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

1,2

Comments

See A185787.

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)

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

Example

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...

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 *)

Prog

(PARI) for(n=1, 50, print1((n-1)*(7*n^2 - 11*n + 6)/6, ", ")) \\ G. C. Greubel, Jul 12 2017

Crossrefs

Cf. A000027, A185787, A079824.

Sequence in context: A294464 A366618 A330781 * A305864 A324027 A035597

Adjacent sequences: A185785 A185786 A185787 * A185789 A185790 A185791

Keyword

nonn,easy

Author

Clark Kimberling, Feb 03 2011

Status

approved