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A064043 Number of length 3 walks on an n-dimensional hypercubic lattice starting at the origin and staying in the nonnegative part.

%I A064043 #33 Dec 26 2024 15:20:22
%S A064043 0,3,18,51,108,195,318,483,696,963,1290,1683,2148,2691,3318,4035,4848,
%T A064043 5763,6786,7923,9180,10563,12078,13731,15528,17475,19578,21843,24276,
%U A064043 26883,29670,32643,35808,39171,42738,46515,50508,54723,59166,63843,68760,73923,79338
%N A064043 Number of length 3 walks on an n-dimensional hypercubic lattice starting at the origin and staying in the nonnegative part.
%H A064043 Harry J. Smith, <a href="/A064043/b064043.txt">Table of n, a(n) for n = 0..1000</a>
%H A064043 D. R. L. Brown, <a href="https://ia.cr/2015/375">Bounds on surmising remixed keys</a>, IACR, Report 2015/375, 2015-2016. See Table 1.
%H A064043 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A064043 a(n) = n*(n^2 + 3*n - 1) = n*A014209(n) = A064044(n, 3).
%F A064043 a(n) = a(n-1) + 3*A002378(n-1) + 6*A001477(n-1) + 3*A000012(n-1).
%F A064043 G.f.: 3*x*(1+2*x-x^2)/(1-x)^4. - _Colin Barker_, Apr 19 2012
%F A064043 E.g.f.: (x^3 + 6*x^2 + 3*x)*exp(x). - _G. C. Greubel_, Jul 20 2017
%F A064043 a(n) = A084990(n)/3. - _Alois P. Heinz_, Jul 21 2017
%p A064043 seq(sum(3*n+n^2-1, k=1..n), n=0..39); # _Zerinvary Lajos_, Jan 28 2008
%t A064043 Table[n*(n^2 + 3n -1), {n,0,50}] (* _G. C. Greubel_, Jul 20 2017 *)
%o A064043 (PARI) a(n) = { n*(n^2 + 3*n - 1) } \\ _Harry J. Smith_, Sep 06 2009
%Y A064043 Number of walks length 0, 1 and 2 are A000012, A001477 and A002378.
%Y A064043 Cf. A084990.
%K A064043 nonn,easy
%O A064043 0,2
%A A064043 _Henry Bottomley_, Aug 23 2001