A377918 - OEIS

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

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OFFSET

1,2

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).

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.

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 *)

CROSSREFS

Cf. A342042, A377912, A377917.