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A072684 Expansion of (2+x+3*x^2+2*x^3+x^4)/(1-x-5*x^2+x^3+3*x^4-x^5). 0
2, 3, 16, 31, 103, 235, 674, 1669, 4526, 11595, 30769, 79885, 210226, 548623, 1439156, 3763159, 9859523, 25800519, 67566130, 176858881, 463073602, 1212259843, 3173871101, 8309086201, 21753819938, 56951673915, 149102333944 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
REFERENCES
S. Ramanujan, Note on a set of Simultaneous Equations, J. Indian Math. Soc., 4(1912), 94-96.
LINKS
S. Ramanujan, Note on a set of Simultaneous Equations, J. Indian Math. Soc., 4(1912), 94-96.
FORMULA
G.f.: (2 + x + 3*x^2 + 2*x^3 + x^4)/(1 - x - 5*x^2 + x^3 + 3*x^4 - x^5).
a(n) = 2*a(n-1) + 3*a(n-2) - 4*a(n-3) + a(n-4) + 3*(-1)^n for all n in Z.
a(n) = (-3/5)*(-1)^n + (-9/2)*F(-n) - F(-n-1) + (13/10)*F(2*n-1) + (23/10)*F(2*n+1) for all n in Z. - Michael Somos, Apr 10 2022
EXAMPLE
G.f. = 2 + 3*x + 16*x^2 + 31*x^3 + 103*x^4 + 235*x^5 + 674*x^6 + 1669*x^7 + 4526*x^8 + ...
MATHEMATICA
CoefficientList[Series[(2+x+3x^2+2x^3+x^4)/(1-x-5x^2+x^3+3x^4-x^5), {x, 0, 40}], x] (* or *) LinearRecurrence[{1, 5, -1, -3, 1}, {2, 3, 16, 31, 103}, 40] (* Harvey P. Dale, Sep 23 2018 *)
a[ n_] := Module[{F = Fibonacci}, (-3/5)*(-1)^n + (-9/2)*F[-n] - F[-n-1] + (13/10)*F[2*n-1] + (23/10)*F[2*n+1] ]; (* Michael Somos, Apr 10 2022 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( (2 + x + 3*x^2 + 2*x^3 + x^4) / (1 - x - 5*x^2 + x^3 + 3*x^4 - x^5) + x * O(x^n), n))};
(PARI) {a(n) = my(F = fibonacci); (-3/5)*(-1)^n + (-9/2)*F(-n) - F(-n-1) + (13/10)*F(2*n-1) + (23/10)*F(2*n+1) }; /* Michael Somos, Apr 10 2022 */
CROSSREFS
Cf. A000045.
Sequence in context: A034382 A034383 A350025 * A344747 A073633 A012357
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Jul 01 2002
STATUS
approved

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Last modified March 28 12:26 EDT 2024. Contains 371254 sequences. (Running on oeis4.)