|
|
A099586
|
|
Constant term in (1+x)^n mod (1+x^4).
|
|
8
|
|
|
1, 1, 1, 0, -4, -14, -34, -68, -116, -164, -164, 0, 560, 1912, 4616, 9232, 15760, 22288, 22288, 0, -76096, -259808, -627232, -1254464, -2141504, -3028544, -3028544, 0, 10340096, 35303296, 85229696, 170459392, 290992384, 411525376, 411525376, 0, -1405035520
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,5
|
|
COMMENTS
|
Equals real part of term (1,1) in M^n, where M = a 2 X 2 matrix [1,1; i,1], where i = sqrt(-1). - Gary W. Adamson, Mar 25 2009
For the definition, see [Erdelyi] and the Shevelev link. - Vladimir Shevelev, Jul 03 2017
a(n) = 0 if and only if n == 4 (mod 8). - Robert Israel, Jul 04 2017
|
|
REFERENCES
|
A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-2*a(n-4) for n>4.
G.f.: -x*(2*x-1)*(x^2-x+1) / (2*x^4-4*x^3+6*x^2-4*x+1).
(End)
a(n) = (1/2)*((2+sqrt(2))^(n/2)*cos(n*Pi/8) + (2-sqrt(2))^(n/2)*cos(3*n*Pi/8)). - G. C. Greubel, Nov 10 2015
a(n) = Sum_{k >= 0}(-1)^k*binomial(n,4*k).
a(n) = round((2+sqrt(2))^(n/2)*cos(Pi*(n)/8)/2), where round(x) is the integer nearest to x.
a(n+m) = a(n)*a(m) - K_4(n)*K_2(m) - K_3(n)*K_3(m) - K_2(n)*K_4(m), where K_2 is A099587, K_3 is A099588 and K_4 is A099589.
(End)
|
|
MAPLE
|
seq(eval(rem((1+x)^n, 1+x^4, x), x=0), n=1..40); # Robert Israel, Jul 03 2017
|
|
MATHEMATICA
|
RecurrenceTable[{a[n] == 4 * a[n - 1] - 6 * a[n - 2] + 4 * a[n - 3] - 2 * a[n - 4], a[1] = 1, a[2] = 1, a[3] = 1, a[4] = 0}, a, {n, 50}] (* G. C. Greubel, Nov 10 2015 *)
a[n_] := HypergeometricPFQ[{(1-n)/4, (2-n)/4, (3-n)/4, -n/4}, {1/4, 1/2, 3/4}, -1]; Array[a, 40] (* Jean-François Alcover, Jul 20 2017, from Vladimir Shevelev's first formula *)
|
|
PROG
|
(PARI) a(n) = polcoeff(((1+x)^n)%(x^4+1), 0)
(PARI) Vec(-x*(2*x-1)*(x^2-x+1)/(2*x^4-4*x^3+6*x^2-4*x+1) + O(x^100)) \\ Colin Barker, Nov 08 2015
(PARI) a(n) = real(([1, 1; I, 1])^n)[1, 1]; \\ Michel Marcus, Nov 08 2015
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|