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

1, 3, 11, 41, 159, 633, 2575, 10657, 44735, 190017, 815231, 3527681, 15378687, 67478401, 297777407, 1320753665, 5884652543, 26326301697, 118211192831, 532574203905, 2406726828031, 10906541371393

Offset: 0

Author

Wouter Meeussen, May 11 2003

Keywords

• nonn

Examples

			{1}, {-2,-1,0}, {-1,0,1,2,3,0,1,2,-1,0,1}
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A084075, A084076, A084078.

Programs

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

Mathematica

Length/@Flatten/@NestList[ # /. k_Integer:>Range[ -Abs[k+1], Abs[k-1]]&, {1}, 8]
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 *)

SageMath

@CachedFunction
def a(n):  # a = A084077
    if (n<3): return (1,3,11)[n]
    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))
[a(n) for n in range(31)] # G. C. Greubel, Nov 23 2022

Formula

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?]

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

a(n) ~ sqrt(52+34*sqrt(2))*(2+2*sqrt(2))^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012