A052928 -id:A052928 - cp's OEIS frontend

A294273 Sum of the sixth powers of the parts in the partitions of n into two parts.

0, 2, 65, 858, 4890, 21244, 67171, 188916, 446964, 994030, 1978405, 3796622, 6735950, 11680408, 19092295, 30745064, 47260136, 71929146, 105409929, 153455810, 216455810, 303993492, 415601835, 566623708, 754740700, 1003708134, 1307797101, 1702747126

Offset: 1

Wesley Ivan Hurt, Oct 26 2017

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,7,-7,-21,21,35,-35,-35,35,21,-21,-7,7,1,-1).

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), A294270 (k=3), A294271 (k=4), A294272 (k=5), this sequence (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).

[(n/42 - n^3/6 + n^5/2 + 1/128*(-63 + (-1)^n)*n^6 + n^7/7) : n in [1..50]]; // Wesley Ivan Hurt, Jul 12 2025

Table[Sum[i^6 + (n - i)^6, {i, Floor[n/2]}], {n, 50}]

concat(0, Vec(x^2*(2 + 63*x + 779*x^2 + 3591*x^3 + 10845*x^4 + 19026*x^5 + 23850*x^6 + 19026*x^7 + 10600*x^8 + 3591*x^9 + 723*x^10 + 63*x^11 + x^12) / ((1 - x)^8*(1 + x)^7) + O(x^40))) \\ Colin Barker, Nov 20 2017

a(n) = Sum_{i=1..floor(n/2)} i^6 + (n-i)^6.
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^2*(2 + 63*x + 779*x^2 + 3591*x^3 + 10845*x^4 + 19026*x^5 + 23850*x^6 + 19026*x^7 + 10600*x^8 + 3591*x^9 + 723*x^10 + 63*x^11 + x^12) / ((1 - x)^8*(1 + x)^7).
a(n) = (n/42 - n^3/6 + n^5/2 + 1/128*(-63 + (-1)^n)*n^6 + n^7/7).
a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) - 21*a(n-4) + 21*a(n-5) + 35*a(n-6) - 35*a(n-7) - 35*a(n-8) + 35*a(n-9) + 21*a(n-10) - 21*a(n-11) - 7*a(n-12) + 7*a(n-13) + a(n-14) - a(n-15) for n>15.
(End)

A294274 Sum of the seventh powers of the parts in the partitions of n into two parts.

Original entry on oeis.org

0, 2, 129, 2444, 18700, 99012, 376761, 1216688, 3297456, 8158550, 18080425, 37847532, 73399404, 136971464, 241561425, 414517952, 680856256, 1095977898, 1703414961, 2607286700, 3877286700, 5697862412, 8172733129, 11613390384, 16164030000, 22330294142

Offset: 1

Wesley Ivan Hurt, Oct 26 2017

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,8,-8,-28,28,56,-56,-70,70,56,-56,-28,28,8,-8,-1,1).

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), A294270 (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), this sequence (k=7), A294275 (k=8), A294276 (k=9), A294279 (k=10).

[n^2*(64 - 224*n^2 + 448*n^4 - 381*n^5 + 96*n^6 + 3*n^5*(-1)^n)/768 : n in [1..50]]; // Wesley Ivan Hurt, Jul 12 2025

Table[Sum[i^7 + (n - i)^7, {i, Floor[n/2]}], {n, 40}]

concat(0, Vec(x^2*(2 + 127*x + 2299*x^2 + 15240*x^3 + 61848*x^4 + 151257*x^5 + 262139*x^6 + 306832*x^7 + 260914*x^8 + 151257*x^9 + 60777*x^10 + 15240*x^11 + 2180*x^12 + 127*x^13+ x^14) / ((1 - x)^9*(1 + x)^8) + O(x^40))) \\ Colin Barker, Nov 20 2017

a(n) = Sum_{i=1..floor(n/2)} i^7 + (n-i)^7.
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^2*(2 + 127*x + 2299*x^2 + 15240*x^3 + 61848*x^4 + 151257*x^5 + 262139*x^6 + 306832*x^7 + 260914*x^8 + 151257*x^9 + 60777*x^10 + 15240*x^11 + 2180*x^12 + 127*x^13+ x^14) / ((1 - x)^9*(1 + x)^8).
a(n) = a(n-1) + 8*a(n-2) - 8*a(n-3) - 28*a(n-4) + 28*a(n-5) + 56*a(n-6) - 56*a(n-7) - 70*a(n-8) + 70*a(n-9) + 56*a(n-10) - 56*a(n-11) - 28*a(n-12) + 28*a(n-13) + 8*a(n-14) - 8*a(n-15) - a(n-16) + a(n-17) for n>17.
(End)
a(n) = n^2*(64 - 224*n^2 + 448*n^4 - 381*n^5 + 96*n^6 + 3*n^5*(-1)^n)/768. - Wesley Ivan Hurt, Jul 12 2025

A294275 Sum of the eighth powers of the parts in the partitions of n into two parts.

