A263719 -id:A263719 - cp's OEIS frontend

A025157 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 3.

1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 22, 25, 28, 32, 36, 41, 46, 52, 58, 66, 73, 82, 91, 102, 113, 126, 139, 155, 171, 190, 209, 232, 255, 282, 310, 342, 375, 413, 452, 497, 544, 596, 651, 713, 778, 850, 927, 1011, 1101, 1200, 1305, 1420, 1544, 1677, 1821, 1977, 2144, 2324, 2519, 2728

Offset: 0

Clark Kimberling

nonn

Also number of partitions of n into distinct parts in which the smallest part is greater than or equal to number of parts. - Vladeta Jovovic, Aug 06 2004

			a(12) = 6 because we have 12 = 11+1 = 10+2 = 9+3 = 8+4 = 7+4+1.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
I. Martinjak and D. Svrtan, New Identities for the Polarized Partitions and Partitions with d-Distant Parts, J. Int. Seq. 17 (2014) # 14.11.4.

Cf. A003114, A096401, A263719.

Column k=3 of A194543.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(n>
      ceil(i*(i+3)/6), 0, b(n, i-1)+b(n-i, min(n-i, i-3))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 26 2022

nn=50; CoefficientList[Series[Sum[x^(j(3j-1)/2)Product[1/(1-x^i), {i, 1, j}], {j, 0, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Jul 21 2013 *)

my(N=99, x='x+O('x^N)); Vec(sum(k=0, N, x^(k*(3*k-1)/2)/prod(j=1, k, 1-x^j))) \\ Seiichi Manyama, Jan 13 2022

G.f.: sum(i>=1, x^(3*A000217(i)-2*i)/product(j=1..i, 1-x^j)). - Jon Perry, Jul 20 2004
G.f.: sum(n>=0, x^(n*(3*n-1)/2)/prod(k=1..n,1-x^k)). - Joerg Arndt, Jan 29 2011
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*n^(3/4)*r*sqrt(Pi*(1+3*r^2))), where r = A263719 = ((9+sqrt(93))/2)^(1/3)/3^(2/3) - (2/(3*(9+sqrt(93))))^(1/3) = 0.682327803828019327369483739711048256891188581898... is the root of the equation r^3 + r = 1 and c = 3*(log(r))^2/2 + polylog(2, 1-r) = 0.566433354765746647188050807325058683443823543741343518... . - Vaclav Kotesovec, Jan 02 2016

Prepended a(0)=1, Joerg Arndt, Jul 21 2013

A092526 Decimal expansion of (2/3)cos( (1/3)arccos(29/2) ) + 1/3, the real root of x^3 - x^2 - 1.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Apr 07 2004

This is the limit x of the ratio N(n+1)/N(n) for n -> infinity of the Narayana sequence N(n) = A000930(n). The real root of x^3 - x^2 - 1. See the formula section. - Wolfdieter Lang, Apr 24 2015
This is the fourth smallest Pisot number. - Iain Fox, Oct 13 2017
Sometimes called the supergolden ratio or Narayana's cows constant, and denoted by the symbol psi. - Ed Pegg Jr, Feb 01 2019

			1.46557123187676802665673122521993910802557756847228570164318311124926...

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.3.
Paul J. Nahin, The Logician and the Engineer, How George Boole and Claude Shannon Created the Information Age, Princeton University Press, Princeton and Oxford, 2013, Chap. 7: Some Combinational Logic Examples, Section 7.1: Channel Capacity, Shannon's Theorem, and Error-Detection Theory, page 120.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Simon Baker, Exceptional digit frequencies and expansions in non-integer bases, arXiv:1711.10397 [math.DS], 2017. See the beta(2) constant pp. 3-4.
H. R. P. Ferguson, On a Generalization of the Fibonacci Numbers Useful in Memory Allocation Schema or All About the Zeroes of Z^k - Z^{k - 1} - 1, k > 0, Fibonacci Quarterly, Volume 14, Number 3, October, 1976 (see Table 2, p. 236).
Ed Pegg Jr., Images based on the supergolden ratio
Michael Penn, What is the super-golden ratio??, YouTube video, 2022.
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 59.
Eric Weisstein's World of Mathematics, Pisot Number.
Eric Weisstein's World of Mathematics, Supergolden Ratio.
Wikipedia, Pisot number
Wikipedia, Supergolden ratio
Index entries for algebraic numbers, degree 3

Cf. A088559, A075778, A076725, A000930, A263719.
Other Pisot numbers: A060006, A086106, A228777, A293506, A293508, A293509, A293557.
Cf. A381124 (numerators of convergents).
Cf. A381125 (denominators of convergents).

RealDigits[(2 Cos[ ArcCos[ 29/2]/3] + 1)/3, 10, 111][[1]] (* Robert G. Wilson v, Apr 12 2004 *)
RealDigits[ Solve[ x^3 - x^2 - 1 == 0, x][[1, 1, 2]], 10, 111][[1]] (* Robert G. Wilson v, Oct 10 2013 *)

allocatemem(932245000); default(realprecision, 20080); x=solve(x=1, 2, x^3 - x^2 - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b092526.txt", n, " ", d));  \\ Harry J. Smith, Jun 21 2009

The real root of x^3 - x^2 - 1. - Franklin T. Adams-Watters, Oct 12 2006
The only real irrational root of x^4-x^2-x-1 (-1 is also a root). [Nahim]
Equals (2/3)*cos( (1/3)*arccos(29/2) ) + 1/3.
Equals 1 + A088559.
Equals (1/6)*(116+12*sqrt(93))^(1/3) + 2/(3*(116+12*sqrt(93))^(1/3)) + 1/3. - Vaclav Kotesovec, Dec 18 2014
Equals 1/A263719. - Alois P. Heinz, Apr 15 2018
Equals (1 + 1/r + r)/3 where r = ((29 + sqrt(837))/2)^(1/3). - Peter Luschny, Apr 04 2020
Equals (1/3)*(1 + ((1/2)*(29 + (3*sqrt(93))))^(1/3) + ((1/2)*(29 - 3*sqrt(93)))^(1/3)). See A075778. - Wolfdieter Lang, Aug 17 2022

A003274 Number of key permutations of length n: permutations {a_i} with |a_i - a_{i-1}| = 1 or 2.

Original entry on oeis.org

1, 1, 2, 6, 12, 20, 34, 56, 88, 136, 208, 314, 470, 700, 1038, 1534, 2262, 3330, 4896, 7192, 10558, 15492, 22724, 33324, 48860, 71630, 105002, 153912, 225594, 330650, 484618, 710270, 1040980, 1525660, 2235994, 3277040, 4802768, 7038832, 10315944, 15118786

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 251 terms from R. H. Hardin)
S. Avgustinovich and S. Kitaev, On uniquely k-determined permutations, Discr. Math., 308 (2008), 1500-1507.
Hugh Denoncourt, Ordinal pattern probabilities for symmetric random walks, arXiv:1907.07172 [math.CO], 2019.
E. S. Page, Systematic generation of ordered sequences using recurrence relations, Computer J., 14 (1971), 150-153.
E. S. Page, Systematic generation of ordered sequences using recurrence relations, The Computer Journal 14 (1971), 150-153. (Annotated scanned copy)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-2,1).

Cf. A069241, A092526, A174700, A263719, A302118.

A003274:=-(1-z+3*z**2-2*z**3+z**5)/(z**3+z-1)/(z-1)**2; # [Conjectured by Simon Plouffe in his 1992 dissertation.]

CoefficientList[Series[-(x^6 - x^5 + x^3 + 2 x^2 - 2 x + 1)/((x^3 + x - 1) (x - 1)^2), {x, 0, 39}], x] (* Michael De Vlieger, Oct 01 2019 *)

For n > 1, a(n) = 2*A069241(n).
G.f.: -(x^6 - x^5 + x^3 + 2*x^2 - 2*x + 1)/((x^3 + x - 1)*(x-1)^2).
Limit_{n->oo} a(n+1)/a(n) = A092526 = 1/A263719. - Alois P. Heinz, Apr 15 2018

Better description and g.f. from Erich Friedman

a(0)=1 prepended and g.f. adapted by Alois P. Heinz, Apr 01 2018

A096401 Number of balanced partitions of n into distinct parts: least part is equal to number of parts.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 10, 11, 13, 14, 17, 18, 21, 23, 26, 28, 32, 35, 39, 43, 48, 53, 59, 65, 72, 80, 88, 97, 107, 118, 129, 142, 155, 171, 186, 204, 222, 244, 265, 290, 315, 345, 374, 409, 443, 484, 524, 571, 618, 673, 727, 790

Offset: 1

Vladeta Jovovic, Aug 06 2004

			a(14)=3 because we have 12+2, 7+4+3 and 6+5+3.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000009, A025157, A237976, A237977, A237979.

