cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Original entry on oeis.org

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

Views

Author

Adi Dani, May 29 2011

Keywords

Comments

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.

Examples

			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).

Links

Crossrefs

Cf. A175110, A191903.

Programs

  • Magma
    [1/2*((n+1)^4+((1+(-1)^n)*1/2)^4): n in [0..40]]; // Vincenzo Librandi, Jun 16 2011
  • Mathematica
    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 *)
  • PARI
    a(n) = ceil(n^4/2); \\ Michel Marcus, Dec 14 2013

Formula

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

Extensions

Better name from Enrique Pérez Herrero, Dec 14 2013

A117216 Number of points in the standard root system version of the D_4 lattice having L_infinity norm n.

Original entry on oeis.org

1, 40, 272, 888, 2080, 4040, 6960, 11032, 16448, 23400, 32080, 42680, 55392, 70408, 87920, 108120, 131200, 157352, 186768, 219640, 256160, 296520, 340912, 389528, 442560, 500200, 562640, 630072, 702688, 780680, 864240, 953560, 1048832, 1150248, 1258000, 1372280
Offset: 0

Views

Author

N. J. A. Sloane, Apr 15 2008

Keywords

Comments

This lattice consists of all points (w,x,y,z) where w,x,y,z are integers with an even sum.
The L_infinity norm of a vector is the largest component in absolute value.
Equals binomial transform of [1, 39, 193, 191, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Feb 05 2010

References

  • J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, Chap. 4.

Links

Crossrefs

Cf. A035878, A105374, A110907, A144965, A175110.

Programs

  • Magma
    I:=[1, 40, 272, 888, 2080]; [n le 5 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 27 2012
  • Mathematica
    CoefficientList[Series[(1+36*x+118*x^2+36*x^3+x^4)/(1-x)^4,{x,0,40}],x]  (* Vincenzo Librandi, Jun 27 2012 *)

Formula

From R. J. Mathar, Feb 03 2010, Feb 13 2010: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>4;
a(n) = 8*n*(1+4*n^2) = 2*A144965(n), n>0 (bisection of A035878 and A105374). (End)
G.f.: (1 + 36*x + 118*x^2 + 36*x^3 + x^4)/(1-x)^4. - Colin Barker, May 24 2012
E.g.f.: 1 + 8*x*(1 + 2*x)*(5 + 2*x)*exp(x). - Elmo R. Oliveira, Aug 18 2025

Extensions

a(2) corrected and sequence extended by R. J. Mathar, Feb 03 2010, Feb 13 2010

A359498 a(n) = ((2*n+1)^8 - 1)/32.

Original entry on oeis.org

0, 205, 12207, 180150, 1345210, 6698715, 25491585, 80090332, 217992420, 530736345, 1181964355, 2447218290, 4768371582, 8825923015, 15632700405, 26652844920, 43950269320, 70371105957, 109764982935, 167250289390, 249528913410, 365256258675, 525472668457, 744102708180
Offset: 0

Views

Author

Jianing Song, Jan 03 2023

Keywords

Comments

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

Links

Crossrefs

Cf. {((2*n+1)^2^k - 1)/2^(k+2)}: A000217 (k=1), A219086 (k=2), this sequence (k=3), A359499 (k=4).

Programs

  • Mathematica
    ((2*Range[0, 25] + 1)^8 - 1)/32 (* Paolo Xausa, Jan 23 2025 *)
  • PARI
    a(n) = ((2*n+1)^8 - 1)/32
  • Python
    def A359498(n): return ((n<<1)+1)**8-1>>5 # Chai Wah Wu, Jan 15 2023

Formula

a(n) = A000217(n) * A219086(n) * A175110(n) = A219086(n) * A175110(n).

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

Original entry on oeis.org

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

Views

Author

Jianing Song, Jan 03 2023

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    Table[((2*n + 1)^16 - 1)/64, {n, 0, 15}] (* Paolo Xausa, Oct 04 2024 *)
  • PARI
    a(n) = ((2*n+1)^16 - 1)/64
  • Python
    def A359499(n): return ((n<<1)+1)**16-1>>6 # Chai Wah Wu, Jan 15 2023

Formula

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

Views

Author

Jianing Song, Jan 15 2023

Keywords

Links

Crossrefs

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).

Programs

  • Mathematica
    ((2*Range[0, 25] + 1)^8 + 1)/2 (* Paolo Xausa, Jan 23 2025 *)
  • PARI
    a(n) = ((2*n+1)^8 + 1)/2
  • Python
    def A359844(n): return ((n<<1)+1)**8+1>>1 # Chai Wah Wu, Jan 15 2023

Formula

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

A180270 Integers of the form (k^12 - k^8 - k^4 + 1)/512.

Original entry on oeis.org

0, 1025, 476073, 27022500, 551536100, 6129324225, 45502479225, 253405810448, 1137920432400, 4322847530025, 14366776735025, 42801847892100, 116415023802948, 293153032943225, 691043521403025, 1538402208782400
Offset: 1

Views

Author

Michel Lagneau, Aug 23 2010

Keywords

Examples

			a(2) = 1025 is in the sequence because (3^12 - 3^8 - 3^4 + 1)/512 = 524800/512 = 1025.

Links

Programs

  • Maple
    for n from 1 by 2 to 60 do: x:= (n^12-n^8 -n^4+1)/512: printf(`%d, `, x):od: # incomplete program which also prints rationals, R. J. Mathar
  • Mathematica
    Select[Table[(k^12-k^8-k^4+1)/512,{k,40}],IntegerQ]  (* Harvey P. Dale, Jan 23 2011 *)

Formula

Integers of the form (k^4+1)*( (k-1)*(k+1)*(k^2+1) )^2/512.
a(n) = ((2*n-1)^4+1)*((n-1)*n*(n^2+(n-1)^2))^2/8 (for k=2n-1).
a(n) = A175110(n-1)*(A001844(n-1)*A000217(n-1))^2. - Bruno Berselli, Sep 21 2010

Extensions

Comment converted to formula by R. J. Mathar, Aug 25 2010
Example corrected and general term of the sequence rewritten by Bruno Berselli, Sep 22 2010
Showing 1-6 of 6 results.

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