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.

A011199 a(n) = (n+1)*(2*n+1)*(3*n+1).

Original entry on oeis.org

1, 24, 105, 280, 585, 1056, 1729, 2640, 3825, 5320, 7161, 9384, 12025, 15120, 18705, 22816, 27489, 32760, 38665, 45240, 52521, 60544, 69345, 78960, 89425, 100776, 113049, 126280, 140505, 155760, 172081, 189504, 208065, 227800, 248745, 270936, 294409, 319200
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A079588.

Programs

  • GAP
    List([0..40], n-> (n+1)*(2*n+1)*(3*n+1) ); # G. C. Greubel, Mar 03 2020
  • Haskell
    a011199 n = product $ map ((+ 1) . (* n)) [1, 2, 3]
-- Reinhard Zumkeller, Jun 08 2015
  • Magma
    [&*[j*n+1:j in [1..3]]: n in [0..40]]; // G. C. Greubel, Mar 03 2020
  • Maple
    seq(mul(j*n+1, j=1..3), n = 0..40); # G. C. Greubel, Mar 03 2020
  • Mathematica
    Product[j*Range[0,40] +1, {j,3}] (* G. C. Greubel, Mar 03 2020 *)
LinearRecurrence[{4,-6,4,-1},{1,24,105,280},40] (* Harvey P. Dale, Apr 21 2020 *)
  • PARI
    vector(41, n, my(m=n-1); prod(j=1,3, j*m+1)) \\ G. C. Greubel, Mar 03 2020
  • Sage
    [product(j*n+1 for j in (1..3)) for n in (0..40)] # G. C. Greubel, Mar 03 2020

Formula

G.f.: (1 + 20*x + 15*x^2)/(x-1)^4. - Alois P. Heinz, Sep 04 2014
a(n) = 6*n^3 + 11*n^2 + 6*n + 1. - Reinhard Zumkeller, Jun 08 2015
E.g.f.: (1 + 23*x + 29*x^2 + 6*x^3)*exp(x). - G. C. Greubel, Mar 03 2020
From Amiram Eldar, Jan 13 2021: (Start)
Sum_{n>=0} 1/a(n) = sqrt(3)*Pi/4 - 4*log(2) + 9*log(3)/4.
Sum_{n>=0} (-1)^n/a(n) = 2*log(2) - (1 - sqrt(3)/2)*Pi. (End)

A100147 Structured icosidodecahedral numbers.

Original entry on oeis.org

1, 30, 135, 364, 765, 1386, 2275, 3480, 5049, 7030, 9471, 12420, 15925, 20034, 24795, 30256, 36465, 43470, 51319, 60060, 69741, 80410, 92115, 104904, 118825, 133926, 150255, 167860, 186789, 207090, 228811, 252000, 276705, 302974, 330855, 360396, 391645, 424650
Offset: 1

Views

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

Comments

Equals row sums of triangle A143254 & binomial transform of [1, 29, 76, 48, 0, 0, 0, ...]. - Gary W. Adamson, Aug 02 2008
Apart from offset, same as A079588.

Links

Crossrefs

Cf. A100146, A100148 for adjacent structured Archimedean solids; and A100145 for more on structured polyhedral numbers.
Cf. also A079588.

Programs

Formula

a(n) = (1/6)*(48*n^3 - 60*n^2 + 18*n).
a(n) = A079588(n-1) = n*(2*n-1)*(4*n-3). - R. J. Mathar, Sep 02 2008
From Jaume Oliver Lafont, Sep 08 2009: (Start)
a(n) = (1+(n-1))*(1+2*(n-1))*(1+4*(n-1)).
G.f.: x*(1 + 26*x + 21*x^2)/(1-x)^4. (End)
E.g.f.: x*(1 + 14*x + 8*x^2)*exp(x). - G. C. Greubel, Oct 18 2018
From Amiram Eldar, Sep 20 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*sqrt(2)*log(sqrt(2)+1)/3 + log(2)/3 - (3 - 2*sqrt(2))*Pi/6. (End)

A101493 Triangle read by rows: T(n,k) = (n+1)*(2*(n+1)-1) - k*(2*k-1).

