A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+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

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

G. C. Greubel, Antidiagonals n = 0..50, flattened

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

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.

Cf. A000384, A002412, A024352, A105020, A128076.

Sequence in context: A092997 A021747 A334631 * A286053 A112594 A188003

Adjacent sequences: A105017 A105018 A105019 * A105021 A105022 A105023

Keyword

nonn,tabl,easy

Author

Andrew S. Plewe and Franklin T. Adams-Watters, Jul 11 2005

Extensions

More terms from Robert G. Wilson v, Jul 11 2005

Status

approved