A344235 Triangle T from the array A(k, n) giving the sums of k+1 consecutive squares starting with n^2, read as upwards antidiagonals, for k >= 0 and n >= 0.

0, 1, 1, 5, 5, 4, 14, 14, 13, 9, 30, 30, 29, 25, 16, 55, 55, 54, 50, 41, 25, 91, 91, 90, 86, 77, 61, 36, 140, 140, 139, 135, 126, 110, 85, 49, 204, 204, 203, 199, 190, 174, 149, 113, 64, 285, 285, 284, 280, 271, 255, 230, 194, 145, 81, 385, 385, 384, 380, 371, 355, 330, 294, 245, 181, 100

Offset

0,4

Comments

Motivated by a proposal from Charlie Marion.

References

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2nd ed.; Addison-Wesley, 1994, pp. 283-290.

Links

Table of n, a(n) for n=0..65.

Formula

A(k, n) = Sum_{j=0..k} (n+j)^2, for k >= 0, n >= 0.

A(k, n) = Sum_{j=0..n+k} j^2 - (2*n-1)*n*(n-1)/3! = S(n+k) - (2*n-1)*n*(n-1)/3!, with S(n+k) = (1/3)*Sum_{j=0..2} binomial(3, j)*B_j*(n+k+1)^(3-j), with the Bernoulli numbers A027641 / A027642 (see Graham et al., pp. 283-290).

Recurrence for sequence of row k: A(k, n) = A(k, n-1) + (k+1)*(2*n + k - 1), n >= 1, with A(k, 0) = (2*k+1)*(k+1)*k/3!, for k >= 0.

Example

The array A(k, n) begins:
k \ n    0   1   2   3   4   5    6    7    8    9   10 ...
-----------------------------------------------------------
0:       0   1   4   9  16  25   36   49   64   81  100 ...
1:       1   5  13  25  41  61   85  113  145  181  221 ...
2:       5  14  29  50  77 110  149  194  245  302  365 ...
3:      14  30  54  86 126 174  230  294  366  446  534 ...
4:      30  55  90 135 190 255  330  415  510  615  730 ...
5:      55  91 139 199 271 355  451  559  679  811  955 ...
6:      91 140 203 280 371 476  595  728  875 1036 1211 ...
7:     140 204 284 380 492 620  764  924 1100 1292 1500 ...
8:     204 285 384 501 636 789  960 1149 1356 1581 1824 ...
9:     285 385 505 645 805 985 1185 1405 1645 1905 2185 ...
...
-----------------------------------------------------------
The triangle T(m, n) begins:
m \ n   0   1   2   3   4   5   6   7   8  9 ...
-----------------------------------------------------------
0:      0
1:      1   1
2:      5   5   4
3:     14  14  13   9
4:     30  30  29  25  16
5:     55  55  54  50  41  25
6:     91  91  90  86  77  61  36
7:    140 140 139 135 126 110  85  49
8:    204 204 203 199 190 174 149 113  64
9:    285 285 284 280 271 255 230 194 145 81
...
----------------------------------------------------------

Crossrefs

Rows of array A, diagonals of T: A000290, A001844, A005918(n+1), A027575, A027578, A027865, A260637, A276026, ...

Columns of array A and T (without leading 0s): A000330, A000330(n+1), A168559(n+1), ...

Sequence in context: A011409 A255240 A168578 * A019253 A019173 A374753

Adjacent sequences: A344232 A344233 A344234 * A344236 A344237 A344238

Keyword

nonn,tabl,easy

Author

Wolfdieter Lang, May 27 2021

Status

approved