A211422 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.
1, 9, 17, 25, 41, 49, 57, 65, 81, 105, 113, 121, 137, 145, 153, 161, 193, 201, 225, 233, 249, 257, 265, 273, 289, 329, 337, 361, 377, 385, 393, 401, 433, 441, 449, 457, 505, 513, 521, 529, 545, 553, 561, 569, 585, 609, 617, 625, 657, 713, 753, 761
Offset: 0
Keywords
A178804 When dealing cards into 3 piles (Left, Center, Right), the number of cards in the n-th card's pile, if dealing in a pattern L, C, R, C, L, C, R, C, L, C, ... [as any thoughtful six-year-old will try to do when sharing a pile of candy among 3 people].
1, 1, 1, 2, 2, 3, 2, 4, 3, 5, 3, 6, 4, 7, 4, 8, 5, 9, 5, 10, 6, 11, 6, 12, 7, 13, 7, 14, 8, 15, 8, 16, 9, 17, 9, 18, 10, 19, 10, 20, 11, 21, 11, 22, 12, 23, 12, 24, 13, 25, 13, 26, 14, 27, 14, 28, 15, 29, 15, 30, 16, 31, 16, 32, 17, 33, 17, 34, 18, 35, 18, 36, 19, 37, 19, 38, 20, 39, 20
Offset: 1
Comments
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
- "TwoPi", A Cool Sequence Problem
- Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).
Programs
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Haskell
import Data.List (transpose) a178804 n = a178804_list !! (n-1) a178804_list = concat $ transpose [a008619_list, a000027_list] -- Reinhard Zumkeller, Nov 15 2014
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Magma
m:=90; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!( x*(1+x+x^3)/((1+x^2)*(x-1)^2*(1+x)^2) )); // G. C. Greubel, Jan 23 2019 -
Mathematica
CoefficientList[Series[x*(1+x+x^3)/((1+x^2)*(x-1)^2*(1+x)^2), {x,0,90}], x] (* G. C. Greubel, Jan 23 2019 *) -
PARI
my(x='x+O('x^90)); Vec(x*(1+x+x^3)/((1+x^2)*(x-1)^2*(1+x)^2)) \\ G. C. Greubel, Jan 23 2019 -
Sage
a=(x*(1+x+x^3)/((1+x^2)*(x-1)^2*(1+x)^2)).series(x, 90).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jan 23 2019
Formula
a(n) = ceiling(n/4) if n is odd, n/2 if n is even.
From R. J. Mathar, Jun 19 2010: (Start)
a(n) = a(n-2) + a(n-4) - a(n-6).
G.f.: x*(1+x+x^3) / ( (1+x^2)*(x-1)^2*(1+x)^2 ). (End)
a(n) = (3n+1-2(-1)^((n+3+(1-n)(-1)^n)/4)+(n-3)(-1)^n)/8. - Wesley Ivan Hurt, Mar 19 2015
Comments
Examples
Links
Crossrefs
Programs
Mathematica
t[n_] := t[n] = Flatten[Table[w^2 + x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]] c[n_] := Count[t[n], 0] t = Table[c[n], {n, 0, 70}] (* A211422 *) (t - 1)/8 (* A120486 *)