A215495 -id:A215495 - cp's OEIS frontend

A217988 Binomial transform of A215495(n).

1, 2, 4, 10, 26, 66, 160, 372, 840, 1864, 4096, 8944, 19424, 41952, 90112, 192576, 409728, 868480, 1835008, 3866368, 8125952, 17038848, 35651584, 74449920, 155191296, 322963456, 671088640, 1392504832, 2885672960, 5972680704, 12348030976, 25501384704

Offset: 0

Paul Curtz, Oct 17 2012

Companion to A218009.
Like any other sequence with a linear recurrence with constant coefficients, this sequence is periodic if read modulo some constant m. These Pisano period lengths for m>=1 are 1, 1, 8, 1, 20, 8, 168, 1, 24, 20, 440, 8, 156, 168, 40, 1, 272, 24, 1368, 20, ... [Curtz's comment reformulated and extended by R. J. Mathar, Oct 23 2012]
Let b(n) = a(n+1)-2*a(n), then b(n+3)-2*b(n+2) = A009545(n+2). - edited by Michel Marcus, Apr 24 2018

			a(n) and successive differences:
1, 2,  4, 10, 26,  66, 160, 372,  840, 1864, 4096, ...
1, 2,  6, 16, 40,  94, 212, 468, 1024, ...
1, 4, 10, 24, 54, 118, 256, ...
3, 6, 14, 30, 64, ...
3, 8, 16, ...
5, 8, ...
3, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-8).

Cf. A214282, A214283.

I:=[1, 2, 4, 10, 26, 66]; [n le 6 select I[n] else 6*Self(n-1) - 14*Self(n-2) + 16*Self(n-3) - 8*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 15 2012

a[n_] := Sum[ Binomial[n, k]*If[ OddQ[k], k, k/2 + Boole[ Mod[k, 4] == 0]], {k, 0, n}]; Table[ a[n], {n, 0, 31}] (* Jean-François Alcover, Oct 17 2012 *)
CoefficientList[Series[(1-4*x+6*x^2-2*x^3-2*x^4+2*x^5)/((1-2*x)^2 * (1 - 2*x + 2*x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 15 2012 *)
LinearRecurrence[{6,-14,16,-8},{1,2,4,10,26,66},40] (* Harvey P. Dale, Aug 14 2018 *)

x='x+O('x^30); Vec((1-4*x+6*x^2-2*x^3-2*x^4+2*x^5)/((1-2*x)^2*(1-2*x+2*x^2))) \\ G. C. Greubel, Apr 23 2018

a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 8*a(n-4) with n > 5.
a(n) = A218009(n) + A146559(n).
G.f.: (1-4*x+6*x^2-2*x^3-2*x^4+2*x^5)/((1-2*x)^2*(1-2*x+2*x^2)). - Bruno Berselli, Oct 22 2012
a(n) = 2^(n-3)*(3*n+2)+((1+i)^n+(1-i)^n)/4, where i=sqrt(-1) and n>1, with a(0)=1, a(1)=2.

A214282 Largest Euler characteristic of a downset on an n-dimensional cube.

Original entry on oeis.org

1, 1, 1, 3, 6, 10, 15, 35, 70, 126, 210, 462, 924, 1716, 3003, 6435, 12870, 24310, 43758, 92378, 184756, 352716, 646646, 1352078, 2704156, 5200300, 9657700, 20058300, 40116600, 77558760, 145422675, 300540195, 601080390, 1166803110, 2203961430, 4537567650, 9075135300, 17672631900

Offset: 1

Terence Tao, Jul 09 2012

nonn

An m-downset is a set of subsets of 1..m such that if S is in the set, so are all subsets of S. The Euler characteristic of a downset is the number of sets in the downset with an even cardinality, minus the number with an odd cardinality.

			G.f. = x + x^2 + x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 15*x^7 + 35*x^8 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Terry Tao, Optimal bounds for an alternating sum on a downset, 2012.