Cf. A006141, A047993, A339446.

G:=sum((x^(m*(3*m-1)/2)-x^(m*(3*m+1)/2))/product(1-x^i,i=1..m),m=1..20): Gser:=series(G,x=0,80): seq(coeff(Gser,x^n),n=1..78); # Emeric Deutsch, Mar 29 2005

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, x^(k*(3*k-1)/2)/prod(j=1, k-1, 1-x^j))) \\ Seiichi Manyama, Jan 15 2022

G.f.: Sum_{m>=1} (x^(m*(3*m-1)/2)-x^(m*(3*m+1)/2))/Product_{i=1..m} (1-x^i).
a(n) = A025157(n) - A237979(n) = A237977(n) - A237976(n) for n > 0. - Seiichi Manyama, Jan 13 2022
a(n) ~ (1 - A263719) * A025157(n). - Vaclav Kotesovec, Jan 15 2022

More terms from Emeric Deutsch, Mar 29 2005

A076725 a(n) = a(n-1)^2 + a(n-2)^4, a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 2, 5, 41, 2306, 8143397, 94592167328105, 13345346031444632841427643906, 258159204435047592104207508169153297050209383336364487461

Offset: 0

Michael Somos, Oct 29 2002

a(n) and a(n+1) are relatively prime for n >= 0.
The number of independent sets on a complete binary tree with 2^(n-1)-1 nodes. - Jonathan S. Braunhut (jonbraunhut(AT)usa.net), May 04 2004. For example, when n=3, the complete binary tree with 2 levels has 2^2-1 nodes and has 5 independent sets so a(3)=5. The recursion for number of independent sets splits in two cases, with or without the root node being in the set.
a(10) has 113 digits and is too large to include.

			a(2) = a(1)^2 + a(0)^4 = 1^2 + 1^4 = 2.
a(3) = a(2)^2 + a(1)^4 = 2^2 + 1^4 = 5.
a(4) = a(3)^2 + a(2)^4 = 5^2 + 2^4 = 41.
a(5) = a(4)^2 + a(3)^4 = 41^2 + 5^4 = 2306.
a(6) = a(5)^2 + a(4)^4 = 2306^2 + 41^4 = 8143397.
a(7) = a(6)^2 + a(5)^4 = 8143397^2 + 2306^4 = 94592167328105.

Robert Israel, Table of n, a(n) for n = 0..13
Karl Petersen, Ibrahim Salama, Tree shift complexity, arXiv:1712.02251 [math.DS], 2017.
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A000283, A005154, A112969, A114793.

A[0]:= 1: A[1]:= 1:
for n from 2 to 10 do
  A[n]:= A[n-1]^2 + A[n-2]^4;
od:
seq(A[i],i=0..10); # Robert Israel, Aug 21 2017

RecurrenceTable[{a[n] == a[n-1]^2 + a[n-2]^4, a[0] ==1, a[1] == 1}, a, {n, 0, 10}] (* Vaclav Kotesovec, Dec 18 2014 *)
NestList[{#[[2]],#[[1]]^4+#[[2]]^2}&,{1,1},10][[All,1]] (* Harvey P. Dale, Jul 03 2021 *)

{a(n) = if( n<2, 1, a(n-1)^2 + a(n-2)^4)}

{a=[0,0];for(n=1,99,iferr(a=[a[2],log(exp(a*[4,0;0,2])*[1,1]~)],E,return([n,exp(a[2]/2^n)])))} \\ To compute an approximation of the constant c1 = exp(lim_{n->oo} (log a(n))/2^n). \\ M. F. Hasler, May 21 2017

a=vector(20); a[1]=1;a[2]=2; for(n=3, #a, a[n]=a[n-1]^2+a[n-2]^4); concat(1, a) \\ Altug Alkan, Apr 04 2018

