OFFSET
1,2
COMMENTS
This is one of many interesting sequences and arrays that stem from the natural number array A000027, of which a northwest corner is as follows:
1....2.....4.....7...11...16...22...29...
3....5.....8....12...17...23...30...38...
6....9....13....18...24...31...39...48...
10...14...19....25...32...40...49...59...
15...20...26....33...41...50...60...71...
21...27...34....42...51...61...72...84...
28...35...43....52...62...73...85...98...
Blocking out all terms below the main diagonal leaves columns whose sums comprise A185787. Deleting the main diagonal and then summing give A185787. Analogous treatments to the left of the main diagonal give A100182 and A101165. Further sequences obtained directly from this array are easily obtained using the following formula for the array: T(n,k)=n+(n+k-2)(n+k-1)/2.
Examples:
row 1: A000124
row 2: A022856
row 3: A016028
row 4: A145018
row 5: A077169
col 1: A000217
col 2: A000096
col 3: A034856
col 4: A055998
col 5: A046691
col 6: A052905
col 7: A055999
diag. (1,5,...) ...... A001844
diag. (2,8,...) ...... A001105
diag. (4,12,...)...... A046092
diag. (7,17,...)...... A056220
diag. (11,23,...) .... A132209
diag. (16,30,...) .... A054000
diag. (22,38,...) .... A090288
diag. (3,9,...) ...... A058331
diag. (6,14,...) ..... A051890
diag. (10,20,...) .... A005893
diag. (15,27,...) .... A097080
diag. (21,35,...) .... A093328
Let S(n,k) denote the n-th partial sum of column k. Then
S(n,k)=n*(n^2+3k*n+3*k^2-6*k+5)/6.
S(n,1)=n(n+1)(n+2)/6
S(n,2)=n(n+1)(n+5)/6
S(n,3)=n(n+2)(n+7)/6
S(n,4)=n(n^2+12n+29)/6
S(n,5)=n(n+5)(n+10)/6
S(n,6)=n(n+7)(n+11)/6
S(n,7)=n(n+10)(n+11)/6
Weight array of T: A144112
Accumulation array of T: A185506
Second rectangular sum array of T: A185507
Third rectangular sum array of T: A185508
Fourth rectangular sum array of T: A185509
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n)=n*(7*n^2-6*n+5)/6.
G.f.: x*(3*x^2+3*x+1)/(1-x)^4. - Vincenzo Librandi, Jul 04 2012
MATHEMATICA
f[n_, k_]:=n+(n+k-2)(n+k-1)/2;
s[k_]:=Sum[f[n, k], {n, 1, k}];
Factor[s[k]]
Table[s[k], {k, 1, 70}] (* A185787 *)
CoefficientList[Series[(3*x^2+3*x+1)/(1-x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Jul 04 2012 *)
PROG
(Magma) [n*(7*n^2-6*n+5)/6: n in [1..50]]; // Vincenzo Librandi, Jul 04 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 03 2011
EXTENSIONS
Edited by Clark Kimberling, Feb 25 2023
STATUS
approved