Cf. A214283.

a214282 n = a007318 (n - 1) (a004524 (n - 1))
-- Reinhard Zumkeller, Jul 14 2012

Table[{Binomial[n - 1, n/2], Binomial[n, n/2], Binomial[n + 1, n/2 + 1], Binomial[n + 2, n/2 + 2]}, {n, 0, 28, 4}] (* Alonso del Arte, Jul 09 2012 *)

a(n)=binomial(n-1,if(n%2,(n+1)\4*2,n/2)) \\ Charles R Greathouse IV, Jul 09 2012

{a(n) = if( n<1, 0, vecmax( Vec((1 - x)^(n-1))))}; /* Michael Somos, Apr 21 2014 */

from math import comb
def A214282(n): return comb(n-1, (n+1>>1)&(-1^(n&1))) # Chai Wah Wu, Jan 31 2024

a(n) = binomial(n - 1, n/2) when n is even, a(n) = binomial(n - 1, (n + 1)/2) when n is 3 mod 4, and a(n) = binomial(n - 1, (n - 1)/2) when n is 1 mod 4.
a(2n) = A001700(n-1). a(4n+1) = A001448(n). a(4n+3) = A186231(n).
a(n) = A214283(n) + A001405(n). - Reinhard Zumkeller, Jul 14 2012
a(n) = A007318(n-1, A004524(n-1)). - Reinhard Zumkeller, Jul 14 2012
a(n+1) = A000108([n/2])*A215495(n). - M. F. Hasler, Aug 25 2012
A214282(n) - A214283(n) is A056040(n) if n is even and A056040(n)/((n+1)/2) otherwise. - Peter Luschny, Jul 08 2016

A212831 a(4n) = 2n, a(2n+1) = 2n+1, a(4n+2) = 2n+2.

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 4, 7, 4, 9, 6, 11, 6, 13, 8, 15, 8, 17, 10, 19, 10, 21, 12, 23, 12, 25, 14, 27, 14, 29, 16, 31, 16, 33, 18, 35, 18, 37, 20, 39, 20, 41, 22, 43, 22, 45, 24, 47, 24, 49, 26, 51, 26, 53, 28, 55, 28, 57, 30, 59, 30, 61, 32, 63, 32, 65, 34, 67, 34, 69, 36, 71, 36, 73, 38, 75

Offset: 0

Paul Curtz, Aug 14 2012

First differences: (1, 1, 1, -1, 3, -1, 3, -3, 5,...) = (1, A186422).
Second differences: (0, 0, -2, 4, -4, 4, -6, 8, ...)  = (-1)^(n+1) * A201629(n).
Interleave the terms with even indices of the companion A215495 and this one to get (A215495(0), A212831(0), A215495(2), A212831(2),...) = (1, 0, 1, 2, 3, 2, 3, 4, 5, 4,...) = A106249, up to the initial term = A083219 = A083220/2.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Cf. A214282, A129756.

[(1/4)*((1 +(-1)^n)*(1 - (-1)^Floor(n/2)) + (3 -(-1)^n)*n): n in [0..50]]; // G. C. Greubel, Apr 25 2018

a[n_] := (1/4)*((-(1 + (-1)^n))*(-1 + (-1)^Floor[n/2]) - (-3 + (-1)^n)*n ); Table[a[n], {n, 0, 84}] (* Jean-François Alcover, Sep 18 2012 *)
LinearRecurrence[{0,1,0,1,0,-1},{0,1,2,3,2,5},80] (* Harvey P. Dale, May 29 2016 *)

A212831(n)=if(bittest(n,0), n, n\2+bittest(n,1)) \\ M. F. Hasler, Oct 21 2012

for(n=0,50, print1((1/4)*((1 +(-1)^n)*(1 - (-1)^floor(n/2)) + (3 -(-1)^n)*n), ", ")) \\ G. C. Greubel, Apr 25 2018

a(n) + A215495(n) = A043547(n).
a(n) = -A214283(n)/A000108([n/2]).
a(n+1) = (A186421(n)=0,1,2,1,4,...) + 1.
a(2*n) = A052928(n+1).
a(n+2) - a(n) = 2, 2, 0, 2. (period 4).
a(n) = a(n-2) +a(n-4) -a(n-6); also holds for A215495(n).
G.f.: x*(1+2*x+2*x^2+x^4) / ( (x^2+1)*(x-1)^2*(1+x)^2 ). - R. J. Mathar, Aug 21 2012
a(n) = (1/4)*((1 +(-1)^n)*(1 - (-1)^floor(n/2)) + (3 -(-1)^n)*n). - G. C. Greubel, Apr 25 2018

Corrected and edited by M. F. Hasler, Oct 21 2012

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A217988 Binomial transform of A215495(n).

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A214282 Largest Euler characteristic of a downset on an n-dimensional cube.

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A212831 a(4*n) = 2*n, a(2*n+1) = 2*n+1, a(4*n+2) = 2*n+2.

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A212831 a(4n) = 2n, a(2n+1) = 2n+1, a(4n+2) = 2n+2.