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%I #61 Feb 14 2023 18:13:37
%S 1,1,7,25,61,121,211,337,505,721,991,1321,1717,2185,2731,3361,4081,
%T 4897,5815,6841,7981,9241,10627,12145,13801,15601,17551,19657,21925,
%U 24361,26971,29761,32737,35905,39271,42841,46621,50617,54835,59281,63961
%N a(n) = n^3 - n + 1.
%C Smallest of n consecutive odd numbers whose sum is n^4. (n^k can be expressed as the sum of n consecutive odd numbers the smallest of which is given by n^(k-1)-n+1.)
%D T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.
%H Harry J. Smith, <a href="/A061600/b061600.txt">Table of n, a(n) for n = 0..1000</a>
%H Leo Tavares, <a href="/A061600/a061600.jpg">Illustration: Diamond Chains</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: (1-3*x+9*x^2-x^3)/(1 - x)^4. a(-n) = -A126420(n). - _Bruno Berselli_, Aug 29 2011
%F a(n) = 1 + Sum_{k=1..n} 3*(k-1)*k. - _Luce ETIENNE_ and _Michel Marcus_, Nov 01 2014
%F E.g.f.: exp(x)*(1 + 3*x^2 + x^3). - _Nikolaos Pantelidis_, Feb 13 2023
%e a(5) = 121 = 5^3 - 5 + 1. We have 121 + 123 + 125 + 127 + 129 = 625 = 5^4.
%t Table[n^3 - n + 1, {n, 0, 40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2012 *)
%t LinearRecurrence[{4,-6,4,-1},{1,1,7,25},50] (* _Harvey P. Dale_, Aug 17 2020 *)
%o (PARI) a(n) = n^3 - n + 1); \\ _Harry J. Smith_, Jul 25 2009
%o (Magma) [n^3 - n + 1: n in [0..40]]; // _Vincenzo Librandi_, Aug 29 2011
%Y Cf. A126420.
%K nonn,easy
%O 0,3
%A _Amarnath Murthy_, May 19 2001
%E Offset changed from 1 to 0 by _Harry J. Smith_, Jul 25 2009
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