A084150 - OEIS

%I #15 Oct 12 2022 09:03:02

%S 0,0,1,3,14,50,199,749,2892,11028,42301,161799,619706,2372006,9081955,

%T 34767953,133109592,509594856,1950956857,7469077707,28594853414,

%U 109473250778,419110475455,1604533706357,6142840740900,23517417426300

%N A Pell related sequence.

%C Binomial transform of expansion of (sinh(sqrt(2)x))^2/4 = (0, 0, 1, 0, 8, 0, 64, ...). Inverse binomial transform of A006668.

%H G. C. Greubel, <a href="/A084150/b084150.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,5,-7).

%F a(n) = ( (1+sqrt(8))^n + (1-sqrt(8))^n - 2 )/16.

%F E.g.f.: (1/4)*exp(x)*( sinh(sqrt(2)*x) )^2.

%F G.f.: x^2 / ( (1-x)*(1-2*x-7*x^2) ). - _R. J. Mathar_, Feb 05 2011

%F a(n) = (A015519(n) - A015519(n-1) - 1)/8 = (A084058(n) - 1)/8. - _G. C. Greubel_, Oct 11 2022

%t LinearRecurrence[{3,5,-7}, {0,0,1}, 41] (* _G. C. Greubel_, Oct 11 2022 *)

%o (Magma) [n le 3 select Floor((n-1)/2) else 3*Self(n-1) +5*Self(n-2) -7*Self(n-3): n in [1..41]]; // _G. C. Greubel_, Oct 11 2022

%o (SageMath)

%o A084058 = BinaryRecurrenceSequence(2,7,1,1)

%o def A084150(n): return (A084058(n) - 1)/8

%o [A084150(n) for n in range(41)] # _G. C. Greubel_, Oct 11 2022

%Y Cf. A006646, A006668, A015519, A084058.

%K easy,nonn

%O 0,4

%A _Paul Barry_, May 16 2003