A377918 a(n) = index in A377912 (or, equally, in A342042) of the first n-digit term.

1, 11, 77, 566, 4197, 31148, 231193, 1716043, 12737453, 94544693, 701765055, 5208903636, 38663477066, 286982552081, 2130149470506, 15811193864583, 117359769764941, 871111674250772, 6465891595866732, 47993564275737877, 356235822660837879, 2644187054283807954, 19626676300599636003

Offset: 1

Author

Sebastian Karlsson and N. J. A. Sloane, Nov 30 2024

Keywords

• nonn
• base

Comments

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

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.

Links

• Ray Chandler, Table of n, a(n) for n = 1..1149
• Index entries for linear recurrences with constant coefficients, signature (6,10,5,-5,-9,-5,-1).

Crossrefs

Cf. A342042, A377912, A377917.

Programs

Maple

A377918 := proc(n) local S; option remember;
S:=[1, 11, 77, 566, 4197, 31148, 231193, 1716043];
if n <= 8 then S[n] else
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;
end;
[seq(A377918(i),i=1..20)];

Mathematica

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

Formula

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).

Recurrence: See Maple code.

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.

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

Extensions

More terms added based on A377917. - N. J. A. Sloane, Dec 01 2024