1,2

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.

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).

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.

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)];

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

Cf. A342042, A377912, A377917.

Sequence in context: A000589 A211830 A322877 * A226708 A206529 A118936

Adjacent sequences: A377915 A377916 A377917 * A377919 A377920 A377921

nonn,base

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

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

approved