Search: a221172 -id:a221172
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A078343
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a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2).
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19
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-1, 2, 3, 8, 19, 46, 111, 268, 647, 1562, 3771, 9104, 21979, 53062, 128103, 309268, 746639, 1802546, 4351731, 10506008, 25363747, 61233502, 147830751, 356895004, 861620759, 2080136522, 5021893803, 12123924128, 29269742059, 70663408246, 170596558551, 411856525348
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OFFSET
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0,2
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COMMENTS
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Inverse binomial transform of -1, 1, 6, 22, 76, 260, ... (see A111566). Binomial transform of -1, 3, -2, 6, -4, 12, -8, 24, -16, ... (see A162255). - R. J. Mathar, Oct 02 2012
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REFERENCES
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H. S. M. Coxeter, 1998, Numerical distances among the circles in a loxodromic sequence, Nieuw Arch. Wisk, 16, pp. 1-9.
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LINKS
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FORMULA
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For the unsigned version: a(1)=1; a(2)=2; a(n) = Sum_{k=2..n-1} (a(k) + a(k-1)).
a(n) is asymptotic to (1/4)*(-2+3*sqrt(2))*(1+sqrt(2))^n.
a(n) = 2*A000129(n) - A000129(n-1) if n > 0; abs(a(n)) = Sum_{k=0..floor(n/2)} (C(n-k-1, k) - C(n-k-1, k-1))2^(n-2k). - Paul Barry, Dec 23 2004
O.g.f.: (1-4*x)/(-1 + 2*x + x^2). - R. J. Mathar, Feb 15 2008
a(n) = -(-1)^n * A048654(-n) = ( (-2+3*sqrt(2))*(1+sqrt(2))^n + (-2-3*sqrt(2))*(1-sqrt(2))^n )/4 for all n in Z. - Michael Somos, Jun 30 2022
E.g.f.: 3*exp(x)*sinh(sqrt(2)*x)/sqrt(2) - exp(x)*cosh(sqrt(2)*x). - Stefano Spezia, May 26 2024
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EXAMPLE
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G.f. = -1 + 2*x + 3*x^2 + 8*x^3 + 19*x^4 + 46*x^5 + 111*x^6 + ... - Michael Somos, Jun 30 2022
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MAPLE
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f:=proc(n) option remember; if n=0 then RETURN(-1); fi; if n=1 then RETURN(2); fi; 2*f(n-1)+f(n-2); end;
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MATHEMATICA
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LinearRecurrence[{2, 1}, {-1, 2}, 40] (* Harvey P. Dale, Apr 15 2019 *)
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PROG
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(Haskell)
a078343 n = a078343_list !! n
a078343_list = -1 : 2 : zipWith (+)
(map (* 2) $ tail a078343_list) a078343_list
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x)/(-1+2*x+x^2))); // G. C. Greubel, Jul 26 2018
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A221173
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a(0)=-3, a(1)=4; thereafter a(n) = 2*a(n-1) + a(n-2).
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5
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-3, 4, 5, 14, 33, 80, 193, 466, 1125, 2716, 6557, 15830, 38217, 92264, 222745, 537754, 1298253, 3134260, 7566773, 18267806, 44102385, 106472576, 257047537, 620567650, 1498182837, 3616933324, 8732049485, 21081032294, 50894114073, 122869260440, 296632634953
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OFFSET
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0,1
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LINKS
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FORMULA
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MATHEMATICA
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LinearRecurrence[{2, 1}, {-3, 4}, 50] (* Harvey P. Dale, Apr 09 2022 *)
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PROG
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(Haskell)
a221173 n = a221173_list !! n
a221173_list = -3 : 4 : zipWith (+)
(map (* 2) $ tail a221173_list) a221173_list
(PARI) Vec(-(10*x-3)/(x^2+2*x-1) + O(x^100)) \\ Colin Barker, Jul 10 2015
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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A221174
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a(0)=-4, a(1)=5; thereafter a(n) = 2*a(n-1) + a(n-2).
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7
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-4, 5, 6, 17, 40, 97, 234, 565, 1364, 3293, 7950, 19193, 46336, 111865, 270066, 651997, 1574060, 3800117, 9174294, 22148705, 53471704, 129092113, 311655930, 752403973, 1816463876, 4385331725, 10587127326, 25559586377, 61706300080, 148972186537, 359650673154
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OFFSET
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0,1
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COMMENTS
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For n >= 3, 2*a(n) is the number of ways to tile this figure of length n-1 with two colors of squares and one color of domino. For n=8, we have here the figure of length n-1=7, and it has 2*a(8) = 2728 different tilings.
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(End)
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LINKS
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FORMULA
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a(n) is the numerator of the continued fraction [4, 2, ..., 2, 4] with n-3 2's in the middle. For denominators, see A048654. - Greg Dresden and Tongjia Rao, Sep 02 2021
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PROG
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(Haskell)
a221174 n = a221174_list !! n
a221174_list = -4 : 5 : zipWith (+)
(map (* 2) $ tail a221174_list) a221174_list
(PARI) Vec(-(13*x-4)/(x^2+2*x-1) + O(x^50)) \\ Colin Barker, Jul 10 2015
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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A221175
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a(0)=-5, a(1)=6; thereafter a(n) = 2*a(n-1) + a(n-2).
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+0
5
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-5, 6, 7, 20, 47, 114, 275, 664, 1603, 3870, 9343, 22556, 54455, 131466, 317387, 766240, 1849867, 4465974, 10781815, 26029604, 62841023, 151711650, 366264323, 884240296, 2134744915, 5153730126, 12442205167, 30038140460, 72518486087, 175075112634
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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LINKS
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FORMULA
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PROG
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(Haskell)
a221175 n = a221175_list !! n
a221175_list = -5 : 6 : zipWith (+)
(map (* 2) $ tail a221175_list) a221175_list
(PARI) Vec(-(16*x-5)/(x^2+2*x-1) + O(x^50)) \\ Colin Barker, Jul 10 2015
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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