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A138137 First differences of A006128. 70
1, 2, 3, 6, 8, 15, 19, 32, 42, 64, 83, 124, 157, 224, 288, 395, 502, 679, 854, 1132, 1422, 1847, 2307, 2968, 3677, 4671, 5772, 7251, 8908, 11110, 13572, 16792, 20439, 25096, 30414, 37138, 44798, 54389, 65386, 78959, 94558, 113687, 135646, 162375, 193133 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Number of parts in the last section of the set of partitions of n (see A135010, A138121).

Sum of largest parts in all partitions in the head of the last section of the set of partitions of n. - Omar E. Pol, Nov 07 2011

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = A006128(n) - A006128(n-1).

a(n) = A000041(n-1) + A138135(n). - Omar E. Pol, Nov 07 2011

a(n) ~ exp(Pi*sqrt(2*n/3)) * (2*gamma + log(6*n/Pi^2)) / (8*sqrt(3)*n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 21 2016

G.f.: Sum_{i>=1} i*x^i * Product_{j=2..i} 1/(1 - x^j). - Ilya Gutkovskiy, Apr 04 2017

EXAMPLE

From Omar E. Pol, Feb 19 2012: (Start)

Illustration of initial terms (n = 1..6) as sums of the first columns from the last sections of the first six natural numbers (or from the first six sections of 6):

.                                           6

.                                           3+3

.                                           4+2

.                                           2+2+2

.                              5              1

.                              3+2              1

.                    4           1              1

.                    2+2           1              1

.            3         1           1              1

.      2       1         1           1              1

.  1     1       1         1           1              1

. --- ----- ------- --------- ----------- --------------

.  1,  2,    3,      6,        8,          15,

...

Also, we can see that the sequence gives the number of parts in each section. For the number of odd/even parts (and more) see A207031, A207032 and also A206563. (End)

From Omar E. Pol, Aug 16 2013: (Start)

The geometric model looks like this:

.                                           _ _ _ _ _ _

.                                          |_ _ _ _ _ _|

.                                          |_ _ _|_ _ _|

.                                          |_ _ _ _|_ _|

.                               _ _ _ _ _  |_ _|_ _|_ _|

.                              |_ _ _ _ _|           |_|

.                     _ _ _ _  |_ _ _|_ _|           |_|

.                    |_ _ _ _|         |_|           |_|

.             _ _ _  |_ _|_ _|         |_|           |_|

.       _ _  |_ _ _|       |_|         |_|           |_|

.   _  |_ _|     |_|       |_|         |_|           |_|

.  |_|   |_|     |_|       |_|         |_|           |_|

.

.   1    2      3        6          8           15

.

(End)

MAPLE

b:= proc(n, i) option remember; local f, g;

      if n=0 then [1, 0]

    elif i<1 then [0, 0]

    elif i>n then b(n, i-1)

    else f:= b(n, i-1); g:= b(n-i, i);

         [f[1]+g[1], f[2]+g[2] +g[1]]

      fi

    end:

a:= n-> b(n, n)[2] -b(n-1, n-1)[2]:

seq(a(n), n=1..50);  # Alois P. Heinz, Feb 19 2012

MATHEMATICA

b[n_, i_] := b[n, i] = Module[{f, g}, Which[n == 0, {1, 0}, i<1, {0, 0}, i>n, b[n, i-1], True, f = b[n, i-1]; g = b[n-i, i]; {f[[1]]+g[[1]], f[[2]]+g[[2]]+g[[1]]}]]; a[n_] := b[n, n][[2]]-b[n-1, n-1][[2]]; Table[a[n], {n, 1, 50}] (* Jean-Fran├žois Alcover, Mar 03 2014, after Alois P. Heinz *)

CROSSREFS

Column 1 of A207031.

Cf. A006128, A135010, A138121, A182703.

Sequence in context: A095162 A075723 A294496 * A319758 A129374 A209405

Adjacent sequences:  A138134 A138135 A138136 * A138138 A138139 A138140

KEYWORD

easy,nonn

AUTHOR

Omar E. Pol, Mar 18 2008

STATUS

approved

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Last modified March 26 19:03 EDT 2019. Contains 321511 sequences. (Running on oeis4.)