A062725 Write 0,1,2,3,4,... in a triangular spiral, then a(n) is the sequence found by reading the terms along the line from 0 in the direction 0,7,...

0, 7, 23, 48, 82, 125, 177, 238, 308, 387, 475, 572, 678, 793, 917, 1050, 1192, 1343, 1503, 1672, 1850, 2037, 2233, 2438, 2652, 2875, 3107, 3348, 3598, 3857, 4125, 4402, 4688, 4983, 5287, 5600, 5922, 6253, 6593, 6942, 7300, 7667, 8043, 8428, 8822, 9225

Offset

0,2

Comments

Central terms of triangle A245300. - Reinhard Zumkeller, Jul 17 2014

Digital root of a(n) = A180597(n). - Gionata Neri, Apr 29 2015

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = n*(9*n+5)/2.

a(n) = 9*n + a(n-1) - 2 with a(0)=0. - Vincenzo Librandi, Aug 07 2010

From Colin Barker, Jul 07 2012: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f.: x*(7+2*x)/(1-x)^3. (End)

a(n) = A218470(9*n+6). - Philippe Deléham, Mar 27 2013

a(n) = a(n-1) + A017245(n-1), a(0)=0. - Gionata Neri, Apr 30 2015

Example

The spiral begins:
.
            15
            / \
          16  14
          /     \
        17   3  13
        /   / \   \
      18   4   2  12
      /   /     \   \
    19   5   0---1  11
    /   /             \
  20   6---7---8---9--10
.

Mathematica

s=0; lst={s}; Do[s+=n++ +7; AppendTo[lst, s], {n, 0, 7!, 9}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
CoefficientList[Series[x (7 + 2 x)/(1 - x)^3, {x, 0, 45}], x] (* Michael De Vlieger, Jan 11 2020 *)

Prog

(Haskell)
a062725 n = n * (9 * n + 5) `div` 2 -- Reinhard Zumkeller, Jul 17 2014
(PARI) a(n) = n*(9*n+5)/2 \\ Charles R Greathouse IV, Apr 30 2015

Crossrefs

Cf. A051682.

Sequence in context: A174590 A162290 A180044 * A147121 A098334 A299255

Adjacent sequences: A062722 A062723 A062724 * A062726 A062727 A062728

Keyword

nonn,easy

Author

Floor van Lamoen, Jul 21 2001

Extensions

Formula that confused indices corrected by R. J. Mathar, Jun 04 2010

Status

approved