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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,...

%I A062708 #51 Aug 18 2024 03:17:13
%S A062708 0,2,13,33,62,100,147,203,268,342,425,517,618,728,847,975,1112,1258,
%T A062708 1413,1577,1750,1932,2123,2323,2532,2750,2977,3213,3458,3712,3975,
%U A062708 4247,4528,4818,5117,5425,5742,6068,6403,6747,7100,7462,7833,8213,8602,9000
%N 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,...
%H A062708 G. C. Greubel, <a href="/A062708/b062708.txt">Table of n, a(n) for n = 0..1000</a>
%H A062708 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a> [broken link]
%H A062708 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A062708 a(n) = n*(9*n-5)/2.
%F A062708 a(n) = 9*n + a(n-1) - 7 (with a(0)=0). - _Vincenzo Librandi_, Aug 07 2010
%F A062708 From _Colin Barker_, Jul 07 2012: (Start)
%F A062708 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A062708 G.f.: x*(2+7*x)/(1-x)^3. (End)
%F A062708 a(n) = A218470(9n+1). - _Philippe Deléham_, Mar 27 2013
%F A062708 E.g.f.:  x*(4 + 9*x)*exp(x)/2. - _G. C. Greubel_, Sep 02 2019
%e A062708 The spiral begins:
%e A062708 .
%e A062708             15
%e A062708             / \
%e A062708           16  14
%e A062708           /     \
%e A062708         17   3  13
%e A062708         /   / \   \
%e A062708       18   4   2  12
%e A062708       /   /     \   \
%e A062708     19   5   0---1  11
%e A062708     /   /             \
%e A062708   20   6---7---8---9--10
%e A062708 .
%e A062708 From _Vincenzo Librandi_, Aug 07 2010: (Start)
%e A062708 a(1) = 9*1 +  0 - 7 =  2;
%e A062708 a(2) = 9*2 +  2 - 7 = 13;
%e A062708 a(3) = 9*3 + 13 - 7 = 33. (End)
%p A062708 seq(n*(9*n-5)/2, n=0..50); # _G. C. Greubel_, Sep 02 2019
%t A062708 Table[n*(9*n-5)/2, {n,0,50}] (* _G. C. Greubel_, Sep 02 2019 *)
%t A062708 nxt[{n_,a_}]:={n+1,9(n+1)+a-7}; NestList[nxt,{0,0},50][[All,2]] (* _Harvey P. Dale_, Apr 11 2022 *)
%o A062708 (PARI) a(n)=n*(9*n-5)/2 \\ _Charles R Greathouse IV_, Jun 17 2017
%o A062708 (Magma) [n*(9*n-5)/2: n in [0..50]]; // _G. C. Greubel_, Sep 02 2019
%o A062708 (Sage) [n*(9*n-5)/2 for n in (0..50)] # _G. C. Greubel_, Sep 02 2019
%o A062708 (GAP) List([0..50], n-> n*(9*n-5)/2); # _G. C. Greubel_, Sep 02 2019
%Y A062708 Cf. A051682.
%Y A062708 Cf. A218470.
%Y A062708 Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this is case k=9).
%K A062708 nonn,easy
%O A062708 0,2
%A A062708 _Floor van Lamoen_, Jul 21 2001