

A105321


Convolution of binomial(1,n) and Gould's sequence A001316.


10



1, 3, 4, 6, 6, 6, 8, 12, 10, 6, 8, 12, 12, 12, 16, 24, 18, 6, 8, 12, 12, 12, 16, 24, 20, 12, 16, 24, 24, 24, 32, 48, 34, 6, 8, 12, 12, 12, 16, 24, 20, 12, 16, 24, 24, 24, 32, 48, 36, 12, 16, 24, 24, 24, 32, 48, 40, 24, 32, 48, 48, 48, 64, 96, 66, 6, 8, 12, 12, 12, 16, 24, 20, 12, 16
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OFFSET

0,2


COMMENTS

A universal function related to the spherical growth of repeated truncations of maps.
a(n) = (number of ones in row n of triangle A249133) = (number of odd terms in row n of triangle A249095) = A000120(A249184(n)).  Reinhard Zumkeller, Nov 14 2014


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata
T. Pisanski and T. W. Tucker, Growth in Repeated Truncations of Maps, Preprint series, Univ. of Ljubljana, Vol. 38 (2000), 696.
T. Pisanski and T. W. Tucker, Growth in Repeated Truncations of Maps, Atti. Sem. Mat. Fis. Univ. Modena, Vol. 49 (2001), 167176.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS  Omar E. Pol, May 29 2010


FORMULA

G.f. (1+x)*Product{k>=0, 1+2x^(2^k)}; a(n) = sum{k=0..n, binomial(1, nk)*sum{j=0..k, binomial(k, j) mod 2}}.
a(n) = 2*A048460(n) for n>=2.  Omar E. Pol, Jan 02 2011
a((2*n1)*2^p) = (2^p+2)*A001316(n1), p >= 0 and n >= 1, with a(0) = 1.  Johannes W. Meijer, Jan 28 2013
a(n) = A001316(n) + A001316(n1) for n > 0.  Reinhard Zumkeller, Nov 14 2014


EXAMPLE

Contribution from Omar E. Pol, May 29 2010: (Start)
If written as a triangle:
1;
3;
4;
6,6;
6,8,12,10;
6,8,12,12,12,16,24,18;
6,8,12,12,12,16,24,20,12,16,24,24,24,32,48,34;
6,8,12,12,12,16,24,20,12,16,24,24,24,32,48,36,12,16,24,24,24,32,48,40,24,32,48,48,48,64,96,66;
(End)


MAPLE

nmax := 74: A001316 := n > if n <= 1 then 0 else 2^add(i, i=convert(n, base, 2)) fi: for p from 0 to ceil(log[2](nmax)) do for n from 1 to nmax/(p+2)+1 do a((2*n1)*2^p) := (2^p+2) * A001316(n1) od: od: a(0) :=1: seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 28 2013


MATHEMATICA

f[n_] := Sum[Binomial[1, n  k]Mod[Binomial[k, j], 2], {k, 0, n}, {j, 0, k}]; Array[f, 75, 0] (* Robert G. Wilson v, Jun 28 2010 *)


PROG

(Haskell)
a105321 n = if n == 0 then 1 else a001316 n + a001316 (n  1)
 Reinhard Zumkeller, Nov 14 2014


CROSSREFS

Cf. A139250, A139251, A173522, A048460.
Cf. A001316, A000120, A249184, A249095, A249133.
Sequence in context: A298808 A033957 A031131 * A217032 A160095 A135319
Adjacent sequences: A105318 A105319 A105320 * A105322 A105323 A105324


KEYWORD

easy,nonn


AUTHOR

Paul Barry, Apr 01 2005


STATUS

approved



