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A365904
Triangle read by rows T(n,k) = n^2 - binomial(k+1,2), n>=1, k<n.
2
1, 4, 3, 9, 8, 6, 16, 15, 13, 10, 25, 24, 22, 19, 15, 36, 35, 33, 30, 26, 21, 49, 48, 46, 43, 39, 34, 28, 64, 63, 61, 58, 54, 49, 43, 36, 81, 80, 78, 75, 71, 66, 60, 53, 45, 100, 99, 97, 94, 90, 85, 79, 72, 64, 55, 121, 120, 118, 115, 111, 106, 100, 93, 85, 76, 66
OFFSET
1,2
COMMENTS
T(n,k) is the number of points in a rhombus that has 1 point in the first row, then 2 in the second, and following until the n-th row with n points, and then n-1 in the following row, n-2 in the following to end with a row with k+1 points.
T(n,0) are the perfect squares (A000290).
T(n,n-1) are the triangular numbers (A000217).
LINKS
Zach Wissner-Gross, Can You Shape the Peloton?, Fiddler on the Proof, Sep 22, 2023.
FORMULA
G.f.: x*(1 + x - 4*x^2*y + x^3*y^2 + x^4*y^2)/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Oct 05 2023
EXAMPLE
Triangle begins:
1;
4, 3;
9, 8, 6;
16, 15, 13, 10;
25, 24, 22, 19, 15;
36, 35, 33, 30, 26, 21;
49, 48, 46, 43, 39, 34, 28;
64, 63, 61, 58, 54, 49, 43, 36;
81, 80, 78, 75, 71, 66, 60, 53, 45;
100, 99, 97, 94, 90, 85, 79, 72, 64, 55;
...
CROSSREFS
Row sums give A004068.
Cf. A214859.
Sequence in context: A240199 A094728 A212001 * A370290 A275473 A131805
KEYWORD
nonn,tabl
AUTHOR
Joan Llobera Querol, Sep 22 2023
STATUS
approved