A359498 -id:A359498 - cp's OEIS frontend

A359499 a(n) = ((2*n+1)^16 - 1)/64.

0, 672605, 2384185791, 519264540150, 28953440450810, 717964529118315, 10397134518487185, 102631380558013916, 760331123057294820, 4506897086994080745, 22352635785031020755, 95822037745015603890, 363797880709171295166, 1246350673076132966615, 3910101151255427324805

Offset: 0

Jianing Song, Jan 03 2023

a(n) and A000217(n) have the same parity.

Jianing Song, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

Cf. {((2*n+1)^2^k - 1)/2^(k+2)}: A000217 (k=1), A219086 (k=2), A359498 (k=3), this sequence (k=4).

Table[((2*n + 1)^16 - 1)/64, {n, 0, 15}] (* Paolo Xausa, Oct 04 2024 *)

PARI
```
a(n) = ((2*n+1)^16 - 1)/64
```

def A359499(n): return ((n<<1)+1)**16-1>>6 # Chai Wah Wu, Jan 15 2023

a(n) = A000217(n) * A219086(n) * A175110(n) * A359844(n) = A219086(n) * A175110(n) * A359844(n) = A359498(n) * A359499(n).

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.

cp's OEIS Frontend

A359499 a(n) = ((2*n+1)^16 - 1)/64.

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