A045761 -id:A045761 - cp's OEIS frontend

A268107 Decimal expansion of 'lambda', a Somos quadratic recurrence constant mentioned by Steven Finch.

3, 9, 9, 5, 2, 4, 6, 6, 7, 0, 9, 6, 7, 9, 9, 4, 6, 5, 5, 2, 5, 0, 3, 3, 4, 7, 4, 3, 3, 2, 2, 5, 8, 3, 3, 2, 2, 1, 7, 3, 6, 9, 8, 5, 4, 6, 7, 5, 9, 9, 6, 8, 9, 7, 7, 3, 6, 7, 0, 0, 5, 2, 8, 9, 4, 8, 5, 3, 0, 7, 3, 7, 0, 2, 7, 1, 2, 5, 9, 3, 4, 5, 6, 6, 3, 4, 9, 2, 0, 9, 8, 2, 0, 2, 5, 7, 5, 7, 4, 3, 3, 3, 1, 0, 0, 1, 6, 6, 7, 5, 1, 6, 4, 6, 9, 6, 4, 7, 4, 5, 6, 4, 4, 9, 8, 4, 5

Offset: 0

Jean-François Alcover, Jan 26 2016

[Quoted from Steven Finch] Another Somos constant lambda = 0.3995246670... arises as follows: If k < lambda, then the sequence h_0 = 0, h_1 = k, h_n = h_(n-1)*(1 + h_(n-1) - h_(n-2)) for n>=2 converges to a limit less than 1; if k > lambda, then the sequence diverges to infinity. This is similar to Grossman's constant.

A heuristical evaluation of lambda = 0.39952466709679946552503347433225833221736985467599... was communicated to me by Jon E. Schoenfield in a private email.

			0.39952466709679946552503347433225833221736985467599689773670052894853...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.10 Quadratic Recurrence Constants, p. 446.

Steven R. Finch, A Deceptively Simple Quadratic Recurrence, arXiv:2409.03510 [math.NT], 2024. See p. 9.
Jon E. Schoenfield, Magma program communicated to J.-F. Alcover.
Eric Weisstein's MathWorld, Somos's Quadratic Recurrence Constant

Cf. A045761, A085835, A112302.

// See the link to Jon E. Schoenfield's program.

n0 (* initial number of terms *) = 2*10^7; iter = 10^5; dn = 10^6; k1 = 0.3; k2 = 0.4; eps = 10^-16; f[k_?NumericQ] := (h0 = 0; h1 = k; h2 = k*(1+k); Do[h0 = h1; h1 = h2; h2 = Min[h1 + (h1-h0), h1*(1+h1-h0)], {iter}]; h2); Clear[g]; g[n0] = k1; g[n = n0+dn] = k2; g[n_] := g[n] = k /. FindRoot[f[k]==1, {k, g[n-dn] }]; While[Print[n, " ", g[n] // RealDigits]; Abs[g[n] - g[n-dn]] > eps, n = n+dn]; lambda = g[n]; RealDigits[lambda][[1]][[1;;9]]

Conjecture: lambda is the radius of convergence of the function Sum_{n>=0} A045761(n)*x^n, that is the constant 1/d computed by Vaclav Kotesovec in A045761.

Extended to 127 digits using Jon E. Schoenfield's evaluation, Aug 27 2016

A360985 Triangle read by rows: T(n,k) is the number of full binary trees with n leaves, each internal node having the heights of its two subtrees weakly increasing left to right, and with k internal nodes having two subtrees of equal height.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 4, 3, 2, 2, 0, 0, 1, 6, 7, 6, 3, 0, 1, 0, 1, 9, 13, 14, 9, 3, 1, 0, 0, 1, 12, 27, 27, 22, 14, 3, 1, 0, 0, 1, 16, 47, 59, 54, 32, 16, 7, 0, 0, 0, 1, 20, 81, 117, 125, 91, 44, 20, 8, 1, 0

Offset: 1

Ludovic Schwob, Feb 27 2023

			The table for T(n,k) begins:
  n\k 0   1   2   3    4    5    6    7    8   9  10  11  12  13  14  15
   1  1
   2  0   1
   3  0   1   0
   4  0   1   0   1
   5  0   1   1   1    0
   6  0   1   2   2    1    0
   7  0   1   4   3    2    2    0
   8  0   1   6   7    6    3    0    1
   9  0   1   9  13   14    9    3    1    0
  10  0   1  12  27   27   22   14    3    1   0
  11  0   1  16  47   59   54   32   16    7   0   0
  12  0   1  20  81  117  125   91   44   20   8   1   0
  13  0   1  25 128  233  272  228  143   61  23   8   2   0
  14  0   1  30 197  439  573  555  389  206  90  21  10   2   0
  15  0   1  36 287  801 1178 1275 1014  621 303 109  32   4   4   0
  16  0   1  42 410 1383 2367 2841 2522 1727 962 421 138  36   7   0   1

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50).

Row sums give A045761.
Column k=2 is A002620.
Cf. A000108, A335833.

T(n)={my(p=x+O(x*x^n), q=p); for(n=2, n, p=y*p^2 + p*(q-p); q+=p); my(v=Vec(q)); vector(#v, n, Vecrev(v[n], n))}
{ my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Mar 24 2023

T(n,2) = A002620(n-3) for all n>=3.
T(n,n-1) = 1 if n is a power of 2, and T(n,n) = 0 otherwise.
T(n,n-2) != 0 if and only if n-1 has exactly one maximal group of consecutive zeros in its binary representation, and in this case T(n,n-2) = 2^(a-1) where a is the number of 1s at the beginning of the binary representation of n-1.
Sum_{k=0..n-1} T(n,k)*2^(n-k-1) = A000108(n-1).

cp's OEIS Frontend

A268107 Decimal expansion of 'lambda', a Somos quadratic recurrence constant mentioned by Steven Finch.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions