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A131177
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Pyramid of squares: a pyramid is built row by row with squares of size 1.
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1
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1, 4, 11, 23, 41, 67, 102, 147, 204, 274, 358, 458, 575, 710, 865, 1041, 1239, 1461, 1708, 1981, 2282, 2612, 2972, 3364, 3789, 4248, 4743, 5275, 5845, 6455, 7106, 7799, 8536, 9318, 10146, 11022, 11947, 13949, 15029, 16163, 17353, 18600, 19905, 21270
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OFFSET
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1,2
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COMMENTS
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At the first step, we put a single square on the row 1. For the second we put a square above the first one on the row 2 and a square on each of its sides on row. At each following step, we begin a new row with one square and add a square at each end of all the previous rows. The term c(n) of the sequence is the total number of squares of any size which can be read in the pyramid.
..........................__
..........__...........__|__|__..
.__....__|__|__.....__|__|__|__|__
|__|..|__|__|__|...|__|__|__|__|__|
The 46 first terms of the sequence are exactly the same than those published in A092498 under a much more complex definition. However it's tempting to suspect a duplicate but I don't have the least idea of how to try to prove it.
In the above array, the first column refers to the height n of the pyramid described on the same row, the following ones to the size of the squares present in the concerned pyramid and the last column gives c(n), total of the line
1 | .1......................1
2 | .4......................4
3 | .9...2.................11
4 | 16...6...1.............23
5 | 25..12...4.............41
6 | 36..20...9...2.........67
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LINKS
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Table of n, a(n) for n=1..44.
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FORMULA
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If n is of the form 3*p c(n) = n*(4*n^2+9*n+3)/18
If n is of the form 3*p+1 c(n) = (n+2)*(4*n^2+n+1)/18
If n is of the form 3*p+2 c(n) = (n+1)*(4*n^2+5*n-2)/18
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EXAMPLE
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For n = 3 there are (5+3+1)=9 squares of size 1 on row 1,2 and 3 and 2 squares of size 2 laying on the floor 1, hence c(3)=11.
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CROSSREFS
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Cf. A092498.
Sequence in context: A009907 A027378 A092498 * A019298 A173702 A014242
Adjacent sequences: A131174 A131175 A131176 * A131178 A131179 A131180
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KEYWORD
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easy,nonn
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AUTHOR
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Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Sep 24 2007
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STATUS
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approved
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