OFFSET
0,3
COMMENTS
A triangular form based on the Hex number recursion: a(n) = 2*a(n-1) - a(n-1) + 6: A003215 form as generalized to Integer m.
A solution for the general type for m held constant: a(n) = 2*a(n-1) - a(n-2) + m, with first two values as {1, 1+m}.
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
T(n, k) = 1 + k*binomial(n+1,2).
EXAMPLE
Triangle begins as:
1;
1, 2;
1, 4, 7;
1, 7, 13, 19;
1, 11, 21, 31, 41;
1, 16, 31, 46, 61, 76;
MAPLE
seq(seq( 1 + k*binomial(n+1, 2), k=0..n), n=0..10); # G. C. Greubel, Nov 21 2019
MATHEMATICA
f[n_Integer] = Module[{a}, a[n]/.RSolve[{a[n]==2*a[n-1]-a[n-2]+m, a[0] ==1, a[1]==1+m}, a[n], n][[1]]//FullSimplify] (* formula of triangle *)
Table[Table[1+k*n*(1+n)/2, {k, 0, n}], {n, 0, 10}]//Flatten
PROG
(PARI) T(n, k) = 1 + k*binomial(n+1, 2); \\ G. C. Greubel, Nov 21 2019
(Magma) [1+k*Binomial(n+1, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 21 2019
(Sage) [[1+k*binomial(n+1, 2) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 21 2019
(GAP) Flat(List([0..10], n-> List([0..n], k-> 1 + k*Binomial(n+1, 2) ))); # G. C. Greubel, Nov 21 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Sep 08 2006
EXTENSIONS
Edited by G. C. Greubel, Nov 21 2019
STATUS
approved