If b(n) = 1 + 1/b(n-1)^2, b(1)=1, then b(n) = a(n)/a(n-1)^2.
Lim_{n->inf} a(n)/a(n-1)^2 = A092526 (constant).
a(n) is asymptotic to c1^(2^n) * c2.
c1 = 1.2897512927198122075..., c2 = 1/A092526 = A263719 = (1/6)*(108 + 12*sqrt(93))^(1/3) - 2/(108 + 12*sqrt(93))^(1/3) = 0.682327803828019327369483739711... is the root of the equation c2*(1 + c2^2) = 1. - Vaclav Kotesovec, Dec 18 2014

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 15 2007

A237979 Number of strict partitions of n such that (least part) > number of parts.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 20, 22, 25, 28, 32, 35, 40, 45, 50, 56, 63, 70, 78, 87, 96, 107, 118, 131, 144, 160, 175, 194, 213, 235, 257, 284, 310, 342, 373, 410, 447, 491, 534, 585, 637, 696, 756, 826, 896, 977, 1060, 1153, 1250, 1359, 1471, 1597, 1729, 1874, 2026, 2195, 2371, 2565

Offset: 1

Clark Kimberling, Feb 18 2014

Also the number of partitions into distinct parts with minimal part >= 2 and difference between parts >= 3. [Joerg Arndt, Mar 31 2014]

			a(9) = 3 counts these partitions:  9, 63, 54.

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

Cf. A000009, A237976, A096401, A237977, A025157.

z = 50; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]]
Table[Count[q[n], p_ /; Min[p] < t[p]], {n, z}]  (* A237976 *)
Table[Count[q[n], p_ /; Min[p] <= t[p]], {n, z}] (* A237977 *)
Table[Count[q[n], p_ /; Min[p] == t[p]], {n, z}] (* A096401 *)
Table[Count[q[n], p_ /; Min[p] > t[p]], {n, z}]  (* A237979 *)
Table[Count[q[n], p_ /; Min[p] >= t[p]], {n, z}] (* A025157 *)

N=66; q='q+O('q^N); Vec(-1+sum(n=0, N, q^(n*(3*n+1)/2) / prod(k=1, n, 1-q^k ) )) \\ Joerg Arndt, Mar 09 2014

G.f. with a(0)=0: sum(n>=0, q^(n*(3*n+1)/2) / prod(k=1..n, 1-q^k ) ). [Joerg Arndt, Mar 09 2014]
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(Pi*(1 + 3*r^2)) * n^(3/4)), where r = A263719 and c = 3*(log(r))^2/2 + polylog(2, 1-r). - Vaclav Kotesovec, Jan 15 2022
a(n) ~ A263719 * A025157(n). - Vaclav Kotesovec, Jan 15 2022

A069241 Number of Hamiltonian paths in the graph on n vertices {1,...,n}, with i adjacent to j iff |i-j| <= 2.

Original entry on oeis.org

1, 1, 1, 3, 6, 10, 17, 28, 44, 68, 104, 157, 235, 350, 519, 767, 1131, 1665, 2448, 3596, 5279, 7746, 11362, 16662, 24430, 35815, 52501, 76956, 112797, 165325, 242309, 355135, 520490, 762830, 1117997, 1638520, 2401384, 3519416, 5157972, 7559393, 11078847

Offset: 0

Don Knuth, Apr 13 2002

Equivalently, the number of bandwidth-at-most-2 arrangements of a straight line of n vertices.

			For example, the six Hamiltonian paths when n=4 are 1234, 1243, 1324, 1342, 2134, 3124.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-2,1).

