A173960 -id:A173960 - cp's OEIS frontend

A173961 Averages of two consecutive even cubes: (n^3 + (n+2)^3)/2.

4, 36, 140, 364, 756, 1364, 2236, 3420, 4964, 6916, 9324, 12236, 15700, 19764, 24476, 29884, 36036, 42980, 50764, 59436, 69044, 79636, 91260, 103964, 117796, 132804, 149036, 166540, 185364, 205556, 227164, 250236, 274820, 300964, 328716, 358124, 389236, 422100

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 03 2010

			(0^3+2^3)/2=4, (2^3+4^3)/2=36, ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005898, A027575, A173960.

I:=[4, 36, 140, 364]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 02 2012

f[n_]:=(n^3+(n+2)^3)/2;Table[f[n],{n,0,5!,2}]
CoefficientList[Series[(4+20*x+20*x^2+4*x^3)/(1-4*x+6*x^2-4*x^3+x^4),{x,0,40}],x] (* Vincenzo Librandi, Jul 02 2012 *)

a(n)=4*n*(2*n^2-3*n+3)-4 \\ Charles R Greathouse IV, Jan 02 2012

G.f.: x*(4+20*x+20*x^2+4*x^3)/(1-4*x+6*x^2-4*x^3+x^4). - Colin Barker, Jan 04 2012
a(n) = 8*n^3 - 12*n^2 + 12*n - 4. - Charles R Greathouse IV, Jan 02 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jul 02 2012
a(n) = 4*A005898(n-1).
E.g.f.: 4 + 4*exp(x)*(-1 + 2*x + 3*x^2 + 2*x^3). - Elmo R. Oliveira, Aug 23 2025

A173962 Averages of two consecutive odd cubes; a(n) = (n^3 + (n+2)^3)/2.

Original entry on oeis.org

14, 76, 234, 536, 1030, 1764, 2786, 4144, 5886, 8060, 10714, 13896, 17654, 22036, 27090, 32864, 39406, 46764, 54986, 64120, 74214, 85316, 97474, 110736, 125150, 140764, 157626, 175784, 195286, 216180, 238514, 262336, 287694, 314636, 343210, 373464, 405446

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 03 2010

			(1^3 + 3^3)/2 = 14, ...

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A027575, A173960, A173961, A271636.

f[n_]:=(n^3+(n+2)^3)/2;Table[f[n],{n,1,6!,2}]
Mean/@Partition[Range[1,81,2]^3,2,1] (* Harvey P. Dale, Apr 20 2015 *)

Vec(2*x*(7*x^2+10*x+7)/(x-1)^4 + O(x^100)) \\ Colin Barker, Jan 17 2015

From Colin Barker, Jan 17 2015: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: 2*x*(7*x^2+10*x+7)/(x-1)^4. (End)
From Elmo R. Oliveira, Aug 23 2025: (Start)
a(n) = 2*n*(4*n^2 + 3) = A271636(n)/2.
E.g.f.: 2*exp(x)*x*(7 + 12*x + 4*x^2). (End)

A173965 Averages of four consecutive cubes.

Original entry on oeis.org

2, 9, 25, 56, 108, 187, 299, 450, 646, 893, 1197, 1564, 2000, 2511, 3103, 3782, 4554, 5425, 6401, 7488, 8692, 10019, 11475, 13066, 14798, 16677, 18709, 20900, 23256, 25783, 28487, 31374, 34450, 37721, 41193, 44872, 48764, 52875, 57211, 61778, 66582, 71629, 76925

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 03 2010

			(0^3+1^3+2^3+3^3)/4 = 9, ...

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A027575, A089071, A173960, A173961, A173962.

f[n_]:=(n^3+(n+1)^3+(n+2)^3+(n+3)^3)/4;Table[f[n],{n,-1,5!}]

From R. J. Mathar, Mar 31 2010: (Start)
a(n) = (2*n-1)*(n^2-n+4)/2 = (2*n-1)*A089071(n+1).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x*(1+x)*(2*x^2-x+2)/(x-1)^4. (End)
E.g.f.: 2 + exp(x)*(-4 + 8*x + 3*x^2 + 2*x^3)/2. - Elmo R. Oliveira, Aug 23 2025

More terms from Elmo R. Oliveira, Aug 23 2025

A272527 Numbers k such that prime(k) - 2 is the average of four consecutive odd squares.

Original entry on oeis.org

9, 14, 20, 28, 36, 56, 67, 94, 124, 155, 173, 192, 213, 230, 253, 344, 395, 475, 504, 534, 596, 725, 759, 795, 1230, 1359, 1449, 1549, 1596, 1647, 1688, 1745, 1798, 2005, 2119, 2164, 2335, 2395, 2457, 2759, 2885, 2952, 3340, 3627, 3696, 3835, 3909, 3987, 4438

Offset: 1

Michel Lagneau, May 02 2016

nonn

The numbers prime(k)- 2 are a subsequence of A173960 (averages of four consecutive odd squares, or numbers of form 4*m^2+8*m+9), and also subsequence of A040976 (numbers prime(n) - 2). So, a(n) are the indices k such prime(k) are of the form 4*m^2+8*m+11 with the corresponding m = {1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 15, 16, 17, 18, 19, 23, 25, 28,...}.

The sequence A173960 and the subsequence prime(a(n)) - 2 appear in a diagonal straight line in the Ulam spiral (see the illustration).

			a(1) = 9 because prime(9) - 2 = 23 - 2 = 21, and (1^2 + 3^2 + 5^2 + 7^2)/4 = 21;
a(2) = 14 because prime(14) - 2 = 43 - 2 = 41, and (3^2 + 5^2 + 7^2 + 9^2)/4 = 41.

Michel Lagneau, Illustration

Cf. A040976, A173960.

for n from 9 to 1000 do:
p:=ithprime(n)-2:
for m from 1 by 2 to p do:
  s:=(m^2+(m+2)^2+(m+4)^2+(m+6)^2)/4:
  if s=p then printf(`%d, `,n):else fi:
od:
od:

PrimePi@ Select[(#^2 + (# + 2)^2 + (# + 4)^2 + (# + 6)^2)/4 &@ Range@ 210 + 2, PrimeQ] (* Michael De Vlieger, May 02 2016 *)

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A173961 Averages of two consecutive even cubes: (n^3 + (n+2)^3)/2.

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A173962 Averages of two consecutive odd cubes; a(n) = (n^3 + (n+2)^3)/2.

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A173965 Averages of four consecutive cubes.

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A272527 Numbers k such that prime(k) - 2 is the average of four consecutive odd squares.

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