|
|
A238846
|
|
Expansion of (1-2*x)/(1-3*x+x^2)^2.
|
|
4
|
|
|
1, 4, 13, 40, 120, 354, 1031, 2972, 8495, 24110, 68016, 190884, 533293, 1484020, 4115185, 11375764, 31358376, 86223942, 236540915, 647556620, 1769374931, 4826148314, 13142564448, 35736448200, 97037995225, 263156279524, 712795854421, 1928547574912, 5212430732760
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Convolution of 1, 1, 2, 5, 13, ... (A001519(n)) with 1, 3, 8, 21, 55, ... (A001906(n+1)).
|
|
LINKS
|
|
|
FORMULA
|
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4) for n>3, a(0)=1, a(1)=4, a(2)=13, a(3)=40.
a(n) = 3*a(n-1) - a(n-2) + A001519(n) for n>1, a(0)=1, a(1)=4.
a(n) = ( (3 + sqrt(5))^(1+n)*(8 - (1 - sqrt(5))*(13 + 5*n)) + (3 - sqrt(5))^(1+n)*(8 - (1 + sqrt(5))*(13 + 5*n)) ) / (25*2^(2+n)). - Bruno Berselli, Mar 06 2014
0 = 2 + a(n)*(a(n+1) - a(n+3)) + a(n+1)*(-6*a(n+1) + 12*a(n+2)) + a(n+2)*(-6*a(n+2) + a(n+3)) for all n in Z. - Michael Somos, Nov 23 2021
|
|
EXAMPLE
|
a(0) = 1*1 = 1;
a(1) = 1*3 + 1*1 = 4;
a(2) = 1*8 + 1*3 + 2*1 = 13;
a(3) = 1*21 + 1*8 + 2*3 + 5*1 = 40;
a(4) = 1*55 + 1*21 + 2*8 + 5*3 + 13*1 = 120; etc. (from first recurrence formula).
a(0) = 3*0 - 0 + 1 = 1;
a(1) = 3*1 - 0 + 1 = 4;
a(2) = 3*4 - 1 + 2 = 13;
a(3) = 3*13 - 4 + 5 = 40;
a(4) = 3*40 - 13 + 13 = 120; etc (from second recurrence formula).
G.f. = 1 + 4*x + 13*x^2 + 40*x^3 + 120*x^4 + 354*x^5 + 1031*x^6 + ... - Michael Somos, Nov 23 2021
|
|
MATHEMATICA
|
LinearRecurrence[{6, -11, 6, -1}, {1, 4, 13, 40}, 30] (* Bruno Berselli, Mar 06 2014 *)
a[ n_] := If[n < 0, SeriesCoefficient[ x^3*(2 - x)/(1 - 3*x + x^2)^2, {x, 0, -n}], SeriesCoefficient[ (1 - 2*x)/(1 - 3*x + x^2)^2, {x, 0, n}]]; (* Michael Somos, Nov 23 2021 *)
|
|
PROG
|
(PARI) {a(n) = if(n<0, polcoeff( x^3*(2-x)/(1-3*x+x^2)^2 + x*O(x^-n), -n), polcoeff( (1-2*x)/(1-3*x+x^2)^2 + x*O(x^n), n))}; /* Michael Somos, Nov 23 2021 */
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|