A377918 - OEIS

%I #33 Dec 02 2024 10:32:12

%S 1,11,77,566,4197,31148,231193,1716043,12737453,94544693,701765055,

%T 5208903636,38663477066,286982552081,2130149470506,15811193864583,

%U 117359769764941,871111674250772,6465891595866732,47993564275737877,356235822660837879,2644187054283807954,19626676300599636003

%N a(n) = index in A377912 (or, equally, in A342042) of the first n-digit term.

%C These are the points in the graph of A342042 where the separate paths come together.

%C The first differences are in A377917, which is the more fundamental sequence. To get this sequence from A377917, add an initial zero, take partial sums, and add 1 to each term.

%H Ray Chandler, <a href="/A377918/b377918.txt">Table of n, a(n) for n = 1..1149</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (6,10,5,-5,-9,-5,-1).

%F G.f. = (x^7+6*x^6+15*x^5+19*x^4+11*x^3-x^2-5*x-1)/((1-x)*(x^6+6*x^5+15*x^4+20*x^3+15*x^2+5*x-1)) (From g.f. for A377917).

%F Recurrence: See Maple code.

%F The smallest root of the denominator of the g.f. is 0.134724138401519... whose reciprocal is (say) c1 = 7.422574840... Then a(n) is asymptotically c2*c1^n, for n >= 0, where c2 = 1.3824387... This is an excellent approximation. It gives a(22) = 0.1962667617*10^20, compared with a(22) = 19626676300599636003.

%F This also enables us to give a formula for the lower envelope of A342042 - see that entry for details.

%p A377918 := proc(n) local S; option remember;

%p S:=[1, 11, 77, 566, 4197, 31148, 231193, 1716043];

%p if n <= 8 then S[n] else

%p 6*A377918(n-1)+10*A377918(n-2)+5*A377918(n-3)-5*A377918(n-4)-9*A377918(n-5)-5*A377918(n-6)-A377918(n-7); fi;

%p end;

%p [seq(A377918(i),i=1..20)];

%t LinearRecurrence[{6, 10, 5, -5, -9, -5, -1}, {1, 11, 77, 566, 4197, 31148, 231193, 1716043}, 25] (* _Paolo Xausa_, Dec 02 2024  *)

%Y Cf. A342042, A377912, A377917.

%K nonn,base

%O 1,2

%A _Sebastian Karlsson_ and _N. J. A. Sloane_, Nov 30 2024

%E More terms added based on A377917. - _N. J. A. Sloane_, Dec 01 2024