 2012 *)

a(n)= 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6), n>6.
a(n) = ((2*n+1)^6-(2*n-1)^6)/2 = 4*n*(12*n^2+1)*(4*n^2+3), n>0. - Bruno Berselli, Dec 27 2010
G.f.: (358*x+5279*x^2+11764*x^3+5279*x^4+358*x^5+1+x^6)/(x-1)^6. - R. J. Mathar, Jan 03 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

A359844 a(n) = ((2*n+1)^8 + 1)/2.

Original entry on oeis.org

1, 3281, 195313, 2882401, 21523361, 107179441, 407865361, 1281445313, 3487878721, 8491781521, 18911429681, 39155492641, 76293945313, 141214768241, 250123206481, 426445518721, 703204309121, 1125937695313, 1756239726961, 2676004630241, 3992462614561, 5844100138801

Offset: 0

Jianing Song, Jan 15 2023

Jianing Song, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. {((2*n+1)^k + 1)/2}: A000012 (k=0), A001477 (k=1), A219086 (k=2), A050492 (k=3), A175110 (k=4), A175113 (k=6), this sequence (k=8).

((2*Range[0, 25] + 1)^8 + 1)/2 (* Paolo Xausa, Jan 23 2025 *)

PARI
```
a(n) = ((2*n+1)^8 + 1)/2
```

def A359844(n): return ((n<<1)+1)**8+1>>1 # Chai Wah Wu, Jan 15 2023

a(n) = A359499(n)/A359498(n) = 16 * A359498(n) + 1.

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A175114 First differences of A175113.

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

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A359844 a(n) = ((2*n+1)^8 + 1)/2.

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