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A084077 Length of list created by n substitutions k -> Range(-abs(k+1), abs(k-1)) starting with {1}.

%I A084077 #18 Nov 27 2022 10:43:12
%S A084077 1,3,11,41,159,633,2575,10657,44735,190017,815231,3527681,15378687,
%T A084077 67478401,297777407,1320753665,5884652543,26326301697,118211192831,
%U A084077 532574203905,2406726828031,10906541371393
%N A084077 Length of list created by n substitutions k -> Range(-abs(k+1), abs(k-1)) starting with {1}.
%H A084077 Vincenzo Librandi, <a href="/A084077/b084077.txt">Table of n, a(n) for n = 0..200</a>
%F A084077 invOGF satisfies n - (1+3*n)*a(n) - 2*n*(1+n)*a(n)^2 - 2*n^2*a(n)^3 = 0. [Is it true?]
%F A084077 Recurrence: (n+3)*(7*n-4)*a(n) = 3*(7*n^2+3*n+2)*a(n-1) + 2*(28*n^2+5*n-9)*a(n-2) + 4*(n-1)*(7*n+3)*a(n-3). - _Vaclav Kotesovec_, Oct 14 2012
%F A084077 a(n) ~ sqrt(52+34*sqrt(2))*(2+2*sqrt(2))^n/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 14 2012
%e A084077 {1}, {-2,-1,0}, {-1,0,1,2,3,0,1,2,-1,0,1}
%t A084077 Length/@Flatten/@NestList[ # /. k_Integer:>Range[ -Abs[k+1], Abs[k-1]]&, {1}, 8]
%t A084077 Flatten[{1,RecurrenceTable[{(n+3)*(7*n-4)*a[n] == 3*(7*n^2+3*n+2)*a[n-1] + 2*(28*n^2+5*n-9)*a[n-2] + 4*(n-1)*(7*n+3)*a[n-3],a[1]==3,a[2]==11,a[3]==41},a,{n,20}]}] (* _Vaclav Kotesovec_, Oct 14 2012 *)
%o A084077 (Magma) I:=[1,3,11]; [n le 3 select I[n] else (3*(7*n^2 -11*n +6)*Self(n-1) + 2*(28*n^2 -51*n +14)*Self(n-2) + 4*(n-2)*(7*n-4)*Self(n-3))/((n+2)*(7*n-11)): n in [1..41]]; // _G. C. Greubel_, Nov 23 2022
%o A084077 (SageMath)
%o A084077 @CachedFunction
%o A084077 def a(n):  # a = A084077
%o A084077     if (n<3): return (1,3,11)[n]
%o A084077     else: return (3*(7*n^2 +3*n +2)*a(n-1) + 2*(28*n^2 +5*n -9)*a(n-2) + 4*(n-1)*(7*n+3)*a(n-3))/((n+3)*(7*n-4))
%o A084077 [a(n) for n in range(31)] # _G. C. Greubel_, Nov 23 2022
%Y A084077 Cf. A084075, A084076, A084078.
%K A084077 nonn
%O A084077 0,2
%A A084077 _Wouter Meeussen_, May 11 2003