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.

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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

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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

A319258 a(n) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12 + ... + (up to n).

Original entry on oeis.org

1, 3, 7, 11, 16, 41, 48, 56, 120, 130, 141, 262, 275, 289, 485, 501, 518, 807, 826, 846, 1246, 1268, 1291, 1820, 1845, 1871, 2547, 2575, 2604, 3445, 3476, 3508, 4532, 4566, 4601, 5826, 5863, 5901, 7345, 7385, 7426, 9107, 9150, 9194, 11130, 11176, 11223
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 16 2018

Keywords

Examples

			a(1) = 1;
a(2) = 1 + 2 = 3;
a(3) = 1 + 2*3 = 7;
a(4) = 1 + 2*3 + 4 = 11;
a(5) = 1 + 2*3 + 4 + 5 = 16;
a(6) = 1 + 2*3 + 4 + 5*6 = 41;
a(7) = 1 + 2*3 + 4 + 5*6 + 7 = 48;
a(8) = 1 + 2*3 + 4 + 5*6 + 7 + 8 = 56;
a(9) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 = 120;
a(10) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 = 130;
a(11) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11 = 141;
a(12) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12 = 262; etc.

Links

Crossrefs

Cf. A093361, A228958, A305189, A319014.

Programs

  • Mathematica
    Table[n (1 + Floor[(n - 2)/3] - Floor[n/3]) + 3 Floor[n/3]^2 (1 + Floor[n/3]) + Floor[(n + 2)/3] (3 Floor[(n + 2)/3] - 1)/2, {n, 50}]
  • PARI
    Vec(x*(1 + 2*x + 4*x^2 + x^3 - x^4 + 13*x^5 - 2*x^6 - x^7 + x^8) / ((1 - x)^4*(1 + x + x^2)^3) + O(x^40)) \\ Colin Barker, Sep 16 2018

Formula

a(n) = n*(1 + floor((n-2)/3) - floor(n/3)) + 3*floor(n/3)^2*(1 + floor(n/3)) + floor((n+2)/3)*(3*floor((n+2)/3) - 1)/2.
From Colin Barker, Sep 16 2018: (Start)
G.f.: x*(1 + 2*x + 4*x^2 + x^3 - x^4 + 13*x^5 - 2*x^6 - x^7 + x^8) / ((1 - x)^4*(1 + x + x^2)^3).
a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) - 3*a(n-6) + 3*a(n-7) + a(n-9) - a(n-10) for n>10.
(End)

A319391 a(n) = (1 + 2)^3 + (4 + 5)^6 + (7 + 8)^9 + (10 + 11)^12 + ... + (up to n).

Original entry on oeis.org

1, 3, 27, 31, 36, 531468, 531475, 531483, 38443890843, 38443890853, 38443890864, 7355865955277484, 7355865955277497, 7355865955277511, 2954320062416788976127, 2954320062416788976143, 2954320062416788976160, 2154028838712789034859190336
Offset: 1

Views

Author

Wesley Ivan Hurt, Sep 18 2018

Keywords

Examples

			a(1) = 1;
a(2) = 1 + 2 = 3;
a(3) = (1 + 2)^3 = 27;
a(4) = (1 + 2)^3 + 4 = 31;
a(5) = (1 + 2)^3 + 4 + 5 = 36;
a(6) = (1 + 2)^3 + (4 + 5)^6 = 531468;
a(7) = (1 + 2)^3 + (4 + 5)^6 + 7 = 531475;
a(8) = (1 + 2)^3 + (4 + 5)^6 + 7 + 8 = 531483;
a(9) = (1 + 2)^3 + (4 + 5)^6 + (7 + 8)^9 = 38443890843;
a(10) = (1 + 2)^3 + (4 + 5)^6 + (7 + 8)^9 + 10 = 38443890853; etc.

Links

Crossrefs

Cf. A305189, A318868, A319014, A319258.

Programs

  • Maple
    f:= proc(n) option remember;
  if n mod 3 = 0 then procname(n-3)+(2*n-3)^n
  else procname(n-1)+n
  fi
end proc:
f(0):= 0:
map(f, [$1..20]); # Robert Israel, Oct 05 2018
  • Mathematica
    Table[Sum[(Floor[i/3] - Floor[(i - 1)/3])*(6*Floor[(i + 2)/3] - 3)^(3*Floor[(i + 2)/3]) + i*(Floor[(i - 1)/3] - Floor[(i - 2)/3]) + i*(Floor[(i + 1)/3] - Floor[i/3]) - (6*Floor[(i + 2)/3] - 3)*(Floor[i/3] - Floor[(i - 1)/3]), {i, n}], {n, 20}]

Formula

a(n) = Sum_{i=1..n} (floor(i/3)-floor((i-1)/3))*(6*floor((i+2)/3)-3)^(3*floor((i+2)/3)) + i*(floor((i-1)/3)-floor((i-2)/3))+i*(floor((i+1)/3)-floor(i/3))-(6*floor((i+2)/3)-3)*(floor(i/3)-floor((i-1)/3)).
If 3|n then a(n) = a(n-3)+(2*n-3)^n, otherwise a(n) = a(n-1)+n. - Robert Israel, Oct 05 2018
Previous Showing 11-13 of 13 results.

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