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A185788
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Sum of the first k-1 numbers in the k-th column of the natural number array A000027, by antidiagonals.
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3
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0, 2, 12, 37, 84, 160, 272, 427, 632, 894, 1220, 1617, 2092, 2652, 3304, 4055, 4912, 5882, 6972, 8189, 9540, 11032, 12672, 14467, 16424, 18550, 20852, 23337, 26012, 28884, 31960, 35247, 38752, 42482, 46444, 50645, 55092, 59792, 64752, 69979, 75480, 81262, 87332, 93697, 100364, 107340, 114632, 122247, 130192, 138474
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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a(n) = (n-1)*(7*n^2 - 11*n + 6)/6. - Corrected by Manfred Arens, Mar 11 2016
G.f.: x^2*(2+4*x+x^2) / (x-1)^4 . - R. J. Mathar, Aug 23 2012
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EXAMPLE
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Start from
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...
Block out all terms starting at and below the main diagonal then sum up the remaining terms.
.....2.....4.....7...11...16...22...29...
...........8....12...17...23...30...38...
................18...24...31...39...48...
.....................32...40...49...59...
..........................50...60...71...
...............................72...84...
....................................98...
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MATHEMATICA
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f[n_, k_]:=n+(n+k-2)(n+k-1)/2;
s[k_]:=Sum[f[n, k], {n, 1, k-1}];
Factor[s[k]]
Table[s[k], {k, 1, 70}]
Table[(n - 1)*(7*n^2 - 11*n + 6)/6, {n, 1, 50}] (* G. C. Greubel, Jul 12 2017 *)
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PROG
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(PARI) for(n=1, 50, print1((n-1)*(7*n^2 - 11*n + 6)/6, ", ")) \\ G. C. Greubel, Jul 12 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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