Original entry on oeis.org

1, 6, 5, 15, 14, 9, 28, 27, 22, 13, 45, 44, 39, 30, 17, 66, 65, 60, 51, 38, 21, 91, 90, 85, 76, 63, 46, 25, 120, 119, 114, 105, 92, 75, 54, 29, 153, 152, 147, 138, 125, 108, 87, 62, 33, 190, 189, 184, 175, 162, 145, 124, 99, 70, 37, 231, 230, 225, 216, 203, 186, 165, 140, 111, 78, 41
Offset: 0

Views

Author

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 21 2005

Keywords

Comments

The triangle is generated from the product B*A of the infinite lower triangular matrices A =
1 0 0 0 ...
1 1 0 0 ...
1 1 1 0 ...
1 1 1 1 ...
... and B =
1 0 0 0 ...
1 5 0 0 ...
1 5 9 0 ...
1 5 9 13 ...
...
T(n+0,0) = n*(2*n-1) = A000384(n) (Hexagonal numbers)
since T(n,n) = 4*n+1 = A016813(n).
T(n,n) = 4*n + 1 = A016813(n);
T(n+1,n) = 8*n + 6 = A017137(n);
T(n+2,n) = 12*n + 3 = A017557(n);
T(n,n)*T(n,0) = (n+1)*(2*n+1)*(4*n+1) = A079588(n).

Examples

			Triangle begins:
   1;
   6,  5;
  15, 14,  9;
  28, 27, 22, 13;
  45, 44, 39, 30, 17;
  66, 65, 60, 51, 38, 21;

Links

Crossrefs

Row sums give 10-gonal pyramidal numbers: n(n+1)(8n-5)/6 = A007585(n+1).
Cf. A101492 (for product A*B), A007585, A000384.

Programs

  • GAP
    Flat(List([0..10],n->List([0..n],k->(n+1)*(2*n+1)-k*(2*k-1)))); # Muniru A Asiru, Mar 05 2019
  • PARI
    T(n,k)=if(k>n,0,(n+1)*(2*(n+1)-1)-k*(2*k-1))
for(i=0,10, for(j=0,i,print1(T(i,j),", "));print())

A258721 a(n) = 24*n^2 + 52*n + 29.

Original entry on oeis.org

29, 105, 229, 401, 621, 889, 1205, 1569, 1981, 2441, 2949, 3505, 4109, 4761, 5461, 6209, 7005, 7849, 8741, 9681, 10669, 11705, 12789, 13921, 15101, 16329, 17605, 18929, 20301, 21721, 23189, 24705, 26269, 27881, 29541, 31249, 33005, 34809, 36661, 38561, 40509
Offset: 0

Views

Author

Reinhard Zumkeller, Jun 08 2015

Keywords

Comments

First differences of A079588.
6*a(n) - 5 is a square. Therefore, this is the quadrisection of the sequence which lists the numbers m such that 6*m-5 is a square (without 1): 1, 5, 9, 21, 29, 49, 61, 89, 105, 141, 161, 205, 229, ... . [Bruno Berselli, Jun 08 2015]

Links

Crossrefs

Cf. A079588.

Programs

  • Haskell
    a258721 n = 4 * n * (6 * n + 13) + 29
  • Magma
    [24*n^2+52*n+29: n in (0..50)] // Bruno Berselli, Jun 08 2015
  • Mathematica
    Table[24 n^2 + 52 n + 29, {n, 0, 50}] (* Bruno Berselli, Jun 08 2015 *)
  • Maxima
    makelist(24*n^2+52*n+29, n, 0, 50); /* Bruno Berselli, Jun 08 2015 */
  • PARI
    vector(50, n, n--; 24*n^2+52*n+29) \\ Bruno Berselli, Jun 08 2015
  • Sage
    [24*n^2+52*n+29 for n in (0..50)] # Bruno Berselli, Jun 08 2015

Formula

G.f.: (29 + 18*x + x^2)/(1 - x)^3.
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.