Original entry on oeis.org

0, 2, 257, 7074, 72354, 469540, 2142595, 7972932, 24684612, 68121958, 167731333, 383769830, 812071910, 1633567432, 3103591687, 5683259528, 9961449608, 16980253770, 27957167625, 45040730666, 70540730666, 108577948908, 163239463563, 241980430540, 351625763020

Offset: 1

Wesley Ivan Hurt, Oct 26 2017

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,9,-9,-36,36,84,-84,-126,126,126,-126,-84,84,36,-36,-9,9,1,-1).

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), A294270 (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), this sequence (k=8), A294276 (k=9), A294279 (k=10).

[-n*(768-5120*n^2+10752*n^4-15360*n^6+11475*n^7-2560*n^8-45*n^7*(-1)^n)/23040 : n in [1..50]]; // Wesley Ivan Hurt, Jul 12 2025

Table[Sum[i^8 + (n - i)^8, {i, Floor[n/2]}], {n, 40}]

concat(0, Vec( x^2*(2 + 255*x + 6799*x^2 + 62985*x^3 + 335905*x^4 + 1094715*x^5 + 2500907*x^6 + 3982845*x^7 + 4690633*x^8 + 3982845*x^9 + 2489581*x^10 + 1094715*x^11 + 331859*x^12 + 62985*x^13 + 6553*x^14 + 255*x^15 + x^16) / ((1 - x)^10*(1 + x)^9) + O(x^40))) \\ Colin Barker, Nov 20 2017

a(n) = Sum_{i=1..floor(n/2)} i^8 + (n-i)^8.
From Colin Barker, Nov 20 2017: (Start)
G.f.: x^2*(2 + 255*x + 6799*x^2 + 62985*x^3 + 335905*x^4 + 1094715*x^5 + 2500907*x^6 + 3982845*x^7 + 4690633*x^8 + 3982845*x^9 + 2489581*x^10 + 1094715*x^11 + 331859*x^12 + 62985*x^13 + 6553*x^14 + 255*x^15 + x^16) / ((1 - x)^10*(1 + x)^9).
a(n) = a(n-1) + 9*a(n-2) - 9*a(n-3) - 36*a(n-4) + 36*a(n-5) + 84*a(n-6) - 84*a(n-7) - 126*a(n-8) + 126*a(n-9) + 126*a(n-10) - 126*a(n-11) - 84*a(n-12) + 84*a(n-13) + 36*a(n-14) - 36*a(n-15) - 9*a(n-16) + 9*a(n-17) + a(n-18) - a(n-19) for n>19.
(End)
a(n) = -n*(768-5120*n^2+10752*n^4-15360*n^6+11475*n^7-2560*n^8-45*n^7*(-1)^n)/23040. - Wesley Ivan Hurt, Jul 12 2025

A294276 Sum of the ninth powers of the parts in the partitions of n into two parts.

Original entry on oeis.org

0, 2, 513, 20708, 282340, 2255148, 12313161, 52928912, 186884496, 576258110, 1574304985, 3942330372, 9092033028, 19736886008, 40357579185, 78935156288, 147520415296, 266495712282, 464467582161, 788155279940, 1299155279940, 2095793274212, 3300704544313

Offset: 1

Wesley Ivan Hurt, Oct 26 2017

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,10,-10,-45,45,120,-120,-210,210,252,-252,-210,210,120,-120,-45,45,10,-10,-1,1).

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), A294270 (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), this sequence (k=9), A294279 (k=10).

[-n^2*(768-2560*n^2+3584*n^4-3840*n^6+2555*n^7-512*n^8-5*n^7*(-1)^n)/5120 : n in [1..50]]; // Wesley Ivan Hurt, Jul 12 2025

