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 1-3 of 3 results.

A104099 a(n) = n * (10*n^2 - 6n + 1) = n * A087348(n).

Original entry on oeis.org

0, 5, 58, 219, 548, 1105, 1950, 3143, 4744, 6813, 9410, 12595, 16428, 20969, 26278, 32415, 39440, 47413, 56394, 66443, 77620, 89985, 103598, 118519, 134808, 152525, 171730, 192483, 214844, 238873, 264630, 292175, 321568, 352869, 386138, 421435
Offset: 0

Views

Author

Roger L. Bagula, Mar 31 2005

Keywords

Links

Programs

  • Maple
    a[0]:=0:a[1]:=5:a[2]:=58:a[3]:=219: for n from 4 to 40 do a[n]:=4*a[n-1]-6*a[n-2]+4*a[n-3]-a[n-4] od: seq(a[n], n=0..40);
  • Mathematica
    Table[n(10n^2-6n+1),{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,5,58,219},40] (* Harvey P. Dale, Sep 01 2018 *)
  • PARI
    a(n)=n*(10*n^2 - 6*n + 1) \\ Charles R Greathouse IV, Oct 18 2022

Formula

Recurrence relation: a(n)=4a(n-1)-6a(n-2)+4a(n-3)-a(n-4) for n>=4; a(0)=0, a(1)=5, a(2)=58, a(3)=219.
O.g.f.: x*(5+38*x+17*x^2)/(-1+x)^4 = 132/(-1+x)^3+60/(-1+x)^4+89/(-1+x)^2+17/(-1+x) . - R. J. Mathar, Dec 05 2007

Extensions

Edited by N. J. A. Sloane, May 20 2006, Jun 06 2007

A153784 4 times heptagonal numbers: a(n) = 2*n*(5*n-3).

Original entry on oeis.org

0, 4, 28, 72, 136, 220, 324, 448, 592, 756, 940, 1144, 1368, 1612, 1876, 2160, 2464, 2788, 3132, 3496, 3880, 4284, 4708, 5152, 5616, 6100, 6604, 7128, 7672, 8236, 8820, 9424, 10048, 10692, 11356, 12040, 12744, 13468, 14212, 14976, 15760, 16564, 17388, 18232, 19096
Offset: 0

Views

Author

Omar E. Pol, Jan 02 2009

Keywords

Comments

Sequence found by reading the line from 0, in the direction 0, 4, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 18 2012

Links

Crossrefs

Cf. A000566, A085787, A087348, A135706, A152745, A152773.

Programs

Formula

a(n) = 10*n^2 - 6*n = 4*A000566(n) = 2*A135706(n).
a(n) = 20*n + a(n-1) - 16 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(n) = A087348(n) - 1, n >= 1. - Omar E. Pol, Jul 18 2012
a(0)=0, a(1)=4, a(2)=28, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Mar 19 2015
From Elmo R. Oliveira, Dec 15 2024: (Start)
G.f.: 4*x*(1 + 4*x)/(1 - x)^3.
E.g.f.: 2*exp(x)*x*(2 + 5*x).
a(n) = A152745(n) - n. (End)

A330082 a(n) = 5*A064038(n+1).

Original entry on oeis.org

0, 5, 15, 15, 25, 75, 105, 70, 90, 225, 275, 165, 195, 455, 525, 300, 340, 765, 855, 475, 525, 1155, 1265, 690, 750, 1625, 1755, 945, 1015, 2175, 2325, 1240, 1320, 2805, 2975, 1575, 1665, 3515, 3705, 1950, 2050, 4305, 4515, 2365, 2475, 5175, 5405, 2820, 2940
Offset: 0

Views

Author

Paul Curtz, Dec 01 2019

Keywords

Comments

Main column of a pentagonal spiral for A026741:
(25)
49 (15) 31
24 29 (15) 8 16
47 14 7 ( 5) 3 17 33
23 27 13 2 ( 0) 1 7 9 17
45 13 6 3 1 4 19 35
22 25 11 5 9 10 18
43 12 23 11 21 37
21 41 20 39 19
a(n) = 5 * A064038(n+1) from a pentagonal spiral.
Compare to A319127 = 6 * A002620 in the hexagonal spiral:
22 23 23 22 (24)
20 12 13 13 (12) 25
21 10 5 4 ( 6) 14 25
21 11 5 1 ( 0) 7 15 24
20 11 4 1 ( 0) 2 7 15 26
18 10 2 3 3 6 14 27
19 8 9 9 8 16 27
19 18 16 17 17 26
30 28 29 29 28

Links

Crossrefs

Cf. A026741, A028895, A064038. A033429, A062786, A087348, A147874, A158447, A168668 are in the spiral.

Programs

  • Mathematica
    A330082[n_]:=5Numerator[n(n+1)/4];Array[A330082,100,0] (* Paolo Xausa, Dec 04 2023 *)
  • PARI
    concat(0, Vec(5*x*(1 + 4*x^3 + x^6) / ((1 - x)^3*(1 + x^2)^3) + O(x^50))) \\ Colin Barker, Dec 08 2019

Formula

a(n) = A026741(A028895(n)).
From Colin Barker, Dec 08 2019: (Start)
G.f.: 5*x*(1 + 4*x^3 + x^6) / ((1 - x)^3*(1 + x^2)^3).
a(n) = 3*a(n-1) - 6*a(n-2) + 10*a(n-3) - 12*a(n-4) + 12*a(n-5) - 10*a(n-6) + 6*a(n-7) - 3*a(n-8) + a(n-9) for n>8.
a(n) = (-5/16 + (5*i)/16)*(((-3-3*i) + (-i)^n + i^(1+n))*n*(1+n)) where i=sqrt(-1).
(End)

Extensions

More terms from Colin Barker, Dec 22 2019
Name corrected by Paolo Xausa, Dec 04 2023
Showing 1-3 of 3 results.

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

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