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 A081489 a(n) = n*(2*n^2 -3*n +7)/6 = C(n, 1) + C(n, 2) + 2*C(n, 3). 10
 1, 3, 8, 18, 35, 61, 98, 148, 213, 295, 396, 518, 663, 833, 1030, 1256, 1513, 1803, 2128, 2490, 2891, 3333, 3818, 4348, 4925, 5551, 6228, 6958, 7743, 8585, 9486, 10448, 11473, 12563, 13720, 14946, 16243, 17613, 19058, 20580, 22181, 23863, 25628 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Diagonal of triangle in A081491. First difference is given by A002522 = n^2 + 1. Second difference is odd numbers given by A005408. With offset 0, this is the binomial transform of (0,1,1,2,0,0,0,...). - Paul Barry, Jul 03 2003 Equals row sums of triangle A144337. - Gary W. Adamson, Sep 18 2008 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 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Jerrold W. Grossman and Serge Kruk, Solution to Problem 11901 in The American Mathematical Monthly, 2016 Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). FORMULA 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 E.g.f.: exp(x)*(x +x^2/2 +x^3/3). O.g.f.: x*(1-x+2*x^2)/(1-x)^4. - Colin Barker, Jul 28 2012 a(n) = 2*n + Sum_{i=1..n} (i^2 - 2*i). - Wesley Ivan Hurt, Feb 25 2014 MAPLE with(combinat):a:=n->sum(fibonacci(3, i), i=0..n): seq(a(n), n=0..42); # Zerinvary Lajos, Mar 20 2008 MATHEMATICA 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 *) PROG (PARI) my(x='x+O(x^50)); Vec(serlaplace(exp(x)*(x+x^2/2+x^3/3))) (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 (Sage) [n*(2*n^2-3*n+7)/6 for n in (1..50)] # G. C. Greubel, Aug 13 2019 (GAP) List([1..50], n-> n*(2*n^2-3*n+7)/6); # G. C. Greubel, Aug 13 2019 CROSSREFS Cf. A002522, A005408, A081490, A081491, A081492, A144337. Sequence in context: A083726 A319006 A212589 * A055278 A293349 A036628 Adjacent sequences:  A081486 A081487 A081488 * A081490 A081491 A081492 KEYWORD nonn,easy AUTHOR Amarnath Murthy, Mar 25 2003 EXTENSIONS More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 29 2003 New name, using the formulas of Paul Barry, Joerg Arndt, Feb 28 2014 STATUS approved

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Last modified April 22 22:19 EDT 2021. Contains 343197 sequences. (Running on oeis4.)