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 1-4 of 4 results.

A268684 a(n) = n*(n + 1)*(4*n - 1)/3.

Original entry on oeis.org

0, 2, 14, 44, 100, 190, 322, 504, 744, 1050, 1430, 1892, 2444, 3094, 3850, 4720, 5712, 6834, 8094, 9500, 11060, 12782, 14674, 16744, 19000, 21450, 24102, 26964, 30044, 33350, 36890, 40672, 44704, 48994, 53550, 58380, 63492, 68894, 74594, 80600, 86920
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 11 2016

Keywords

Comments

Partial sums of A002939.
a(n) is the maximum value obtainable by partitioning the set {x in the natural numbers | 1 <= x <= 2n} into pairs, taking the products of all such pairs, and taking the sum of all such products. - Thomas Anton, Oct 20 2020

Examples

			a(0) = 0;
a(1) = 0 + 1*2 = 2;
a(2) = 0 + 1*2 + 3*4 = 14;
a(3) = 0 + 1*2 + 3*4 + 5*6 = 44;
a(4) = 0 + 1*2 + 3*4 + 5*6 + 7*8 = 100;
a(5) = 0 + 1*2 + 3*4 + 5*6 + 7*8 + 9*10 = 190, etc.

Links

Crossrefs

Cf. A001477, A002939, A005408, A005843, A135036.

Programs

  • Magma
    [n*(n + 1)*(4*n - 1)/3: n in [0..40]]; // Vincenzo Librandi, Feb 11 2016
  • Mathematica
    Table[n (n + 1) ((4 n - 1)/3), {n, 0, 40}] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {0, 2, 14, 44}, 40]
CoefficientList[Series[2 x (3 x + 1) / (x - 1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 11 2016 *)
  • PARI
    a(n)=n*(n+1)*(4*n-1)/3 \\ Charles R Greathouse IV, Jul 26 2016

Formula

G.f.: 2*x*(3*x + 1)/(x - 1)^4.
a(n) = Sum_{k = 0..n} 2*k*(2*k - 1).
Sum_{n>=1} 1/a(n) = -3*(2*Pi - 12*log(2) + 1)/5 = 0.620748515723854...
a(n) mod 2 = 0.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(1 + 2*sqrt(2)*Pi - 2*(3 + sqrt(2))*log(2) + 4*sqrt(2)*log(2-sqrt(2)))/5. - Amiram Eldar, Nov 05 2020

A268685 a(n) = 3*(n + 1)*(n + 2)*(3*n + 1)*(3*n + 4)/4.

Original entry on oeis.org

6, 126, 630, 1950, 4680, 9576, 17556, 29700, 47250, 71610, 104346, 147186, 202020, 270900, 356040, 459816, 584766, 733590, 909150, 1114470, 1352736, 1627296, 1941660, 2299500, 2704650, 3161106, 3673026, 4244730, 4880700, 5585580, 6364176, 7221456, 8162550
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 11 2016

Keywords

Comments

a(n) is the total volume of the family of (n+1) rectangular prisms, where the k-th prism has dimensions (3k) X (3k-1) X (3k-2). - Wesley Ivan Hurt, Oct 02 2018

Examples

			a(0) = 1*2*3 = 6;
a(1) = 1*2*3 + 4*5*6 = 126;
a(2) = 1*2*3 + 4*5*6 + 7*8*9 = 630;
a(3) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 = 1950;
a(4) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 = 4680;
a(5) = 1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 + 16*17*18 = 9576, etc.

Links

Crossrefs

Cf. A000027, A054776, A116689, A135036.
Trisection of A319014 and A319867.

Programs

  • Magma
    [3*(n + 1)*(n + 2)*(3*n + 1)*(3*n + 4)/4: n in [0..40]]; // Vincenzo Librandi, Feb 11 2016
  • Mathematica
    Table[3 (n + 1) (n + 2) (3 n + 1) ((3 n + 4)/4), {n, 0, 32}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {6, 126, 630, 1950, 4680}, 32]
CoefficientList[Series[6 (10 x^2 + 16 x + 1) / (1 - x)^5, {x, 0, 33}], x] (* Vincenzo Librandi, Feb 11 2016 *)
  • PARI
    a(n) = 3*(n+1)*(n+2)*(3*n+1)*(3*n+4)/4 \\ Felix Fröhlich, Jun 07 2016

