|
%I #26 Dec 26 2022 09:47:58
%S 1,5,18,40,71,111,160,218,285,361,446,540,643,755,876,1006,1145,1293,
%T 1450,1616,1791,1975,2168,2370,2581,2801,3030,3268,3515,3771,4036,
%U 4310,4593,4885,5186,5496,5815,6143,6480,6826,7181,7545,7918,8300,8691,9091,9500,9918,10345,10781,11226,11680,12143,12615
%N a(n) = (9*n^2 - n)/2 + 1.
%C Diagonal of triangular spiral in A051682. The other 5 diagonals are given by A140064, A117625, A081267, A064225, A006137. See the link as well.
%C First differences are given by A017209.
%C 72*a(n) - 71 is a perfect square. - _Klaus Purath_, Jan 14 2022
%H Colin Barker, <a href="/A276819/b276819.txt">Table of n, a(n) for n = 0..1000</a>
%H Yuriy Sibirmovsky, <a href="/A276819/a276819.png">Six diagonals of the triangular spiral</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = (9*n^2 - n)/2 + 1.
%F a(n) = a(n-1) + 9*n - 5 with a(0) = 1.
%F From _Colin Barker_, Sep 18 2016: (Start)
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
%F G.f.: (1 + 2*x + 6*x^2)/(1 - x)^3. (End)
%F From _Klaus Purath_, Jan 14 2022: (Start)
%F a(n) = A006137(n) - n.
%F A003215(a(n)) - A003215(a(n)-3) = A002378(9*n-1). (End)
%F E.g.f.: exp(x)*(2 + 8*x + 9*x^2)/2. - _Stefano Spezia_, Dec 25 2022
%t Table[(9*n^2-n)/2+1, {n,0,100}]
%o (PARI) Vec((1+2*x+6*x^2)/(1-x)^3 + O(x^60)) \\ _Colin Barker_, Sep 18 2016
%o (PARI) a(n) = (9*n^2 - n)/2 + 1; \\ _Altug Alkan_, Sep 18 2016
%Y Cf. A002378, A003215, A006137, A017209, A051682, A064225, A081267, A117625, A140064.
%K nonn,easy
%O 0,2
%A _Yuriy Sibirmovsky_, Sep 18 2016
|
