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 1-10 of 12 results. Next

A016959 a(n) = (6*n + 4)^3.

Original entry on oeis.org

64, 1000, 4096, 10648, 21952, 39304, 64000, 97336, 140608, 195112, 262144, 343000, 438976, 551368, 681472, 830584, 1000000, 1191016, 1404928, 1643032, 1906624, 2197000, 2515456, 2863288, 3241792
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			a(0) = (6*0 + 4)^3 = 4^3 = 64.

Links

Crossrefs

Cf. A002117, A016911, A016923, A016935, A016947, A016971.

Programs

  • Magma
    [(6*n+4)^3: n in [0..40]]; // Vincenzo Librandi, May 06 2011
  • Mathematica
    CoefficientList[Series[8*(x^3 + 60*x^2 + 93*x + 8)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 27 2013 *)
(6*Range[0,30]+4)^3 (* or *) LinearRecurrence[{4,-6,4,-1},{64,1000,4096,10648},30] (* Harvey P. Dale, Nov 22 2018 *)

Formula

G.f.: 8*(x^3 + 60*x^2 + 93*x + 8)/(1-x)^4. - Vincenzo Librandi, Jan 27 2013
Sum_{n>=0} 1/a(n) = -Pi^3 / (324*sqrt(3)) + 13*zeta(3)/216. - Amiram Eldar, Oct 02 2020
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 02 2020

A016971 a(n) = (6*n + 5)^3.

Original entry on oeis.org

125, 1331, 4913, 12167, 24389, 42875, 68921, 103823, 148877, 205379, 274625, 357911, 456533, 571787, 704969, 857375, 1030301, 1225043, 1442897, 1685159, 1953125, 2248091, 2571353, 2924207, 3307949
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			a(0) = (6*0 + 5)^3 = 5^3 = 125.

Links

Crossrefs

Cf. A002117, A016911, A016923, A016935, A016947, A016959.

Programs

Formula

Sum_{n>=0} 1/a(n) = -Pi^3/(36*sqrt(3)) + 91*zeta(3)/216. - Amiram Eldar, Oct 02 2020
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 02 2020
a(n) = (125+831*x+339*x^2+x^3)/(-1+x)^4. - Wesley Ivan Hurt, Oct 02 2020

A016948 a(n) = (6*n + 3)^4.

Original entry on oeis.org

81, 6561, 50625, 194481, 531441, 1185921, 2313441, 4100625, 6765201, 10556001, 15752961, 22667121, 31640625, 43046721, 57289761, 74805201, 96059601, 121550625, 151807041, 187388721, 228886641, 276922881, 332150625, 395254161, 466948881, 547981281, 639128961
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000583, A016756, A016945, A016946, A016947.

Programs

  • Magma
    [(6*n+3)^4: n in [0..50]]; // Vincenzo Librandi, May 05 2011
  • Mathematica
    a[n_] := (6*n + 3)^4; Array[a, 50, 0] (* Amiram Eldar, Mar 30 2022 *)

Formula

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^4 = A016946(n)^2.
a(n) = 3^4*A016756(n).
Sum_{n>=0} 1/a(n) = Pi^4/7776. (End)

A016949 a(n) = (6*n + 3)^5.

Original entry on oeis.org

243, 59049, 759375, 4084101, 14348907, 39135393, 90224199, 184528125, 345025251, 601692057, 992436543, 1564031349, 2373046875, 3486784401, 4984209207, 6956883693, 9509900499, 12762815625, 16850581551, 21924480357, 28153056843, 35723051649, 44840334375, 55730836701
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016757, A016945, A016946, A016947, A016948.
Subsequence of A000584.

Programs

  • Magma
    [(6*n+3)^5: n in [0..50]]; // Vincenzo Librandi, May 05 2011
  • Mathematica
    a[n_] := (6*n + 3)^5; Array[a, 50, 0] (* Amiram Eldar, Mar 30 2022 *)

Formula

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^5.
a(n) = 3^5*A016757(n).
Sum_{n>=0} 1/a(n) = 31*zeta(5)/7776.
Sum_{n>=0} (-1)^n/a(n) = 5*Pi^5/373248. (End)

A016911 a(n) = (6*n)^3.

Original entry on oeis.org

0, 216, 1728, 5832, 13824, 27000, 46656, 74088, 110592, 157464, 216000, 287496, 373248, 474552, 592704, 729000, 884736, 1061208, 1259712, 1481544, 1728000, 2000376, 2299968, 2628072, 2985984, 3375000, 3796416, 4251528, 4741632, 5268024, 5832000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Volume of a cube with side 6*n. - Wesley Ivan Hurt, Jul 05 2014

Examples

			a(1) = (6*1)^3 = 216.

Links

Crossrefs

Cf. A000578, A002117, A016923, A016935, A016947, A016959, A016971.
Cf. similar sequences listed in A244725.

