A210964 Column 10 of square array A195825. Also column 1 of triangle A210954. Also 1 together with the row sums of triangle A210954.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 13, 13, 13, 13, 13, 14, 16, 21, 27, 32, 34, 35, 35, 35, 35, 35, 36, 38, 44, 54, 67, 77, 83, 85, 86, 86, 86, 87, 89, 95, 107, 128, 152, 173, 185, 191, 193, 194, 195

Offset: 0

Author

Omar E. Pol, Jun 16 2012

Keywords

• nonn

Comments

Note that this sequence contains five plateaus: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4, 4, 4, 4, 4], [13, 13, 13, 13, 13, 13, 13], [35, 35, 35, 35, 35], [86, 86, 86]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..3000 from Vaclav Kotesovec)
• Vaclav Kotesovec, Graph - The asymptotic ratio
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A000041, A006950, A036820, A057077, A195818, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A210944, A210954, A211970, A211971.

Cf. A247133.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[1 / ((1 - x^(12*k)) * (1 - x^(12*k-1)) * (1 - x^(12*k-11))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 08 2015 *)

Formula

Expansion of 1 / f(-x, -x^11) in powers of x where f() is a Ramanujan theta function. - Michael Somos, Jan 10 2015

Partitions of n into parts of the form 12*k, 12*k+1, 12*k+11. - Michael Somos, Jan 10 2015

Euler transform of period 12 sequence [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, ...]. - Michael Somos, Jan 10 2015

G.f.: Product_{k>0} 1 / ((1 - x^(12*k)) * (1 - x^(12*k - 1)) * (1 - x^(12*k - 11))).

Convolution inverse of A247133.

a(n) ~ sqrt(2)*(1+sqrt(3)) * exp(Pi*sqrt(n/6)) / (8*n). - Vaclav Kotesovec, Nov 08 2015

a(n) = (1/n)*Sum_{k=1..n} A284372(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

a(n) = a(n-1) + a(n-11) - a(n-14) - a(n-34) + + - - (with the convention a(n) = 0 for negative n), where 1, 11, 14, 34, ... is the sequence of generalized 14-gonal numbers A195818. - Peter Bala, Dec 10 2020