Table[Sum[i^9 + (n - i)^9, {i, Floor[n/2]}], {n, 40}]

concat(0, Vec(x^2*(2 + 511*x + 20175*x^2 + 256522*x^3 + 1770948*x^4 + 7464688*x^5 + 21796206*x^6 + 45087574*x^7 + 69569484*x^8 + 79813090*x^9 + 69501528*x^10 + 45087574*x^11 + 21722580*x^12 + 7464688*x^13 + 1756842*x^14 + 256522*x^15 + 19674*x^16 + 511*x^17+ x^18) / ((1 - x)^11*(1 + x)^10) + O(x^40))) \\ Colin Barker, Nov 21 2017

a(n) = Sum_{i=1..floor(n/2)} i^9 + (n-i)^9.
From Colin Barker, Nov 21 2017: (Start)
G.f.: x^2*(2 + 511*x + 20175*x^2 + 256522*x^3 + 1770948*x^4 + 7464688*x^5 + 21796206*x^6 + 45087574*x^7 + 69569484*x^8 + 79813090*x^9 + 69501528*x^10 + 45087574*x^11 + 21722580*x^12 + 7464688*x^13 + 1756842*x^14 + 256522*x^15 + 19674*x^16 + 511*x^17+ x^18) / ((1 - x)^11*(1 + x)^10).
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) - 45*a(n-4) + 45*a(n-5) + 120*a(n-6) - 120*a(n-7) - 210*a(n-8) + 210*a(n-9) + 252*a(n-10) - 252*a(n-11) - 210*a(n-12) + 210*a(n-13) + 120*a(n-14) - 120*a(n-15) - 45*a(n-16) + 45*a(n-17) + 10*a(n-18) - 10*a(n-19) - a(n-20) + a(n-21) for n>21.
(End)
a(n) = -n^2*(768-2560*n^2+3584*n^4-3840*n^6+2555*n^7-512*n^8-5*n^7*(-1)^n)/5120. - Wesley Ivan Hurt, Jul 12 2025

A294279 Sum of the tenth powers of the parts in the partitions of n into two parts.

Original entry on oeis.org

0, 2, 1025, 61098, 1108650, 10933324, 71340451, 354864276, 1427557524, 4924107550, 14914341925, 40912232702, 102769130750, 240910097848, 529882277575, 1107606410024, 2206044295976, 4225524980826, 7792505423049, 13933571680850, 24163571680850, 40869390083652

Offset: 1

Wesley Ivan Hurt, Oct 26 2017

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,11,-11,-55,55,165,-165,-330,330,462,-462,-462,462,330,-330,-165,165,55,-55,-11,11,1,-1).

Sum of k-th powers of the parts in the partitions of n into two parts for k=0..10: A052928 (k=0), A093353 (k=1), A226141 (k=2), A294270 (k=3), A294271 (k=4), A294272 (k=5), A294273 (k=6), A294274 (k=7), A294275 (k=8), A294276 (k=9), this sequence (k=10).

[n*(5120-33792*n^2+67584*n^4-67584*n^6+56320*n^8-33759*n^9+6144*n^10+33*n^9*(-1)^n)/67584 : n in [1..50]]; // Wesley Ivan Hurt, Jul 13 2025

f:= proc(n)
if n::even then (1/66)*n*(6*n^10-(16863/512)*n^9+55*n^8-66*n^6+66*n^4-33*n^2+5)
  else (1/66*(n-1))*n*(2*n-1)*(n^2-n-1)*(3*n^6-9*n^5+2*n^4+11*n^3+3*n^2-10*n-5)
fi end proc:
map(f, [$1..50]); # Robert Israel, Oct 27 2017

Table[Sum[i^10 + (n - i)^10, {i, Floor[n/2]}], {n, 30}]

a(n) = Sum_{i=1..floor(n/2)} i^10 + (n-i)^10.
From Robert Israel, Oct 27 2017: (Start)
a(2*k) = (6144*k^10-16863*k^9+14080*k^8-4224*k^6+1056*k^4-132*k^2+5)*k/33.
a(2*k+1) = (6144*k^10+16896*k^9+14080*k^8-4224*k^6+1056*k^4-132*k^2+5)*k/33.
G.f.: x^2*(x^20+1023*x^19+59039*x^18+1036299*x^17+9117154*x^16+48940320*x^15
+178348744*x^14+465661416*x^13+907378474*x^12+1340492142*x^11+1528402822*x^10
+1340492142*x^9+908233636*x^8+465661416*x^7+178756096*x^6+48940320*x^5
+9163981*x^4+1036299*x^3+60051*x^2+1023*x+2)/((x^2-1)^11*(x-1)). (End)
a(n) = n*(5120-33792*n^2+67584*n^4-67584*n^6+56320*n^8-33759*n^9+6144*n^10+33*n^9*(-1)^n)/67584. - Wesley Ivan Hurt, Jul 13 2025
a(n) = a(n-1) + 11*a(n-2) - 11*a(n-3) - 55*a(n-4) + 55*a(n-5) + 165*a(n-6) - 165*a(n-7) - 330*a(n-8) + 330*a(n-9) + 462*a(n-10) - 462*a(n-11) - 462*a(n-12) + 462*a(n-13) + 330*a(n-14) - 330*a(n-15) - 165*a(n-16) + 165*a(n-17) + 55*a(n-18) - 55*a(n-19) - 11*a(n-20) + 11*a(n-21) + a(n-22) - a(n-23). - Wesley Ivan Hurt, Jul 13 2025

A168273 a(n) = 2*n + (-1)^n - 1.