Formula

G.f.: -6*(10*x^2 + 16*x + 1)/(x - 1)^5.
a(n) = Sum_{k = 0..n} (3*k + 1)(3*k + 2)(3*k + 3).
Sum {n>=0} 1/a(n) = 2*(sqrt(3)*Pi + 9*log(3) - 14)/15 = 0.1771878254287521...
a(n) mod 6 = 0.
a(n) = 6*A116689(n+1). - R. J. Mathar, Jun 07 2016
E.g.f.: 3*exp(x)*(8 + 160*x +256*x^2 + 96*x^3 + 9*x^4)/4. - Stefano Spezia, Apr 18 2023
Sum_{n>=0} (-1)^n/a(n) = 28/15 - 8*Pi/(15*sqrt(3)) - 16*log(2)/15. - Amiram Eldar, Apr 30 2023

A268484 a(n) = (n + 1)*(4*n^2 + 14*n + 9)/3.

Original entry on oeis.org

3, 18, 53, 116, 215, 358, 553, 808, 1131, 1530, 2013, 2588, 3263, 4046, 4945, 5968, 7123, 8418, 9861, 11460, 13223, 15158, 17273, 19576, 22075, 24778, 27693, 30828, 34191, 37790, 41633, 45728, 50083, 54706, 59605, 64788, 70263, 76038, 82121, 88520, 95243
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 12 2016

Keywords

Examples

			a(0) = 1*3 = 3;
a(1) = 1*3 + 3*5 = 18;
a(2) = 1*3 + 3*5 + 5*7 = 53;
a(3) = 1*3 + 3*5 + 5*7 + 7*9 = 116, etc.

Links

Crossrefs

Cf. A000466, A005408, A007290, A135036.

Programs

  • Mathematica
    Table[(n + 1) ((4 n^2 + 14 n + 9)/3), {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {3, 18, 53, 116}, 40]
  • PARI
    a(n)=(n+1)*(4*n^2+14*n+9)/3 \\ Charles R Greathouse IV, Jul 26 2016

Formula

G.f.: (3 + 6*x - x^2)/(x - 1)^4.
a(n) = Sum_{k = 0..n} (2*k + 1)*(2*k + 3) = Sum_{k = 0..n} A005408(k)*A005408(k + 1).
Sum_{n>=0} 1/a(n) = 0.4315109123788144393864...

A268579 Expansion of (1 + 6*x + x^2 + 12*x^3 - 2*x^4)/((1 - x)^4*(1 + x)^3).

Original entry on oeis.org

1, 7, 11, 41, 48, 120, 130, 262, 275, 485, 501, 807, 826, 1246, 1268, 1820, 1845, 2547, 2575, 3445, 3476, 4532, 4566, 5826, 5863, 7345, 7385, 9107, 9150, 11130, 11176, 13432, 13481, 16031, 16083, 18945, 19000, 22192, 22250, 25790, 25851, 29757, 29821
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 21 2016

Keywords

Examples

			a(0) = 1;
a(1) = 1 + 2*3 = 7;
a(2) = 1 + 2*3 + 4 = 11;
a(3) = 1 + 2*3 + 4 + 5*6 = 41;
a(4) = 1 + 2*3 + 4 + 5*6 + 7 = 48;
a(5) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 = 120;
a(6) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 = 130;
a(7) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12= 262;
a(8) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12 + 13 = 275;
a(9) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12 + 13 + 14*15 = 485, etc.

Links

Crossrefs

Cf. A000027, A001651, A008585, A016777, A135036, A152743.

Programs

  • Mathematica
    Table[Sum[(6 k + (-1)^k + 3) ((3 k - (-1)^k (3 k + 1) + 5)/16), {k, 0, n}], {n, 0, 42}]
Table[1 + (n (6 n^2 + 27 n + 35) - (9 n^2 + 15 n + 2) (-1)^n + 2)/16, {n, 0, 42}]
LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {1, 7, 11, 41, 48, 120, 130}, 43]
  • PARI
    Vec((1 + 6*x + x^2 + 12*x^3 - 2*x^4)/((1 - x)^4*(1 + x)^3) + O(x^50)) \\ Michel Marcus, Feb 21 2016

Formula

G.f.: (1 + 6*x + x^2 + 12*x^3 - 2*x^4)/((1 - x)^4*(1 + x)^3).
a(n) = Sum_{k = 0..n} (6*k + (-1)^k +3)*(3*k - (-1)^k*(3*k + 1) + 5)/16.
a(n) = 1 + (n*(6*n^2 + 27*n + 35) - (9*n^2 + 15*n + 2)*(-1)^n + 2)/16.
Showing 1-4 of 4 results.

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