A062708 - OEIS

%I #51 Aug 18 2024 03:17:13

%S 0,2,13,33,62,100,147,203,268,342,425,517,618,728,847,975,1112,1258,

%T 1413,1577,1750,1932,2123,2323,2532,2750,2977,3213,3458,3712,3975,

%U 4247,4528,4818,5117,5425,5742,6068,6403,6747,7100,7462,7833,8213,8602,9000

%N 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,...

%H G. C. Greubel, <a href="/A062708/b062708.txt">Table of n, a(n) for n = 0..1000</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a> [broken link]

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

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

%F a(n) = 9*n + a(n-1) - 7 (with a(0)=0). - _Vincenzo Librandi_, Aug 07 2010

%F From _Colin Barker_, Jul 07 2012: (Start)

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

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

%F a(n) = A218470(9n+1). - _Philippe Deléham_, Mar 27 2013

%F E.g.f.:  x*(4 + 9*x)*exp(x)/2. - _G. C. Greubel_, Sep 02 2019

%e The spiral begins:

%e .

%e             15

%e             / \

%e           16  14

%e           /     \

%e         17   3  13

%e         /   / \   \

%e       18   4   2  12

%e       /   /     \   \

%e     19   5   0---1  11

%e     /   /             \

%e   20   6---7---8---9--10

%e .

%e From _Vincenzo Librandi_, Aug 07 2010: (Start)

%e a(1) = 9*1 +  0 - 7 =  2;

%e a(2) = 9*2 +  2 - 7 = 13;

%e a(3) = 9*3 + 13 - 7 = 33. (End)

%p seq(n*(9*n-5)/2, n=0..50); # _G. C. Greubel_, Sep 02 2019

%t Table[n*(9*n-5)/2, {n,0,50}] (* _G. C. Greubel_, Sep 02 2019 *)

%t nxt[{n_,a_}]:={n+1,9(n+1)+a-7}; NestList[nxt,{0,0},50][[All,2]] (* _Harvey P. Dale_, Apr 11 2022 *)

%o (PARI) a(n)=n*(9*n-5)/2 \\ _Charles R Greathouse IV_, Jun 17 2017

%o (Magma) [n*(9*n-5)/2: n in [0..50]]; // _G. C. Greubel_, Sep 02 2019

%o (Sage) [n*(9*n-5)/2 for n in (0..50)] # _G. C. Greubel_, Sep 02 2019

%o (GAP) List([0..50], n-> n*(9*n-5)/2); # _G. C. Greubel_, Sep 02 2019

%Y Cf. A051682.

%Y Cf. A218470.

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

%K nonn,easy

%O 0,2

%A _Floor van Lamoen_, Jul 21 2001