A212831 -id:A212831 - cp's OEIS frontend

A218009 Binomial transform of A212831(n).

0, 1, 4, 12, 30, 70, 160, 364, 824, 1848, 4096, 8976, 19488, 42016, 90112, 192448, 409472, 868224, 1835008, 3866880, 8126976, 17039872, 35651584, 74447872, 155187200, 322959360, 671088640, 1392513024, 2885689344, 5972697088, 12348030976, 25501351936, 52613316608

Offset: 0

Paul Curtz, Oct 18 2012

Companion to A217988.
Considering a(n+1) - 2*a(n) = 1,2,4,6,10,20,44,96,200,... = b(n), is
b(n+3) - 2*b(n+2) = -2,-2,0,4,8,8,0,-16,-32,-32,0,... = -A009545(n+2).

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. A212831, A214282, A214283, A217988, A009545.

I:=[0, 1, 4, 12, 30, 70]; [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[(1/4)*Binomial[n, k]*((-(1 + (-1)^k))*(-1 + (-1)^Floor[k/2]) - (-3 + (-1)^k)*k), {k, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 18 2012 *)
CoefficientList[Series[x*(1 - 2*x + 2*x^2 - 2*x^3 + 2*x^4)/((1 - 2*x)^2*(1 - 2*x + 2*x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 15 2012 *)

G.f.: x*(1-2*x+2*x^2-2*x^3+2*x^4)/((1-2*x)^2*(1-2*x+2*x^2)). - Bruno Berselli, Oct 18 2012
a(n) = 2^(n-3)*(3*n+2)+(1+i)^(n-4)+(1-i)^(n-4), where i=sqrt(-1) and n>1, with a(0)=0, a(1)=1. - Bruno Berselli, Oct 18 2012
a(n) = 6*a(n-1) -14*a(n-2) +16*a(n-3) -8*a(n-4).

A214283 Smallest Euler characteristic of a downset on an n-dimensional cube.

Original entry on oeis.org

0, -1, -2, -3, -4, -10, -20, -35, -56, -126, -252, -462, -792, -1716, -3432, -6435, -11440, -24310, -48620, -92378, -167960, -352716, -705432, -1352078, -2496144, -5200300, -10400600, -20058300, -37442160, -77558760, -155117520, -300540195

Offset: 1

Terence Tao, Jul 09 2012

sign

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.

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

Cf. A000108, A001405, A004525, A007318, A212831, A214282.

a214283 1 = 0
a214283 n = - a007318 (n - 1) (a004525 n)
-- Reinhard Zumkeller, Jul 14 2012

A212831[n_] := (1/4)*((-(1+(-1)^n))*(-1+(-1)^Floor[n/2]) - (-3+(-1)^n)*n); a[n_] := -CatalanNumber[Floor[(n-1)/2]]*A212831[n-1]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Nov 06 2012, after Paul Curtz *)

a(n)=-binomial(n-1,if(n%2,if(n%4==3,n-1,n+1),n)/2) \\ Charles R Greathouse IV, Jul 10 2012

from math import comb
def A214283(n): return -comb(n-1,(n>>1)|(n&1)) # Chai Wah Wu, Jan 31 2024

a(n=2k) = -binomial(n-1,n/2) = -binomial(2k-1,k),
a(n=4k+3) = -binomial(n-1,(n-1)/2) = -binomial(4k+2,2k+1),
a(n=4k+1) = -binomial(n-1,(n+1)/2) = -binomial(4k,2k+1).
a(n) = A214282(n) - A001405(n). - Reinhard Zumkeller, Jul 14 2012
For n > 1: a(n) = - A007318(n-1, A004525(n)). - Reinhard Zumkeller, Jul 14 2012
a(n+1) = -A000108(n/2) * A212831(n). - Paul Curtz, Nov 04 2012

A212849 Numbers whose sum of prime factors (counted with multiplicity) is a triangular number.

