A099429 A Jacobsthal-Lucas convolution.
0, 0, 2, 3, 12, 25, 66, 147, 344, 765, 1710, 3751, 8196, 17745, 38234, 81915, 174768, 371365, 786438, 1660239, 3495260, 7340025, 15379122, 32156323, 67108872, 139810125, 290805086, 603979767, 1252698804, 2594876065, 5368709130, 11095332171, 22906492256
Offset: 0
Examples
G.f. = 2*x^2 + 3*x^3 + 12*x^4 + 25*x^5 + 66*x^6 + 147*x^7 + 344*x^8 + ... If we concatenate the lexicographically ordered bit strings of length 4, we obtain the expression 0000000100100011010001010110011110001001101010111100110111101111, and we see that the substring 11 appears a total of a(4) = 12 times, with overlapping substrings not being considered as being separate. - _John M. Campbell_, Jan 18 2019
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-4).
Programs
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Magma
m:=40; R
:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!( x^2*(2-x)/(1-x-2*x^2)^2 )); // G. C. Greubel, Feb 25 2019 -
Mathematica
CoefficientList[Series[x^2*(2-x)/(1-x-2x^2)^2, {x, 0, 32}], x] (* Michael De Vlieger, Jan 18 2019 *) -
PARI
{a(n) = if( n>=0, polcoeff( x^2*(2-x)/((1+x)*(1-2*x))^2 + x*O(x^n), n), polcoeff( x*(1-2*x)/((1+x)*(2-x))^2 + x*O(x^-n), -n) )}; /* Michael Somos, Jun 02 2014 */ -
Sage
(x^2*(2-x)/(1-x-2*x^2)^2).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 25 2019
Formula
G.f.: x^2*(2-x)/(1-x-2*x^2)^2. [Typo corrected by Colin Barker, Jun 16 2012]
a(n) = Sum_{k=0..n} J(n-k)*(2^(k-1) -(-1)^k +0^k/2).
a(n) = Sum_{k=0..n+1} J(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2.
a(n) = A036289(n)/6 +(-1)^n*n/3. - R. J. Mathar, Sep 21 2012
a(-n) = (-2)^(-n-1) * A193449(n) for all n in Z. - Michael Somos, Jun 02 2014
Comments