A105020 - OEIS

%I #32 Mar 15 2023 11:40:27

%S 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,

%T 45,48,49,15,28,39,48,55,60,63,64,17,32,45,56,65,72,77,80,81,19,36,51,

%U 64,75,84,91,96,99,100,21,40,57,72,85,96,105,112,117,120,121

%N Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.

%C 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

%C 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

%C 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

%D R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

%H G. C. Greubel, <a href="/A105020/b105020.txt">Antidiagonals n = 0..50, flattened</a>

%F 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

%F a(n) = A128076(n+1) * A105020(n+1). - _Wesley Ivan Hurt_, Jan 07 2022

%F From _G. C. Greubel_, Mar 15 2023: (Start)

%F Sum_{k=0..n} T(n, k) = A002412(n+1).

%F 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)

%e Array begins:

%e   1  4  9 16 25 36  49  64  81 100 ...

%e   3  8 15 24 35 48  63  80  99 120 ...

%e   5 12 21 32 45 60  77  96 117 140 ...

%e   7 16 27 40 55 72  91 112 135 160 ...

%e   9 20 33 48 65 84 105 128 153 180 ...

%e   ...

%e Triangle begins:

%e    1;

%e    3,  4;

%e    5,  8,  9;

%e    7, 12, 15, 16;

%e    9, 16, 21, 24, 25;

%e   11, 20, 27, 32, 35, 36;

%e   13, 24, 33, 40, 45, 48, 49;

%e   15, 28, 39, 48, 55, 60, 63, 64;

%e   17, 32, 45, 56, 65, 72, 77, 80, 81;

%e   19, 36, 51, 64, 75, 84, 91, 96, 99, 100;

%t t[n_, m_]:= (n^2 - m^2); Flatten[Table[t[i, j], {i,12}, {j,i-1,0,-1}]]

%t (* to view table *) Table[t[i, j], {j,0,6}, {i,j+1,10}]//TableForm (* _Robert G. Wilson v_, Jul 11 2005 *)

%t Table[(k+1)*(2*n-k+1), {n,0,15}, {k,0,n}]//Flatten (* _Roger L. Bagula_, Aug 05 2008 *)

%o (Magma) [(k+1)*(2*n-k+1): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Mar 15 2023

%o (SageMath)

%o def A105020(n,k): return (k+1)*(2*n-k+1)

%o flatten([[A105020(n,k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Mar 15 2023

%Y Rows give A000290, A005563, A028347, A028560, A028566, A098603, A098847, A098848, A098849, A098850.

%Y Columns give A005408, A008586, A016945, A008590, A017329, A008594, A008598, A008602, A008606, A000567.

%Y Diagonals give A033428, A045944, A067725.

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

%K nonn,tabl,easy

%O 0,2

%A _Andrew S. Plewe_ and _Franklin T. Adams-Watters_, Jul 11 2005

%E More terms from _Robert G. Wilson v_, Jul 11 2005