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 1-2 of 2 results.

A100178 Structured hexagonal diamond numbers (vertex structure 5).

Original entry on oeis.org

1, 8, 29, 72, 145, 256, 413, 624, 897, 1240, 1661, 2168, 2769, 3472, 4285, 5216, 6273, 7464, 8797, 10280, 11921, 13728, 15709, 17872, 20225, 22776, 25533, 28504, 31697, 35120, 38781, 42688, 46849, 51272, 55965, 60936, 66193, 71744, 77597, 83760, 90241, 97048, 104189
Offset: 1

Views

Author

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

Keywords

Comments

Row 1 of the convolution array A213838. - Clark Kimberling, Jul 05 2012

Links

Crossrefs

Cf. A000578 (alternate vertex), A000447 (structured diamonds) A100145 (for more on structured numbers).
Cf. A019558, A167471, A213838.

Programs

  • Magma
    [(1/6)*(8*n^3-6*n^2+4*n): n in [1..40]]; // Vincenzo Librandi, Aug 03 2011
  • Mathematica
    LinearRecurrence[{4, -6, 4, -1}, {1, 8, 29, 72}, 50] (* Paolo Xausa, Aug 06 2025 *)

Formula

a(n) = (1/6)*(8*n^3 - 6*n^2 + 4*n).
G.f.: x*(1+4*x+3*x^2)/(1-4*x+6*x^2-4*x^3+x^4). - Colin Barker, Jan 04 2012
From Elmo R. Oliveira, Aug 28 2025: (Start)
E.g.f.: exp(x)*x*(4*x^2 + 9*x + 3)/3.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.
a(n) = A167471(n)/16 = A019558(n)/48. (End)

A167498 a(n) = 6+32*n^2+8*n*(7+8*n^2)/3.

Original entry on oeis.org

6, 78, 342, 926, 1958, 3566, 5878, 9022, 13126, 18318, 24726, 32478, 41702, 52526, 65078, 79486, 95878, 114382, 135126, 158238, 183846, 212078, 243062, 276926, 313798, 353806, 397078, 443742, 493926, 547758, 605366, 666878, 732422, 802126, 876118, 954526, 1037478, 1125102
Offset: 0

Views

Author

Paul Curtz, Nov 05 2009

Keywords

Comments

Binomial transform of quasi-finite sequence 6,72,192,128,0,(0 continued).
a(n) mod 10 is periodic with period length 5: repeat 6,8,2,6,8.

Links

Crossrefs

Cf. A166911, A167471.

Programs

  • Magma
    [6+32*n^2+8*n*(7+8*n^2)/3: n in [0..50] ]; // Vincenzo Librandi, Aug 06 2011
  • Mathematica
    LinearRecurrence[{4,-6,4,-1}, {6, 78, 342, 926}, 100] (* G. C. Greubel, Jun 14 2016 *)

Formula

a(n) = A166464(n) + A166464(2n+1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 128.
G.f.: ( 6+54*x+66*x^2+2*x^3 ) / (x-1)^4 . - R. J. Mathar, Jul 01 2011
Showing 1-2 of 2 results.

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