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A183978 1/4 the number of (n+1) X 2 binary arrays with all 2 X 2 subblock sums the same. 4
4, 6, 9, 15, 25, 45, 81, 153, 289, 561, 1089, 2145, 4225, 8385, 16641, 33153, 66049, 131841, 263169, 525825, 1050625, 2100225, 4198401, 8394753, 16785409, 33566721, 67125249, 134242305, 268468225, 536920065, 1073807361, 2147581953, 4295098369 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Column 1 of A183986

Based on the conjectured recursion formula, it is also the number of notches in a sheet of paper when you fold it n times and cut off the four corners (see A274230). - Philippe Gibone, Jul 06 2016

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..46

Robert Israel, Proof of empirical formulas for A183978

FORMULA

Empirical: a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4)

Based on the conjectured recursion formula, we may prove (by a tedious induction) that a(n) = (2^ceiling(n/2) + 1) * (2^floor(n/2) + 1) = A051032(n) * A051032(n-1) for n >= 1. - Philippe Gibone, Jul 06 2016, corrected by Robert Israel, May 21 2019

Empirical: G.f.: -x*(4-6*x-9*x^2+12*x^3) / ( (x-1)*(2*x-1)*(2*x^2-1) ). - R. J. Mathar, Jul 15 2016

Empirical formulas verified (see link): Robert Israel, May 21 2019.

EXAMPLE

Some solutions for 5X2

..0..1....1..0....1..0....1..1....0..1....1..0....1..0....0..1....0..1....0..1

..0..0....1..0....1..0....1..0....1..0....1..0....1..0....0..1....1..0....0..1

..1..0....1..0....0..1....1..1....0..1....0..1....0..1....1..0....0..1....1..0

..0..0....1..0....1..0....0..1....1..0....1..0....0..1....1..0....0..1....0..1

..1..0....1..0....1..0....1..1....1..0....0..1....0..1....1..0....1..0....0..1

MAPLE

seq((1+2^floor((n-1)/2))*(1+2^ceil((n-1)/2)), n=1..20); # Robert Israel, May 21 2019

CROSSREFS

Cf. A274230.

Conjectured to be the main diagonal of A274636.

Sequence in context: A118691 A087718 A033476 * A118696 A065856 A136357

Adjacent sequences:  A183975 A183976 A183977 * A183979 A183980 A183981

KEYWORD

nonn

AUTHOR

R. H. Hardin, Jan 08 2011

STATUS

approved

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Last modified February 27 09:40 EST 2020. Contains 332301 sequences. (Running on oeis4.)