Programs

  • Magma
    [(6*n)^3: n in [0..40]]; // Vincenzo Librandi, May 03 2011
  • Magma
    I:=[0,216,1728,5832]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 05 2014
  • Maple
    A016911:=n->216*n^3: seq(A016911(n), n=0..40); # Wesley Ivan Hurt, Jul 05 2014
  • Mathematica
    Table[216 n^3, {n, 0, 40}] (* or *) CoefficientList[Series[216 x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 05 2014 *)

Formula

G.f.: 216*x*(1 + 4*x + x^2)/(1 - x)^4. - Vincenzo Librandi, Jul 05 2014
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, Jul 05 2014
a(n) = 216 * A000578(n). - Wesley Ivan Hurt, Jul 05 2014
Sum_{n>=1} 1/a(n) = zeta(3)/216. - Amiram Eldar, Oct 02 2020

A016950 a(n) = (6*n + 3)^6.

Original entry on oeis.org

729, 531441, 11390625, 85766121, 387420489, 1291467969, 3518743761, 8303765625, 17596287801, 34296447249, 62523502209, 107918163081, 177978515625, 282429536481, 433626201009, 646990183449, 941480149401, 1340095640625, 1870414552161, 2565164201769, 3462825991689
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016758, A016945, A016946, A016947, A016948, A016949.
Subsequence of A001014.

Programs

Formula

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^6 = A016946(n)^3 = A016947(n)^2.
a(n) = 3^6*A016758(n).
Sum_{n>=0} 1/a(n) = Pi^6/699840. (End)

A016951 a(n) = (6*n + 3)^7.

Original entry on oeis.org

2187, 4782969, 170859375, 1801088541, 10460353203, 42618442977, 137231006679, 373669453125, 897410677851, 1954897493193, 3938980639167, 7446353252589, 13348388671875, 22876792454961, 37725479487783, 60170087060757, 93206534790699, 140710042265625, 207616015289871
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016759, A016945, A016946, A016947, A016948, A016949, A016950.
Subsequence of A001015.

Programs

  • Magma
    [(6*n+3)^7: n in [0..40]]; // Vincenzo Librandi, May 05 2011
  • Mathematica
    a[n_] := (6*n + 3)^7; Array[a, 50, 0] (* Amiram Eldar, Mar 30 2022 *)

Formula

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^7.
a(n) = 3^7*A016759(n).
Sum_{n>=0} 1/a(n) = 127*zeta(7)/279936.
Sum_{n>=0} (-1)^n/a(n) = 61*Pi^7/403107840. (End)

A016952 a(n) = (6*n + 3)^8.

Original entry on oeis.org

6561, 43046721, 2562890625, 37822859361, 282429536481, 1406408618241, 5352009260481, 16815125390625, 45767944570401, 111429157112001, 248155780267521, 513798374428641, 1001129150390625, 1853020188851841, 3282116715437121, 5595818096650401, 9227446944279201
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016760, A016945, A016946, A016947, A016948, A016949, A016950, A016951.
Subsequence of A001016.

Programs

  • Magma
    [(6*n+3)^8: n in [0..40]]; // Vincenzo Librandi, May 05 2011
  • Mathematica
    a[n_] := (6*n + 3)^8; Array[a, 50, 0] (* Amiram Eldar, Mar 30 2022 *)

Formula

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^8 = A016946(n)^4 = A016948(n)^2.
a(n) = 3^8*A016760(n).
Sum_{n>=0} 1/a(n) = 17*Pi^8/1058158080. (End)

A016953 a(n) = (6*n + 3)^9.

Original entry on oeis.org

19683, 387420489, 38443359375, 794280046581, 7625597484987, 46411484401953, 208728361158759, 756680642578125, 2334165173090451, 6351461955384057, 15633814156853823, 35452087835576229, 75084686279296875, 150094635296999121, 285544154243029527, 520411082988487293
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016761, A016945, A016946, A016947, A016948, A016949, A016950, A016951, A016952.
Subsequence of A001017.

Programs

Formula

a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10). - Harvey P. Dale, Jan 19 2012
From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^9 = A016947(n)^3.
a(n) = 3^9*A016761(n).
Sum_{n>=0} 1/a(n) = 511*zeta(9)/10077696.
Sum_{n>=0} (-1)^n/a(n) = 277*Pi^9/162533081088. (End)

A016954 a(n) = (6n+3)^10.

Original entry on oeis.org

59049, 3486784401, 576650390625, 16679880978201, 205891132094649, 1531578985264449, 8140406085191601, 34050628916015625, 119042423827613001, 362033331456891249, 984930291881790849, 2446194060654759801, 5631351470947265625, 12157665459056928801
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A008454, A016762, A016945, A016946, A016947, A016948, A016949, A016950, A016951, A016952, A016953.
Subsequence of A008454.

Programs

Formula

From Wesley Ivan Hurt, Aug 22 2016: (Start)
G.f.: 59049*(1 + 59038*x + 9116141*x^2 + 178300904*x^3 + 906923282*x^4 + 1527092468*x^5 + 906923282*x^6 + 178300904*x^7 + 9116141*x^8 + 59038*x^9 + x^10)/(1-x)^11.
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11) for n>10.
a(n) = A008454(A016945(n)). (End)
From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016946(n)^5 = A016949(n)^2.
a(n) = 3^10*A016762(n).
Sum_{n>=0} 1/a(n) = 31*Pi^10/171421608960. (End)
Showing 1-10 of 12 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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