Cf. A003274, A000930, A092526, A263719, A302119.

a:= n-> (Matrix([[1,1,1,0,1]]). Matrix(5, (i,j)-> if i=j-1 then 1 elif j=1 then [3,-3,2,-2,1][i] else 0 fi)^n)[1,3]: seq(a(n), n=0..50); # Alois P. Heinz, Sep 09 2008

a[0] = a[1] = a[2] = 1; a[3] = 3; a[4] = 6; a[n_] := a[n] = 3a[n-1] - 3a[n-2] + 2a[n-3] - 2a[n-4] + a[n-5]; Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Feb 13 2015 *)
CoefficientList[Series[(3+x+x^2)/(1-x-x^3)-(2-x)/(1-x)^2,{x,0,60}],x] (* or *) LinearRecurrence[{3,-3,2,-2,1},{1,1,1,3,6},60] (* Harvey P. Dale, Apr 07 2019 *)

a(n) = A003274(n)/2, n > 1.
a(n) = 3*s(n) + s(n-1) + s(n-2) - 2 - n, where s(n) = A000930(n).
G.f.: (3+x+x^2)/(1-x-x^3) - (2-x)/(1-x)^2.
Lim_{n->infinity} a(n+1)/a(n) = A092526 = 1/A263719. - Alois P. Heinz, Apr 15 2018

A274112 Number of equivalence classes of ballot paths of length n for the string ddu.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 8, 12, 17, 23, 35, 52, 75, 105, 157, 232, 337, 480, 712, 1049, 1529, 2199, 3248, 4777, 6976, 10092, 14869, 21845, 31937, 46377, 68222, 100159, 146536, 213328, 313487, 460023, 673351, 981976, 1441999, 2115350, 3097326, 4522529, 6637879, 9735205, 14257734, 20836827, 30572032, 44829766, 65666593

Offset: 0

N. J. A. Sloane, Jun 17 2016

Gheorghe Coserea, Table of n, a(n) for n = 0..300
K. Manes, A. Sapounakis, I. Tasoulas, P. Tsikouras, Equivalence classes of ballot paths modulo strings of length 2 and 3, arXiv:1510.01952 [math.CO], 2015, Section 3.4.

Cf. A274110-A274115, A263719.

A274112 := proc(n)
    add( (n-4*i+1)/(n-3*i+1)*binomial(n-2*i,i),i=0..n/4) ;
end proc:
seq(A274112(n),n=0..50) ; # R. J. Mathar, Jun 20 2016

a[n_] := Sum[(n - 4*i + 1)/(n - 3*i + 1)*Binomial[n - 2*i, i], {i, 0, n/4} ];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 27 2017, after R. J. Mathar *)

x='x; y='y;
Fxy = x*(x^3+x-1)*y^2 + (2*x-1)*y + 1;
seq(N) = {
  my(y0 = 1 + O('x^N), y1=0);
  for (k = 1, N,
    y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy, y), y, y0);
    if (y1 == y0, break()); y0 = y1);
  Vec(y0);
};
seq(51)  \\ Gheorghe Coserea, Jan 05 2017

G.f. y satisfies: 0 = x*(x^3+x-1)*y^2 + (2*x-1)*y + 1. - Gheorghe Coserea, Jan 05 2017
G.f.: 1/(1 - x - x^4/(1 - x^4/(1 - x^4/(1 - x^4/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Jul 26 2017
a(n) ~ 3 * (1+r^2)^(n+1) / (7 + 4*r + 8*r^2), where r = A263719 = ((9+sqrt(93))/2)^(1/3)/3^(2/3) - (2/(3*(9+sqrt(93))))^(1/3) = 0.682327803828019327369483739711048256891188581898... is the real root of the equation r^3 + r = 1. - Vaclav Kotesovec, Nov 27 2017

a(0)=1 prepended by Gheorghe Coserea, Jan 05 2017

A341534 Number of possible final configurations in a biased cake-cutting procedure for n people.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 7, 8, 11, 12, 13, 14, 16, 18, 19, 20, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 35, 37, 38, 39, 40, 41, 43, 45, 47, 48, 49, 50, 51, 52, 53, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Luc Rousseau, Feb 13 2021

nonn

A cake of size 1 is to be divided between n people. The cutter iterates the following procedure: to obtain a new piece of the cake, he cuts the biggest piece into two subpieces (if several pieces are biggest ex aequo, he cuts just one of them); the cut is biased in the proportions t versus 1-t, where t is a constant real number between 1/2 and 1. We will assume that t is transcendental. After the procedure is applied, each piece's size is clearly t^x*(1-t)^y for some (x,y), ordered pair of nonnegative integers. The obtained ordered multiset of t^x*(1-t)^y polynomials depends on t and n; we shall call this the f(t,n) configuration. By definition a(n) is Card({f(t,n); t varies}), i.e., the number of configurations that the procedure for n people can possibly generate, when one does not know the value of t before it starts.