Original entry on oeis.org

1, 3, 8, 9, 21, 25, 26, 30, 32, 36, 38, 44, 68, 86, 105, 106, 112, 115, 125, 126, 138, 150, 155, 160, 164, 178, 180, 186, 187, 192, 195, 203, 206, 208, 216, 231, 234, 243, 266, 275, 290, 299, 302, 305, 323, 330, 338, 343, 348, 352, 365, 366, 380, 396, 404, 413

Offset: 1

Jonathan Vos Post, May 28 2012

This is to A212831 Numbers whose sum of prime factors is a square (counted with multiplicity) as A000290 squares are to A000217 triangular numbers.

			sopfr(21) = sum of primes dividing 21 (with repetition) = 10, which is the  4th triangular number, so 21 is in this sequence.
The number 1 is here because the sum of its prime factors is 0, which is a triangular number.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000217, A001414, A212831.

triangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; fQ[n_] := Module[{p, e}, {p, e} = Transpose[FactorInteger[n]]; triangularQ[Dot[p, e]]]; Join[{1}, Select[Range[2, 500], fQ]] (* T. D. Noe, May 30 2012 *)

sopfr(n) = my(f=factor(n)); sum(k=1, matsize(f)[1], f[k, 1]*f[k, 2]); \\ A001414
isok(n) = ispolygonal(sopfr(n), 3); \\ Michel Marcus, May 02 2018

{k such that A001414(k) = sopfr(k) is in A000217}.

A246416 A permutation of essentially the duplicate nonnegative numbers: a(4n) = n + 1/2 - (-1)^n/2, a(2n+1) = a(4n+2) = 2n+1.

Original entry on oeis.org

0, 1, 1, 3, 2, 5, 3, 7, 2, 9, 5, 11, 4, 13, 7, 15, 4, 17, 9, 19, 6, 21, 11, 23, 6, 25, 13, 27, 8, 29, 15, 31, 8, 33, 17, 35, 10, 37, 19, 39, 10, 41, 21, 43, 12, 45, 23, 47, 12, 49, 25, 51, 14, 53, 27, 55, 14, 57, 29, 59, 16, 61, 31, 63, 16

Offset: 0

Paul Curtz, Sep 14 2014

A permutation of A004526 (n > 0).
0 is at its own place. Distance between the two (2*k+1)'s: 2*k+1 terms. 0 is in position 0, the first 1 in position 1, the second 1 in position 2, the first 2 in position 4, the second 2 in position 8. Hence, r(n) = 0, 1, 2, 4, 8, 3, 6, 12, 16, 5, 10, 20, 24, ..., a permutation of A001477. See A225055. The recurrence r(n) = r(n-4) + r(n-8) - r(n-12) is the same as for a(n).
A061037(n+3) is divisible by a(n+5) (= 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 3, ...). Hence a link, via A212831 and A214282, between the Catalan numbers A000108 and the Balmer series.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,0,0,1,0,0,0,-1).

Cf. A000108, A004526, A052928, A061037, A176895, A212831, A214282, A226279, A225055, A247617.

I:=[0,1,1,3,2,5,3,7,2,9,5,11,4,13,7,15,4,17,9,19,6,21,11,23]; [n le 24 select I[n] else 3*Self(n-8)-3*Self(n-16)+Self(n-24): n in [1..80]]; // Vincenzo Librandi, Oct 15 2014

A246416:=n->n*(1+floor((2-n)/4)+floor((n-2)/4))/2+n*(1+floor((1-n)/2)+floor((n-1)/2))+(-n-2+2*(-1)^(n/4))*(ceil(n/4)-floor(n/4)-1)/4: seq(A246416(n), n=0..50); # Wesley Ivan Hurt, Sep 14 2014

