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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A386858 a(n) = floor(5*n^2/8).

Original entry on oeis.org

0, 2, 5, 10, 15, 22, 30, 40, 50, 62, 75, 90, 105, 122, 140, 160, 180, 202, 225, 250, 275, 302, 330, 360, 390, 422, 455, 490, 525, 562, 600, 640, 680, 722, 765, 810, 855, 902, 950, 1000, 1050, 1102, 1155, 1210, 1265, 1322, 1380, 1440, 1500, 1562, 1625, 1690, 1755
Offset: 1

Views

Author

Chai Wah Wu, Aug 05 2025

Keywords

Links

Crossrefs

Cf. A032526, A028895.

Programs

Formula

a(2n) = A032526(n).
a(2n+1) = A028895(n).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n > 6.
G.f.: -x^2*(2*x^2 + x + 2)/((x - 1)^3*(x + 1)*(x^2 + 1)).
Sum_{n>=2} 1/a(n) = 2/5 + Pi^2/60 + tan(Pi/(2*sqrt(5)))*Pi/(2*sqrt(5)). - Amiram Eldar, Aug 15 2025

A338544 a(n) = (5*floor((n-1)/2)^2 + (4+(-1)^n)*floor((n-1)/2)) / 2.

Original entry on oeis.org

0, 0, 0, 4, 5, 13, 15, 27, 30, 46, 50, 70, 75, 99, 105, 133, 140, 172, 180, 216, 225, 265, 275, 319, 330, 378, 390, 442, 455, 511, 525, 585, 600, 664, 680, 748, 765, 837, 855, 931, 950, 1030, 1050, 1134, 1155, 1243, 1265, 1357, 1380, 1476, 1500, 1600, 1625, 1729, 1755, 1863
Offset: 0

Views

Author

Wesley Ivan Hurt, Nov 01 2020

Keywords

Comments

Sum of the largest side lengths of all integer-sided triangles with perimeter 3n whose side lengths are in arithmetic progression (for example, when n=5 there are two triangles with perimeter 3*5 = 15 whose side lengths are in arithmetic progression: [3,5,7] and [4,5,6]; thus a(5) = 7+6 = 13).

Links

Crossrefs

Cf. A028895, A050187, A147875, A211539.

Programs

  • Mathematica
    Table[(5 Floor[(n - 1)/2]^2 + Floor[(n - 1)/2] (4 + (-1)^n))/2, {n, 0, 100}]

Formula

A147875 UNION A028895.
From Stefano Spezia, Nov 01 2020: (Start)
G.f.: x^3*(4 + x)/((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4. (End)
16*a(n) = -14*n-1+10*n^2+(-1)^n-6*(-1)^n*n . - R. J. Mathar, Aug 19 2022

A375220 T(n,k) is the number of permutations of the multiset {1, 1, 2, 2, ..., n, n} with k occurrences of fixed pairs (j,j), where T(n,k), n >= 2, 0 <= k <= n-2 is a triangle read by rows.

Original entry on oeis.org

5, 74, 15, 2193, 296, 30, 101644, 10965, 740, 50, 6840085, 609864, 32895, 1480, 75, 630985830, 47880595, 2134524, 76755, 2590, 105, 76484389121, 5047886640, 191522380, 5692064, 153510, 4144, 140, 11792973495032, 688359502089, 22715489880, 574567140, 12807144, 276318, 6216, 180
Offset: 2

Views

Author

Hugo Pfoertner, Aug 08 2024

Keywords

Examples

			The triangle begins
          5,
         74,       15,
       2193,      296,      30,
     101644,    10965,     740,    50,
    6840085,   609864,   32895,  1480,   75,
  630985830, 47880595, 2134524, 76755, 2590, 105

Crossrefs

Cf. A000217, A000680, A028895, A116218, A374980 (column 0), A375222 (column 1), A375223.
Cf. A375219 (similar for triples in the multiset).

Programs

  • PARI
    \\ using functions mima and a375219 from A375219, row n of triangle:
a375219(n,sizeb=2)

Formula

T(n,n) = 1, T(n,n-1) = 0 (terms not in DATA),
T(n,n-2) = 5*n*(n-1)/2 = 5*A000217(n-1) = A028895(n-1),
Sum_{j=0..n-2} T(n,j) = (2*n)!/(2^n) - 1 = A000680(n) - 1,
Sum_{j=1..n-2} T(n,j) = A375223(n) - 1.
Previous Showing 31-33 of 33 results.

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