|
%I #55 Sep 08 2022 08:45:09
%S 1,3,8,18,35,61,98,148,213,295,396,518,663,833,1030,1256,1513,1803,
%T 2128,2490,2891,3333,3818,4348,4925,5551,6228,6958,7743,8585,9486,
%U 10448,11473,12563,13720,14946,16243,17613,19058,20580,22181,23863,25628
%N a(n) = n*(2*n^2 -3*n +7)/6 = C(n, 1) + C(n, 2) + 2*C(n, 3).
%C Diagonal of triangle in A081491.
%C First difference is given by A002522 = n^2 + 1. Second difference is odd numbers given by A005408.
%C With offset 0, this is the binomial transform of (0,1,1,2,0,0,0,...). - _Paul Barry_, Jul 03 2003
%C Equals row sums of triangle A144337. - _Gary W. Adamson_, Sep 18 2008
%C a(n) = sum of first (n-1) squares + n; e.g., a(5) = 35 = (1 + 4 + 9 + 16 + 5). - _Gary W. Adamson_, Aug 27 2010
%C Equals the number of functions from {1,2,...,n} to {1,2,...,n} that occur as compositions of U(x) = min{x+1,n} and D(x) = max{x-1,1}, including the identity function (the empty composition). Problem 11901 in The American Mathematical Monthly, volume 123, page 399, April 2016), proposed by _Don Knuth_, asked for the count of such functions (solution submitted to Monthly by Jerrold W. Grossman and Serge Kruk, August 21, 2016). - _Jerrold Grossman_, Aug 21 2016
%H Vincenzo Librandi, <a href="/A081489/b081489.txt">Table of n, a(n) for n = 1..1000</a>
%H Jerrold W. Grossman and Serge Kruk, <a href="/A081489/a081489.pdf">Solution to Problem 11901 in The American Mathematical Monthly</a>, 2016
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = n*(2*n^2 -3*n +7)/6 = C(n, 1) + C(n, 2) + 2*C(n, 3). - _Paul Barry_, Jul 03 2003
%F E.g.f.: exp(x)*(x +x^2/2 +x^3/3).
%F O.g.f.: x*(1-x+2*x^2)/(1-x)^4. - _Colin Barker_, Jul 28 2012
%F a(n) = 2*n + Sum_{i=1..n} (i^2 - 2*i). - _Wesley Ivan Hurt_, Feb 25 2014
%p with(combinat):a:=n->sum(fibonacci(3,i), i=0..n): seq(a(n), n=0..42); # _Zerinvary Lajos_, Mar 20 2008
%t Table[n*(2*n^2-3*n+7)/6, {n,50}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 07 2008, modified by _G. C. Greubel_, Aug 13 2019 *)
%o (PARI) my(x='x+O(x^50)); Vec(serlaplace(exp(x)*(x+x^2/2+x^3/3)))
%o (Magma) I:=[1,3,8,18]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Feb 28 2014
%o (Sage) [n*(2*n^2-3*n+7)/6 for n in (1..50)] # _G. C. Greubel_, Aug 13 2019
%o (GAP) List([1..50], n-> n*(2*n^2-3*n+7)/6); # _G. C. Greubel_, Aug 13 2019
%Y Cf. A002522, A005408, A081490, A081491, A081492, A144337.
%K nonn,easy
%O 1,2
%A _Amarnath Murthy_, Mar 25 2003
%E More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 29 2003
%E New name, using the formulas of _Paul Barry_, _Joerg Arndt_, Feb 28 2014
|
