A085027 -id:A085027 - cp's OEIS frontend

A070262 5th diagonal of triangle defined in A051537.

5, 3, 21, 2, 45, 15, 77, 6, 117, 35, 165, 12, 221, 63, 285, 20, 357, 99, 437, 30, 525, 143, 621, 42, 725, 195, 837, 56, 957, 255, 1085, 72, 1221, 323, 1365, 90, 1517, 399, 1677, 110, 1845, 483, 2021, 132, 2205, 575, 2397, 156, 2597, 675, 2805, 182, 3021, 783

Offset: 1

Amarnath Murthy, May 09 2002

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A061037. [From R. J. Mathar, Sep 29 2008]
Cf. A010873, A051537.
Cf. A000466, A002378, A003185, A085027.

[LCM(n + 4, n)/GCD(n + 4, n): n in [1..50]]; // G. C. Greubel, Sep 20 2018

Table[ LCM[i + 4, i] / GCD[i + 4, i], {i, 1, 60}]
LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1},{5,3,21,2,45,15,77,6,117,35,165,12},90] (* Harvey P. Dale, Jul 13 2019 *)

Vec(x*(5 + 3*x + 21*x^2 + 2*x^3 + 30*x^4 + 6*x^5 + 14*x^6 - 3*x^8 - x^9 - 3*x^10) / ((1 - x)^3*(1 + x)^3*(1 + x^2)^3) + O(x^60)) \\ Colin Barker, Mar 27 2017

a(n) = lcm(n+4,n)/gcd(n+4,n); \\ Altug Alkan, Sep 20 2018

a(n) = lcm(n + 4, n) / gcd(n + 4, n).
From Colin Barker, Mar 27 2017: (Start)
G.f.: x*(5 + 3*x + 21*x^2 + 2*x^3 + 30*x^4 + 6*x^5 + 14*x^6 - 3*x^8 - x^9 - 3*x^10) / ((1 - x)^3*(1 + x)^3*(1 + x^2)^3).
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12) for n>12. (End)
From Luce ETIENNE, May 10 2018: (Start)
a(n) = n*(n+4)*4^((5*(n mod 4)^3 - 24*(n mod 4)^2 + 31*(n mod 4)-12)/6).
a(n) = n*(n+4)*(37-27*cos(n*Pi)-6*cos(n*Pi/2))/64. (End)
From Amiram Eldar, Oct 08 2023: (Start)
Sum_{n>=1} 1/a(n) = 11/6.
Sum_{n>=1} (-1)^n/a(n) = 7/6.
Sum_{k=1..n} a(k) ~ (37/192) * n^3. (End)

Edited by Robert G. Wilson v, May 10 2002

A173039 Odd numerators of the fractions (1/4-1/n^2), n>= -2.

Original entry on oeis.org

-3, -1, -3, 5, 3, 21, 45, 15, 77, 117, 35, 165, 221, 63, 285, 357, 99, 437, 525, 143, 621, 725, 195, 837, 957, 255, 1085, 1221, 323, 1365, 1517, 399, 1677, 1845, 483, 2021, 2205, 575, 2397, 2597, 675

Offset: 1

Paul Curtz, Nov 21 2010

sign

Odd numbers in A061037, which is extended to negative n, and uses -1 to represent -1/0 at n=0. Trisections are apparently A003185, A000466 and A085027.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 3, 0, 0, -3, 0, 0, 1).

Conjecture: a(n)= +3*a(n-3) -3*a(n-6) +a(n-9). G.f.: ( 3+x+3*x^2-14*x^3-6*x^4-30*x^5-21*x^6-3*x^7-5*x^8 ) / ( (x-1)^3*(1+x+x^2)^3 ). - R. J. Mathar, Dec 02 2010

A241747 Triangle read by rows: T(n,k) = (4n+3)(4*k+3).

Original entry on oeis.org

9, 21, 49, 33, 77, 121, 45, 105, 165, 225, 57, 133, 209, 285, 361, 69, 161, 253, 345, 437, 529, 81, 189, 297, 405, 513, 621, 729, 93, 217, 341, 465, 589, 713, 837, 961, 105, 245, 385, 525, 665, 805, 945, 1085, 1225, 117, 273, 429, 585, 741, 897, 1053, 1209, 1365, 1521

Offset: 0

Vincenzo Librandi, Apr 29 2014

A016838(n) first diagonal.
A085027(n) second diagonal.
A017629(n) column k=0.
Row sums give the second bisection of A002414: 9, 70, 231, 540, 1045, 1794, 2835, 4216, ... [Bruno Berselli, May 08 2014]

			Triangle begins:
n\k |   0     1     2    3    4    5     6     7     8     9
----|--------------------------------------------------------
0   |   9;
1   |  21,   49;
2   |  33,   77,  121;
3   |  45,  105,  165, 225;
4   |  57,  133,  209, 285, 361;
5   |  69,  161,  253, 345, 437, 529;
6   |  81,  189,  297, 405, 513, 621,  729;
7   |  93,  217,  341, 465, 589, 713,  837,  961;
8   | 105,  245,  385, 525, 665, 805,  945, 1085, 1225;
9   | 117,  273,  429, 585, 741, 897, 1053, 1209, 1365, 1521;
.....

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A002145, A004767, A016838, A017629, A085027.

[(4*n+3)*(4*k+3): k in [0..n], n in [0..15]]; /* or, as triangle: */ [[(4*n+3)*(4*k+3): k in [0..n]]: n in [0..10]];

t[n_, k_] := (4 n + 3) (4 k + 3); Table[t[n, k], {n, 0, 10}, {k, n}] // Flatten

Edited by Alois P. Heinz and Bruno Berselli, May 08 2014

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A070262 5th diagonal of triangle defined in A051537.

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A173039 Odd numerators of the fractions (1/4-1/n^2), n>= -2.

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A241747 Triangle read by rows: T(n,k) = (4n+3)(4*k+3).

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cp's OEIS Frontend

A070262 5th diagonal of triangle defined in A051537.

Views

Author

Keywords

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Magma

Mathematica

PARI

PARI

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Extensions

A173039 Odd numerators of the fractions (1/4-1/n^2), n>= -2.

Views

Author

Keywords

Comments

Links

Formula

A241747 Triangle read by rows: T(n,k) = (4*n+3)*(4*k+3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A241747 Triangle read by rows: T(n,k) = (4n+3)(4*k+3).