Table[n (1 + Floor[(2 - n)/4] + Floor[(n - 2)/4])/2 + n (1 + Floor[(1 - n)/2] + Floor[(n - 1)/2]) + (-n - 2 + 2 (-1)^(n/4)) (Ceiling[n/4] - Floor[n/4] - 1)/4, {n, 0, 50}] (* Wesley Ivan Hurt, Sep 14 2014 *)
a[n_] := Switch[Mod[n, 4], 0, n/4-(-1)^(n/4)/2+1/2, 1|3, n, 2, n/2]; Table[a[n], {n, 0, 64}] (* Jean-François Alcover, Oct 09 2014 *)
LinearRecurrence[{0,0,0,1,0,0,0,1,0,0,0,-1},{0,1,1,3,2,5,3,7,2,9,5,11},70] (* Harvey P. Dale, Mar 23 2015 *)

a(n)=if(n%4,n/(2-n%2),if(n%8,1,0)+n/4) \\ Charles R Greathouse IV, Sep 14 2014

a(n) = 3*a(n-8) - 3*a(n-16) + a(n-24).
a(n+4) = a(n) + period 8: repeat [2, 4, 2, 4, 0, 4, 2, 4].
a(n+8) = a(n) + period 4: repeat [2, 8, 4, 8] (= 2 * A176895).
a(2n) = A212831(n).
a(n) = n*(1+floor((2-n)/4)+floor((n-2)/4))/2+n*(1+floor((1-n)/2)+floor((n-1)/2))+(-n-2+2*(-1)^(n/4))*(ceiling(n/4)-floor(n/4)-1)/4. - Wesley Ivan Hurt, Sep 14 2014
a(n) = a(n-4) + a(n-8) - a(n-12). - Charles R Greathouse IV, Sep 14 2014
G.f.: x*(x^10+x^9+3*x^8+4*x^6+2*x^5+4*x^4+2*x^3+3*x^2+x+1) / ((x-1)^2*(x+1)^2*(x^2+1)^2*(x^4+1)). - Colin Barker, Sep 15 2014

A212918 Numbers whose sum of prime factors (counted with multiplicity) is a pentagonal number (A000326).

Original entry on oeis.org

1, 5, 6, 35, 42, 50, 57, 60, 64, 72, 81, 85, 102, 121, 124, 182, 188, 201, 232, 260, 261, 267, 308, 312, 351, 440, 452, 495, 519, 528, 594, 645, 649, 688, 735, 741, 774, 784, 805, 854, 861, 875, 882, 901, 966, 1025, 1027, 1045, 1050, 1081, 1105, 1112, 1119

Offset: 1

Jonathan Vos Post, May 31 2012

This is to pentagonal numbers A000326 as A000290 squares are to A212831 numbers whose sum of prime factors is a square (counted with multiplicity) and as A000217 triangular numbers are to A212849 Numbers whose sum of prime factors (counted with multiplicity) is a triangular number.

			a(3) = 35 because sopfr(35) = sum of prime factors of 35 = 5 + 7 = 12, and  12 is the 3rd pentagonal number.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000326, A001414, A212831, A212849.

pentagonalQ[n_] := IntegerQ[(1 + Sqrt[1 + 24*n])/6]; pfs[n_] := Module[{p, e}, {p, e} = Transpose[FactorInteger[n]]; Dot[p, e]]; Select[Range[1500], pentagonalQ[pfs[#]] &] (* T. D. Noe, May 31 2012 *)

sopfr(n) = my(f=factor(n)); sum(k=1, matsize(f)[1], f[k, 1]*f[k, 2]); \\ A001414
isok(n) = ispolygonal(sopfr(n), 5); \\ Michel Marcus, May 02 2018

{k such that A001414(k) = sopfr(k) is in A000326(j) = j*(3*j-1)/2 for some integer j}.

Corrected by T. D. Noe, May 31 2012

cp's OEIS Frontend

A218009 Binomial transform of A212831(n).

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A214283 Smallest Euler characteristic of a downset on an n-dimensional cube.

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A212849 Numbers whose sum of prime factors (counted with multiplicity) is a triangular number.

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A246416 A permutation of essentially the duplicate nonnegative numbers: a(4n) = n + 1/2 - (-1)^n/2, a(2n+1) = a(4n+2) = 2n+1.

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A212918 Numbers whose sum of prime factors (counted with multiplicity) is a pentagonal number (A000326).

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