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%I #32 Feb 26 2024 01:57:12
%S 1,0,4,5,11,14,22,27,37,44,56,65,79,90,106,119,137,152,172,189,211,
%T 230,254,275,301,324,352,377,407,434,466,495,529,560,596,629,667,702,
%U 742,779,821,860,904,945,991,1034,1082,1127,1177,1224,1276,1325,1379,1430
%N a(0) = 1; for n >= 1, a(n) = n^2 - a(n-1).
%C A sequence given by a recurrence that is almost polynomial; it cannot be expressed as a polynomial, but is bounded by n^2.
%C If we let a(0) = 0, the triangular numbers result; a typo led to the new sequence.
%H Vincenzo Librandi, <a href="/A125577/b125577.txt">Table of n, a(n) for n = 0..1000</a>
%F O.g.f.: (-1+2*x-4*x^2+x^3)/((-1+x)^3*(1+x)). a(n) = -n-1+(-1)^n+A000217(n+1). - _R. J. Mathar_, Dec 05 2007
%F a(n) = n*(n+1)/2 + (-1)^n = A000217(n) + (-1)^n. - _Franklin T. Adams-Watters_, Jul 13 2014
%F E.g.f.: exp(x)*(x+x^2/2) + exp(-x). - _Franklin T. Adams-Watters_, Jul 13 2014
%e a(0)=1, so a(1) = 1^2 - 1 = 0; a(2) = 2^2 - 0 = 4; a(3) = 9 - 4 = 5; etc.
%t a[0] := 1 a[n_] := n^2 - a[n - 1]
%t CoefficientList[Series[(-1 + 2 x - 4 x^2 + x^3)/((-1 + x)^3 (1 + x)), {x, 0, 50}], x] (* _Vincenzo Librandi_, May 19 2014 *)
%o (Python)
%o a = 1
%o for n in range(1,77):
%o print(a, end=',')
%o a = n*n - a
%o (Magma) [1] cat [n le 1 select n-1 else n^2-Self(n-1): n in [1..50]]; // _Vincenzo Librandi_, May 19 2014
%Y Cf. A000217.
%K nonn,easy
%O 0,3
%A John C. George (John.George(AT)ENMU.edu), Jan 03 2007
%E Name corrected by _Alex Ratushnyak_, Aug 03 2012
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