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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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See A185787.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1)
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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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Cf. A000027, A185787, A079824.
Sequence in context: A192385 A294464 A330781 * A305864 A324027 A035597
Adjacent sequences: A185785 A185786 A185787 * A185789 A185790 A185791
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling, Feb 03 2011
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STATUS
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approved
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