|
Search: id:A105020
|
|
|
| A105020 |
|
Array read by antidiagonals: row n (n >= 0) contains the numbers m^2-n^2, m >= n+1. |
|
+0
1
|
|
| 1, 3, 4, 5, 8, 9, 7, 12, 15, 16, 9, 16, 21, 24, 25, 11, 20, 27, 32, 35, 36, 13, 24, 33, 40, 45, 48, 49, 15, 28, 39, 48, 55, 60, 63, 64, 17, 32, 45, 56, 65, 72, 77, 80, 81, 19, 36, 51, 64, 75, 84, 91, 96, 99, 100, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 121, 23, 44, 63, 80
(list; table; graph; listen)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
A "Goldbach Conjecture" for this sequence: when there are n terms between consecutive odd integers (2n+1) and (2n+3) for n>0, at least one will be the product of 2 primes (not necessarily distinct). Example: n=3 for consecutive odd integers a(7)=7 and a(11)=9 and of the 3 sequence entries a(8)=12, a(9)=15 and a(10)=16 between them, one is the product of 2 primes a(9)=15=3*5. - Michael Hiebl (michael.hiebl(AT)gmail.com), Jul 15 2007
A024352 gives distinct values in the array, minus the first row (1, 4, 9, 16, etc.). a(n) gives all solutions to the equation x^2 + xy = n, with y mod 2 = 0, x > 0, y >= 0. - Andrew Plewe (aplewe(AT)sbcglobal.net), Oct 19 2007
Alternatively, triangular sequence of coefficients of Dynkin diagram weights for the Cartan groups C_n: t(n,m)=m*(2*n - m). Row sums are A002412. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Aug 05 2008
|
|
REFERENCES
|
R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.
|
|
EXAMPLE
|
Array begins:
1 4 9 16 25 36 49 64 81 100 ...
3 8 15 24 35 48 63 80 99 120 ...
5 12 21 32 45 60 77 96 117 140 ...
7 16 27 40 55 72 91 112 135 160 ...
9 20 33 48 65 84 105 128 153 180 ...
...
Triangle begins:
{1},
{3, 4},
{5, 8, 9},
{7, 12, 15, 16},
{9, 16, 21, 24, 25},
{11, 20, 27, 32, 35, 36},
{13, 24, 33, 40, 45, 48, 49},
{15, 28, 39, 48, 55, 60, 63, 64},
{17, 32, 45, 56, 65, 72, 77, 80, 81},
{19, 36, 51, 64, 75, 84, 91, 96, 99, 100}
|
|
MATHEMATICA
|
t[n_, m_] := (n^2 - m^2); Flatten[ Table[ t[i, j], {i, 12}, {j, i - 1, 0, -1}]] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 11 2005)
(* to view table *) Table[t[i, j], {j, 0, 6}, {i, j + 1, 10}] // TableForm
Clear[T, n, m, a] T[n_, m_] = m*(2*n - m ); a = Table[Table[T[n, m], {m, 1, n}], {n, 1, 10}]; Flatten[a] - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Aug 05 2008
|
|
CROSSREFS
|
Rows give A000290, A005563, A028347, A028560, A028566, A098603, A098847, A098848, A098849, A098850.
Columns give A005408, A008586, A016945, A008590, A017329, A008594, A008598, A008602, A008606, A000567.
Diagonals give A033428, A045944, A067725.
Sequence in context: A028267 A092997 A021747 this_sequence A112594 A120519 A100614
Adjacent sequences: A105017 A105018 A105019 this_sequence A105021 A105022 A105023
|
|
KEYWORD
|
nonn,tabl,easy
|
|
AUTHOR
|
Andrew Plewe (aplewe(AT)sbcglobal.net) and Franklin T. Adams-Watters (franktaw(AT)netscape.net), Jul 11 2005
|
|
EXTENSIONS
|
More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 11 2005
|
|
|
Search completed in 0.003 seconds
|
