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

0,2

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).

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

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
.
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)

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 *)

Prog

(PARI) a(n)=n*(9*n-5)/2 \\ Charles R Greathouse IV, Jun 17 2017
(Magma) [n*(9*n-5)/2: n in [0..50]]; // G. C. Greubel, Sep 02 2019
(Sage) [n*(9*n-5)/2 for n in (0..50)] # G. C. Greubel, Sep 02 2019
(GAP) List([0..50], n-> n*(9*n-5)/2); # G. C. Greubel, Sep 02 2019

Crossrefs

Cf. A051682.

Cf. A218470.

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

Sequence in context: A084828 A100512 A051474 * A296293 A190816 A084910

Adjacent sequences: A062705 A062706 A062707 * A062709 A062710 A062711

Keyword

nonn,easy

Author

Floor van Lamoen, Jul 21 2001

Status

approved