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A178626 Convolution of Pell(n) and 10^n.

%I A178626 #22 Sep 08 2022 08:45:54
%S A178626 1,12,125,1262,12649,126560,1265769,12658098,126581965,1265822028,
%T A178626 12658226021,126582274070,1265822774161,12658227822392,
%U A178626 126582278418945,1265822784660282,12658227847739509,126582278480139300
%N A178626 Convolution of Pell(n) and 10^n.
%C A178626 Row sums of:
%C A178626 1
%C A178626 10....2
%C A178626 100...20...5
%C A178626 1000..200..50..12
%C A178626 10000.2000.500.120.29
%C A178626 or:
%C A178626 1
%C A178626 10....2
%C A178626 100...21...4
%C A178626 1000..210..44..8
%C A178626 10000.2100.441.92.16
%C A178626 or without the first column:
%C A178626 1
%C A178626 10
%C A178626 100......1
%C A178626 1000.....12
%C A178626 10000....124....1
%C A178626 100000...1248...14
%H A178626 G. C. Greubel, <a href="/A178626/b178626.txt">Table of n, a(n) for n = 1..995</a>
%H A178626 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (12,-19,-10).
%F A178626 a(n) = 10*a(n-1) + Pell(n) with a(1)= Pell(1) = 1.
%F A178626 From _R. J. Mathar_, May 31 2010: (Start)
%F A178626 a(n) = +12*a(n-1) -19*a(n-2) -10*a(n-3).
%F A178626 G.f.: x/ ( (10*x-1)*(x^2+2*x-1) ). (End)
%F A178626 a(n) = (2^(3+n)*5^(1+n) + (1-sqrt(2))^n*(-20+11*sqrt(2)) - (1+sqrt(2))^n*(20+11*sqrt(2))) / 316. - _Colin Barker_, Jan 29 2019
%t A178626 LinearRecurrence[{12, -19, -10}, {1, 12, 125}, 30] (* _G. C. Greubel_, Jan 28 2019 *)
%o A178626 (PARI) my(x='x+O('x^30)); Vec( x/((10*x-1)*(x^2+2*x-1)) ) \\ _G. C. Greubel_, Jan 28 2019
%o A178626 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( x/((10*x-1)*(x^2+2*x-1)) )); // _G. C. Greubel_, Jan 28 2019
%o A178626 (Sage) a=(x/((10*x-1)*(x^2+2*x-1))).series(x, 30).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, Jan 28 2019
%o A178626 (GAP) a:=[1,12,125];; for n in [4..30] do a[n]:=12*a[n-1]-19*a[n-2] - 10*a[n-3]; od; a; # _G. C. Greubel_, Jan 28 2019
%Y A178626 Cf. A021083, A000129, A038207, A009965.
%K A178626 nonn,easy
%O A178626 1,2
%A A178626 _Mark Dols_, May 31 2010
%E A178626 a(10), a(11) corrected and sequence extended by _R. J. Mathar_, May 31 2010