The problem boils down to a geometrical one, where one has to compare the Y-coordinates of rotated lattice points, the angle of rotation depending on t: tan(angle)=log(1-t)/log(t). See SVG link.

			=========================================================================
  1 |    2    |          3          |                4                  |
=========================================================================
    |         |                     | t^3 > 1-t > t*(1-t) > t^2*(1-t)   |
    |         | t^2 > 1-t > t*(1-t) +-----------------------------------+
    |         |                     | 1-t > t^3 > t*(1-t) > t^2*(1-t)   |
  1 | t > 1-t +---------------------+-----------------------------------+
    |         | 1-t > t^2 > t*(1-t) | t^2 > t*(1-t) = t*(1-t) > (1-t)^2 |
=========================================================================
n=1: the whole cake is the only piece, a(1) = 1.
n=2: the first division necessarily divided 1 into t and 1-t; t and 1-t are necessarily ordered this way: t > 1-t. a(2) = 1.
n=3: the second division necessarily divided t into t^2 and t*(1-t); t*(1-t) is necessarily smaller than both t^2 and 1-t; but either t^2 or 1-t may be the biggest: it depends on whether t < 1/phi or t > 1/phi, where phi denotes the golden ratio; so there are two cases and a(3) = 2.
n=4:
  * in the case when t^2 > 1-t, the third division divided t^2 into t^3 and t^2*(1-t); t^2*(1-t) is necessarily smaller than t*(1-t) which is necessarily smaller than both t^3 and 1-t; but either t^3 or 1-t may be the biggest: it depends on whether t < 1/psi or t > 1/psi, where psi denotes the constant described in A092586 (sometimes called the supergolden ratio); so there are two subcases;
  * in the case when 1-t > t^2, the third division divided 1-t into t*(1-t) and (1-t)^2; the order of the elements is fully determined without requiring new assumptions on t, so there is just one subcase;
  * gathering all subcases contributions yields a(4) = 3.

Luc Rousseau, Java program
Luc Rousseau, SVG illustration for n=1..10

Cf. A094214, A001622 (1/phi, phi).

Cf. A263719, A092586 (1/psi, psi).

Java
```
// See Rousseau link.
```

A376707 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} 1/(1 - x^j)^3.

Original entry on oeis.org

1, 1, 3, 7, 13, 24, 41, 70, 114, 186, 293, 459, 703, 1067, 1593, 2359, 3447, 4998, 7175, 10222, 14445, 20277, 28263, 39156, 53922, 73843, 100587, 136331, 183890, 246909, 330094, 439453, 582738, 769782, 1013169, 1328805, 1736942, 2263018, 2939280, 3806072, 4914221

Offset: 0

Vaclav Kotesovec, Oct 02 2024

nonn

Cf. A000009, A143184, A376710.

Cf. A263719.

nmax = 40; CoefficientList[Series[Sum[x^(k*(k+1)/2)/Product[1-x^j, {j, 1, k}]^3, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]

a(n) ~ r^(1/6) * (log(r)^2 + 6*polylog(2, 1-r))^(3/4) * exp(sqrt(2*(log(r)^2 + 6*polylog(2, 1-r))*n)) / (2^(11/4) * Pi^(3/2) * sqrt(1 + 2*r) * n^(5/4)), where r = 1 - A263719 = 0.3176721961719... is the real root of the equation r = (1-r)^3.

cp's OEIS Frontend

A025157 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 3.

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A092526 Decimal expansion of (2/3)*cos( (1/3)*arccos(29/2) ) + 1/3, the real root of x^3 - x^2 - 1.

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A003274 Number of key permutations of length n: permutations {a_i} with |a_i - a_{i-1}| = 1 or 2.

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A096401 Number of balanced partitions of n into distinct parts: least part is equal to number of parts.

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A076725 a(n) = a(n-1)^2 + a(n-2)^4, a(0) = a(1) = 1.

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A237979 Number of strict partitions of n such that (least part) > number of parts.

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A092526 Decimal expansion of (2/3)cos( (1/3)arccos(29/2) ) + 1/3, the real root of x^3 - x^2 - 1.

A376707 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} 1/(1 - x^j)^3.