OFFSET
1,2
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened).
FORMULA
a(n) = A212014(n)/2.
Let R = floor(sqrt(8*n+1)) and S = floor(R/2) + R mod 2; then a(n) = binomial(S,3) + n + (n-binomial(S,2))*(S*(S+3)-2*n-2)/4. - Gerald Hillier, Jan 16 2018
T(n,k) = n*(n+1)*(n+2)/6 - (n-k)*(n-k+1)/2. - Davide Rotondo, Mar 10 2025
G.f.: x*y*(1 - x + x^2*(1 - 3*y) - x^5*y^3 + x^3*y*(1 + y) - x^4*y*(1 - 2*y))/((1 - x)^4*(1 - x*y)^4). - Stefano Spezia, Mar 10 2025
EXAMPLE
Example 1: written as a triangle in which row i is related to the (i-1)st level of nucleus. Triangle begins:
1;
3, 4;
7, 9, 10;
14, 17, 19, 20;
25, 29, 32, 34, 35;
41, 46, 50, 53, 55, 56;
63, 69, 74, 78, 81, 83, 84;
92, 99, 105, 110, 114, 117, 119, 120;
129, 137, 144, 150, 155, 159, 162, 164, 165;
175, 184, 192, 199, 205, 210, 214, 217, 219, 220;
...
Column 1 gives positive terms of A004006. Right border gives positive terms of A000292. Row sums give positive terms of A006325.
Example 2: written as an irregular triangle in which row j is related to the j-th shell of nucleus. Note that in this case row 4 has only one term. Triangle begins:
1;
3, 4;
7, 9, 10;
14;
17, 19, 20, 25;
29, 32, 34, 35, 41;
46, 50, 53, 55, 56, 63;
69, 74, 78, 81, 83, 84, 92;
99, 105, 110, 114, 117, 119, 120, 129;
137, 144, 150, 155, 159, 162, 164, 165, 175;
184, 192, 199, 205, 210, 214, 217, 219, 220, 231;
...
MATHEMATICA
Accumulate[Flatten[Range[Range[15], 1, -1]]] (* Paolo Xausa, Mar 15 2025 *)
PROG
(J)
row =: monad define
d=.>y
< |. (+/d)-d
)
; }. row"0 <\ +/\ 1+i.11 NB. Vanessa McHale (vamchale(AT)gmail.com), Mar 01 2025
(PARI) row(n) = vector(n, k, n*(n+1)*(n+2)/6 - (n-k)*(n-k+1)/2); \\ Michel Marcus, Mar 10 2025
CROSSREFS
KEYWORD
AUTHOR
Omar E. Pol, Jul 15 2012
EXTENSIONS
More terms from Michel Marcus, Mar 10 2025
STATUS
approved