A062708 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 0,2,...

0, 2, 13, 33, 62, 100, 147, 203, 268, 342, 425, 517, 618, 728, 847, 975, 1112, 1258, 1413, 1577, 1750, 1932, 2123, 2323, 2532, 2750, 2977, 3213, 3458, 3712, 3975, 4247, 4528, 4818, 5117, 5425, 5742, 6068, 6403, 6747, 7100, 7462, 7833, 8213, 8602, 9000

Offset: 0

Author

Floor van Lamoen, Jul 21 2001

Keywords

• nonn
• easy

Examples

			The spiral begins:
.
            15
            / \
          16  14
          /     \
        17   3  13
        /   / \   \
      18   4   2  12
      /   /     \   \
    19   5   0---1  11
    /   /             \
  20   6---7---8---9--10
.
From _Vincenzo Librandi_, Aug 07 2010: (Start)
a(1) = 9*1 +  0 - 7 =  2;
a(2) = 9*2 +  2 - 7 = 13;
a(3) = 9*3 + 13 - 7 = 33. (End)
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Milan Janjic, Two Enumerative Functions [broken link]
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A051682.

Cf. A218470.

Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this is case k=9).

Programs

GAP

List([0..50], n-> n*(9*n-5)/2); # G. C. Greubel, Sep 02 2019

Magma

[n*(9*n-5)/2: n in [0..50]]; // G. C. Greubel, Sep 02 2019

Maple

seq(n*(9*n-5)/2, n=0..50); # G. C. Greubel, Sep 02 2019

Mathematica

Table[n*(9*n-5)/2, {n,0,50}] (* G. C. Greubel, Sep 02 2019 *)
nxt[{n_,a_}]:={n+1,9(n+1)+a-7}; NestList[nxt,{0,0},50][[All,2]] (* Harvey P. Dale, Apr 11 2022 *)

PARI

a(n)=n*(9*n-5)/2 \\ Charles R Greathouse IV, Jun 17 2017

Sage

[n*(9*n-5)/2 for n in (0..50)] # G. C. Greubel, Sep 02 2019

Formula

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

a(n) = 9*n + a(n-1) - 7 (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*(2+7*x)/(1-x)^3. (End)

a(n) = A218470(9n+1). - Philippe Deléham, Mar 27 2013

E.g.f.: x*(4 + 9*x)*exp(x)/2. - G. C. Greubel, Sep 02 2019