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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A173154 a(n) = n^3/6 + 3*n^2/4 + 7*n/3 + 7/8 + (-1)^n/8.

Original entry on oeis.org

1, 4, 10, 19, 33, 52, 78, 111, 153, 204, 266, 339, 425, 524, 638, 767, 913, 1076, 1258, 1459, 1681, 1924, 2190, 2479, 2793, 3132, 3498, 3891, 4313, 4764, 5246, 5759, 6305, 6884, 7498, 8147, 8833, 9556, 10318, 11119, 11961, 12844, 13770, 14739, 15753, 16812, 17918, 19071, 20273, 21524
Offset: 0

Views

Author

Paul Curtz, Feb 11 2010

Keywords

Comments

Generated by reading the table shown in A172002 down the diagonal starting at 1.
The inverse binomial transform yields 1, 3, 3, 0, 2, -4, 8, -16, 32, -64, 128, -256, 512, -1024, ... with a pattern of powers of 2.

Links

Programs

  • Magma
    [n^3/6 + 3*n^2/4 + 7*n/3 + 7/8 + (-1)^n/8: n in [0..50]]; // Vincenzo Librandi, Aug 05 2011
  • Mathematica
    Table[n^3/6+(3n^2)/4+(7n)/3+7/8+(-1)^n/8,{n,0,50}] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{1,4,10,19,33},50] (* Harvey P. Dale, Jan 04 2012 *)

Formula

G.f.: ( 1 + x - x^3 + x^4 ) / ( (1+x)*(x-1)^4 ).
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
a(n+4) - a(n) = 4*A152948(n+5) = 4*A089071(n+5).
First differences: a(n+1) - a(n) = A061925(n+2).
Second differences: a(n+2) - 2*a(n+1) + a(n) = n + 5/2 + (-1)^n/2 = 3, 3, 5, 5, 7, 7, 9, 9, ... , duplicated A144396.

A172049 Irregular triangle T(n,k) = 2k-1 with A008794(n+2) values in row n.

Original entry on oeis.org

1, 1, 1, 3, 5, 7, 1, 3, 5, 7, 1, 3, 5, 7, 9, 11, 13, 15, 17, 1, 3, 5, 7, 9, 11, 13, 15, 17, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43
Offset: 1

Views

Author

Paul Curtz, Jan 24 2010

Keywords

Comments

These are the values A172002(2*n)-A172002(2*n-1) arranged in a table by adding line breaks.
The last (and largest) numbers in the lines are in A056220(floor(n+1)/2).

Examples

			1;
1;
1, 3, 5, 7;
1, 3, 5, 7;
1, 3, 5, 7, 9, 11, 13, 15, 17;
1, 3, 5, 7, 9, 11, 13, 15, 17;
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31;
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31;
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43,...

Crossrefs

Cf. A158405.

A173991 Bayley-Thomsen-Bohr periodic table(s) (1882-1895-1922) adapted by Scerri (1997).

Original entry on oeis.org

1, 2, 6, 7, 5, 8, 4, 9, 3, 10, 14, 15, 13, 16, 12, 17, 11, 18, 27, 28, 26, 29, 25, 30, 24, 31, 23, 32, 22, 33, 21, 34, 20, 35, 19, 36, 45, 46, 44, 47, 43, 48, 42, 49, 41, 50, 40, 51, 39, 52, 38, 53, 37, 54, 70, 71, 69, 72, 68, 73, 67, 74, 66, 75, 65, 76, 64, 77, 63, 78, 62, 79
Offset: 1

Views

Author

Paul Curtz, Mar 04 2010

Keywords

Comments

This a compact (no spaces) symmetric table of (7 rows, 32 columns) 118 elements. Also from Mendeleyev-Moseley-Seaborg. A permutation of the numbers from 1 to 118. The writing is the same as A172002, from Janet table.
Writing begins from central (two) columns. Number of terms by columns: 2,2,2,2,2,2,2,4,4,4,4,4,6,6,6,7,7,6,6,6,4,4,4,4,4,2,2,2,2,2,2,2; by rows: 2,8,8,18,18,32,32 (see A093907 and A137583).

Formula

a(n)= A172002(n+2) - 2.
Previous Showing 11-13 of 13 results.

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