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 A266505 a(n) = 2*a(n - 2) + a(n - 4) with a(0) = -1, a(1) = 1, a(2) = 3, a(3) = 5. 3
 -1, 1, 3, 5, 5, 11, 13, 27, 31, 65, 75, 157, 181, 379, 437, 915, 1055, 2209, 2547, 5333, 6149, 12875, 14845, 31083, 35839, 75041, 86523, 181165, 208885, 437371, 504293, 1055907, 1217471, 2549185, 2939235, 6154277, 7095941, 14857739, 17131117, 35869755, 41358175, 86597249, 99847467 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n)/A266504(n) converges to sqrt(2). Alternatively, bisection of A266506. Alternatively, A135532(n) and A048655(n) interlaced. Alternatively, A255236(n-1), A054490(n), A038762(n) and A101386(n) interlaced. Also see comments in A222786 regarding a curious relationship between this sequence, A266504, A266506 and laminated lattice kissing numbers with an average number of spheres/dimension equal to either a pronic number or twice a square. Let b(n) = (a(n) - (a(n) mod 2))/2, that is b(n) = {-1, 0, 1, 2, 2, 5, 6, 13, 15, 32, 37, 78, 90, ...}. Then: A006451(n) = {b(4n+0) U b(4n+1)} gives n in N such that triangular(n) + 1 is square; A216134(n) = {b(4n+2) U b(4n+3)} gives n in N such that triangular(n) follows form n^2 + n + 1 (twice a triangular number + 1). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (0,2,0,1). FORMULA G.f.: (-1 + 3*x)*(1 + x)^2/(1 - 2 x^2 - x^4). a(n) = (-(1+sqrt(2))^floor(n/2)*(-1)^n - sqrt(8)*(1-sqrt(2))^floor(n/2) - (1-sqrt(2))^floor(n/2)*(-1)^n + sqrt(8)*(1+sqrt(2))^floor(n/2))/2. a(n) = 3*(((1+sqrt(2))^floor(n/2)-(1-sqrt(2))^floor(n/2))/sqrt(8)) - (-1)^n*(((1+sqrt(2))^(floor(n/2)-(-1)^n)-(1-sqrt(2))^(floor(n/2)-(-1)^n))/sqrt(8)). a(n) = (3*A000129(floor(n/2)) - A000129(n-(-1)^n)), where A000129 gives the Pell numbers. a(n) = sqrt(2*A266504(n))^2 - 7(-1)^A266504(n))*sgn(2*n -1), where A266504 gives all x in N such that 2*x^2 - 7(-1)^x = y^2. This sequence gives associated y values. a(2n) = (-(1 + sqrt(2))^n - sqrt(8)*(1 - sqrt(2))^n - (1 - sqrt(2))^n + sqrt(8)*(1 + sqrt(2))^n)/2 = a(2n) = A135532(n). a(2n) = 3*(((1+sqrt(2))^n-(1-sqrt(2))^n)/sqrt(8)) - (((1+sqrt(2))^(n-1)-(1-sqrt(2))^(n-1))/sqrt(8)) = A135532(n). a(2n+1) = (+(1 + sqrt(2))^n - sqrt(8)*(1 - sqrt(2))^n + (1 - sqrt(2))^n + sqrt(8)*(1 + sqrt(2))^n)/2 = a(2n + 1) = A048655(n). a(2n+1) = 3*(((1+sqrt(2))^n-(1-sqrt(2))^n)/sqrt(8)) + (((1+sqrt(2))^(n+1)-(1-sqrt(2))^(n+1))/sqrt(8)) = A048655(n). a(4n + 0) = 6*a(4n - 4) - a(4n - 8) = A255236(n-1). a(4n + 1) = 6*a(4n - 3) - a(4n - 7) = A054490(n). a(4n + 2) = 6*a(4n - 2) - a(4n - 6) = A038762(n). a(4n + 3) = 6*a(4n - 1) - a(4n - 5) = A101386(n). (sqrt(2*(a(2n + 1) )^2 + 14*(-1)^floor(n/2)))/2 = A266504(n). (a(2n + 1) + a(2n))/8 = A000129(n), where A000129 gives the Pell numbers. a(2n + 1) - a(2n) = A002203(n), where A002203 gives the companion Pell numbers. (a(2n + 2) + a(2n + 1))/2 = A000129(n+2). (a(2n + 2) - a(2n + 1))/2 = A000129(n-1). MAPLE a:=proc(n) option remember; if n=0 then -1 elif n=1 then 1 elif n=2 then 3 elif n=3 then 5 else 2*a(n-2)+a(n-4); fi; end:  seq(a(n), n=0..50); # Wesley Ivan Hurt, Jan 01 2016 MATHEMATICA LinearRecurrence[{0, 2, 0, 1}, {-1, 1, 3, 5}, 70] (* Vincenzo Librandi, Dec 31 2015 *) Table[SeriesCoefficient[(-1 + 3 x) (1 + x)^2/(1 - 2 x^2 - x^4), {x, 0, n}], {n, 0, 42}] (* Michael De Vlieger, Dec 31 2015 *) PROG (MAGMA) I:=[-1, 1, 3, 5]; [n le 4 select I[n] else 2*Self(n-2)+Self(n-4): n in [1..70]]; // Vincenzo Librandi, Dec 31 2015 (PARI) x='x+O('x^30); Vec((-1+3*x)*(1+x)^2/(1-2*x^2-x^4)) \\ G. C. Greubel, Jul 26 2018 CROSSREFS Cf. A000129, A001333, A002203, A002965, A006451, A006452, A002965, A038761, A038762, A048654, A048655, A054490, A078343, A098586, A098790, A100525, A101386, A135532, A216134, A216162, A253811, A255236, A266504, A266505, A266507. Sequence in context: A093572 A317650 A240731 * A118132 A089167 A188345 Adjacent sequences:  A266502 A266503 A266504 * A266506 A266507 A266508 KEYWORD sign,easy AUTHOR Raphie Frank, Dec 30 2015 STATUS approved

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Last modified May 10 00:45 EDT 2021. Contains 343747 sequences. (Running on oeis4.)