A185788 Sum of the first k-1 numbers in the k-th column of the natural number array A000027, by antidiagonals.
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
Offset: 1
Examples
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...
Links
- 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)
Programs
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Mathematica
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 *) -
PARI
for(n=1,50, print1((n-1)*(7*n^2 - 11*n + 6)/6, ", ")) \\ G. C. Greubel, Jul 12 2017
Formula
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
Comments