Original entry on oeis.org

0, 4, 4, 8, 8, 12, 12, 16, 16, 20, 20, 24, 24, 28, 28, 32, 32, 36, 36, 40, 40, 44, 44, 48, 48, 52, 52, 56, 56, 60, 60, 64, 64, 68, 68, 72, 72, 76, 76, 80, 80, 84, 84, 88, 88, 92, 92, 96, 96, 100, 100, 104, 104, 108, 108, 112, 112, 116, 116, 120, 120, 124, 124, 128, 128, 132

Offset: 1

Vincenzo Librandi, Nov 22 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A052928, A008586.

[2*n-1 + (-1)^n: n in [1..70]] // Vincenzo Librandi, May 14 2013

CoefficientList[Series[4 x / ((1+x) (1-x)^2), {x, 0, 70}], x] (* Vincenzo Librandi, May 14 2013 *)
LinearRecurrence[{1,1,-1},{0,4,4},50] (* G. C. Greubel, Jul 16 2016 *)

a(n)=2*n-1+(-1)^n \\ Charles R Greathouse IV, Oct 07 2015

From  R. J. Mathar, Jan 05 2011: (Start)
G.f.: 4*x^2/( (1+x)*(1-x)^2).
a(n) = 2*A052928(n).
a(n) = A008586(floor(n/2)). (End)
a(n) = 2*n - 2*(n mod 2). - Wesley Ivan Hurt, Jun 30 2013
E.g.f.: (1 + (2*x - 1)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 16 2016

A072705 Triangle of number of unimodal compositions of n into exactly k distinct terms.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 2, 0, 0, 1, 4, 0, 0, 0, 1, 4, 4, 0, 0, 0, 1, 6, 4, 0, 0, 0, 0, 1, 6, 8, 0, 0, 0, 0, 0, 1, 8, 12, 0, 0, 0, 0, 0, 0, 1, 8, 16, 8, 0, 0, 0, 0, 0, 0, 1, 10, 20, 8, 0, 0, 0, 0, 0, 0, 0, 1, 10, 28, 16, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 32, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 40, 40, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Henry Bottomley, Jul 04 2002

Also the number of compositions of n into exactly k distinct terms whose negation is unimodal. - Gus Wiseman, Mar 06 2020

			Rows start: 1; 1,0; 1,2,0; 1,2,0,0; 1,4,0,0,0; 1,4,4,0,0,0; 1,6,4,0,0,0,0; 1,6,8,0,0,0,0,0; etc. T(6,3)=4 since 6 can be written as 1+2+3, 1+3+2, 2+3+1, or 3+2+1 but not 2+1+3 or 3+1+2.
From _Gus Wiseman_, Mar 06 2020: (Start)
Triangle begins:
  1
  1  0
  1  2  0
  1  2  0  0
  1  4  0  0  0
  1  4  4  0  0  0
  1  6  4  0  0  0  0
  1  6  8  0  0  0  0  0
  1  8 12  0  0  0  0  0  0
  1  8 16  8  0  0  0  0  0  0
  1 10 20  8  0  0  0  0  0  0  0
  1 10 28 16  0  0  0  0  0  0  0  0
  1 12 32 24  0  0  0  0  0  0  0  0  0
  1 12 40 40  0  0  0  0  0  0  0  0  0  0
  1 14 48 48 16  0  0  0  0  0  0  0  0  0  0
(End)

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A060016, A072574, A072704. Row sums are A072706.
Column k = 2 is A052928.
Unimodal compositions are A001523.
Unimodal sequences covering an initial interval are A007052.
Strict compositions are A032020.
Non-unimodal strict compositions are A072707.
Unimodal compositions covering an initial interval are A227038.
Numbers whose prime signature is not unimodal are A332282.
Cf. A059204, A107429, A115981, A329398, A332285, A332286, A332578, A332870, A332874.

