login
A105020
Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.
5
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
OFFSET
0,2
COMMENTS
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, 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 S. Plewe, 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, Aug 05 2008
REFERENCES
R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.
LINKS
FORMULA
a(n) = r^2 - (r^2 + r - m)^2/4, where r = round(sqrt(m)) and m = 2*n+2. - Wesley Ivan Hurt, Sep 04 2021
a(n) = A128076(n+1) * A105020(n+1). - Wesley Ivan Hurt, Jan 07 2022
From G. C. Greubel, Mar 15 2023: (Start)
Sum_{k=0..n} T(n, k) = A002412(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*((1+(-1)^n)*A000384((n+2)/2) - (1- (-1)^n)*A000384((n+1)/2)). (End)
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}]]
(* to view table *) Table[t[i, j], {j, 0, 6}, {i, j+1, 10}]//TableForm (* Robert G. Wilson v, Jul 11 2005 *)
Table[(k+1)*(2*n-k+1), {n, 0, 15}, {k, 0, n}]//Flatten (* Roger L. Bagula, Aug 05 2008 *)
PROG
(Magma) [(k+1)*(2*n-k+1): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 15 2023
(SageMath)
def A105020(n, k): return (k+1)*(2*n-k+1)
flatten([[A105020(n, k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Mar 15 2023
KEYWORD
nonn,tabl,easy
AUTHOR
EXTENSIONS
More terms from Robert G. Wilson v, Jul 11 2005
STATUS
approved