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, 1,
      expand(b(n, i-1) +`if`(i>n, 0, x*b(n-i, i-1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i)*ceil(2^(i-1)), i=1..n))(b(n$2)):
seq(T(n), n=1..14);  # Alois P. Heinz, Mar 26 2014

b[n_, i_] := b[n, i] = If[n > i*(i+1)/2, 0, If[n == 0, 1, Expand[b[n, i-1] + If[i > n, 0, x*b[n-i, i-1]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i]* Ceiling[2^(i-1)], {i, 1, n}]][b[n, n]]; Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],UnsameQ@@#&&unimodQ[#]&]],{n,12},{k,n}] (* Gus Wiseman, Mar 06 2020 *)

T(n,k) = 2^(k-1)*A060016(n,k) = T(n-k,k)+2*T(n-k,k-1) [starting with T(0,0)=0, T(0,1)=0 and T(n,1)=1 for n>0].

A137501 The even numbers repeated, with alternating signs.

Original entry on oeis.org

0, 0, 2, -2, 4, -4, 6, -6, 8, -8, 10, -10, 12, -12, 14, -14, 16, -16, 18, -18, 20, -20, 22, -22, 24, -24, 26, -26, 28, -28, 30, -30, 32, -32, 34, -34, 36, -36, 38, -38, 40, -40, 42, -42, 44, -44, 46, -46, 48, -48, 50, -50, 52, -52, 54, -54, 56, -56, 58, -58, 60, -60, 62, -62, 64, -64, 66, -66, 68, -68, 70, -70, 72, -72, 74

Offset: 0

Carlos Alberto da Costa Filho (cacau_dacosta(AT)hotmail.com), Apr 22 2008

The general formula for alternating sums of powers of even integers is in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,1)-(-1)^k P(n,2k+1))/2. Here n=1 and k shifted one place, thus a(k) = (P(1,1)-(-1)^(k-1) P(1,2(k-1)+1))/2. - Peter Luschny, Jul 12 2009

With just one 0 at the beginning, this is a permutation of all the even integers. - Alonso del Arte, Jun 24 2012

Index entries for linear recurrences with constant coefficients, signature (-1,1,1).

Cf. A052928, A064455, A123684, A153641.

den:= n -> (n-1/2+1/2*(-1)^n)*(-1)^n: seq(den(n),n=-10..10);
a := n -> (1+(-1)^n*(2*n-1))/2; # Peter Luschny, Jul 12 2009

Flatten[Table[{2n, -2n}, {n, 0, 39}]] (* Alonso del Arte, Jun 24 2012 *)
With[{enos=2*Range[0,40]},Riffle[enos,-enos]] (* Harvey P. Dale, Oct 12 2014 *)

a(n) = ( n - (1/2) + (1/2)*(-1)^n )*(-1)^n.
From R. J. Mathar, Feb 14 2010: (Start)
a(n) = -a(n-1) + a(n-2) + a(n-3).
G.f.: 2*x^2/((1-x) * (1+x)^2). (End)
a(n) = A064455(n) - A123684(n). - Jaroslav Krizek, Mar 22 2011

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

A118102 Triangle read by rows giving successive states of cellular automaton generated by "Rule 94" initiated with a single ON (black) cell.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0

Offset: 0

Eric W. Weisstein, Apr 12 2006

Totals for row k, starting with k = 0, are {1, 3, 4, 6, 6, 8, 8, 10, 10, 12, 12, ...}, i.e., {1, 3} followed by A052928(k + 3). - Michael De Vlieger, Oct 08 2015

			From _Michael De Vlieger_, Oct 08 2015: (Start)
First 12 rows, replacing "0" with ".", ignoring "0" outside of range of 1's for better visibility of ON cells:
                        1
                      1 1 1
                    1 1 . 1 1
                  1 1 1 . 1 1 1
                1 1 . 1 . 1 . 1 1
              1 1 1 . 1 . 1 . 1 1 1
            1 1 . 1 . 1 . 1 . 1 . 1 1
          1 1 1 . 1 . 1 . 1 . 1 . 1 1 1
        1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1
      1 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 1
    1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1
  1 1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1 1
1 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 1
(End)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..9999
Michael De Vlieger, Visualization of rows 0 - 31 of Rule 94
Eric Weisstein's World of Mathematics, Rule 94
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata

Cf. A052928.

This sequence, A118101 and A071033 are equivalent descriptions of the Rule 94 automaton.

clip[lst_] := Block[{p = Flatten@ Position[lst, 1]}, Take[lst, {Min@ p, Max@ p}]]; FromDigits[#, 2] & /@ Map[clip, CellularAutomaton[94, {{1}, 0}, 24]] ; clip /@ CellularAutomaton[94, {{1}, 0}, 9] // Flatten (* Michael De Vlieger, Oct 08 2015 *)

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