A099764 -id:A099764 - cp's OEIS frontend

A136359 Perfect squares in A133459; or perfect squares that are the sums of two nonzero pentagonal pyramidal numbers.

36, 81, 144, 289, 484, 576, 625, 676, 3600, 7396, 9801, 14400, 35344, 40000, 40804, 44100, 45796, 56644, 59049, 71824, 112896, 121104, 172225, 226576, 231361, 254016, 274576, 290521, 319225, 362404, 480249, 495616, 518400, 527076, 535824

Offset: 1

Alexander Adamchuk, Dec 25 2007

nonn

Corresponding numbers m such that m^2 = a(n) are listed in A136360.
Note that some numbers in A136360 are also perfect squares. The corresponding numbers k such that m = k^2 are listed in A136361.
Includes all nonzero members of A099764: this occurs when the two pentagonal pyramidal numbers are both equal to i^2*(i+1)/2 where i+1 is a square. - Robert Israel, Feb 04 2020

			A133459 begins {2, 7, 12, 19, 24, 36, 41, 46, 58, 76, 80, 81, 93, 115, 127, 132, 144, 150, 166, 197, 201, 202, 214, 236, 252, 271, 289, ...}.
Thus a(1) = 36, a(2) = 81, a(3) = 144, a(4) = 289 that are the perfect squares in A133459.

Robert Israel, Table of n, a(n) for n = 1..985

Cf. A136360, A136361, A133459, A002311, A002411, A053721, A099764.

N:= 200: # for terms up to N^2*(N+1)/2.
PP:= [seq(i^2*(i+1)/2, i=1..N)]:
PP2:= sort(convert(select(`<=`,{seq(seq(PP[i]+PP[j],j=i..N),i=1..N)},PP[-1]),list)):
select(issqr,PP2); # Robert Israel, Feb 04 2020

Select[ Intersection[ Flatten[ Table[ i^2*(i+1)/2 + j^2*(j+1)/2, {i,1,300}, {j,1,i} ] ] ], IntegerQ[ Sqrt[ # ] ] & ]

a(n) = A136360(n)^2.

A169801 a(n) = ((n-1)^2n^2(n+1)^2)/6 - 2Sum_{l=2..n}Sum_{k=2..n}(n-k+1)(n-l+1)(k-1)(l-1).

Original entry on oeis.org

0, 0, 4, 64, 400, 1600, 4900, 12544, 28224, 57600, 108900, 193600, 327184, 529984, 828100, 1254400, 1849600, 2663424, 3755844, 5198400, 7075600, 9486400, 12545764, 16386304, 21160000, 27040000, 34222500, 42928704, 53406864, 65934400, 80820100, 98406400

Offset: 0

N. J. A. Sloane, May 19 2010, May 22 2010

Created in an attempt to repair a formula in A045996, which however turned out to be correct after all.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

f[n_] := ((n - 1)^2*n^2*(n + 1)^2)/6 - 2*Sum[(n - k + 1)*(n - l + 1)*(k - 1) (l - 1), {k, 2, n}, {l, 2, n}]; Array[f, 31] (* Robert G. Wilson v, May 23 2010 *)

a(n) = A099764(n-1) - 2*Sum_{l=2..n}Sum_{k=2..n}(n-k+1)*(n-l+1)*(k-1)*(l-1) = A099764(n-1)/9 = 4*A001249(n-2). - R. J. Mathar, May 23 2010

G.f.: 4*x^2*(1+x)*(x^2+8*x+1)/(1-x)^7. - R. J. Mathar, May 23 2010

cp's OEIS Frontend

A136359 Perfect squares in A133459; or perfect squares that are the sums of two nonzero pentagonal pyramidal numbers.

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A169801 a(n) = ((n-1)^2n^2(n+1)^2)/6 - 2Sum_{l=2..n}Sum_{k=2..n}(n-k+1)(n-l+1)(k-1)(l-1).

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cp's OEIS Frontend

A136359 Perfect squares in A133459; or perfect squares that are the sums of two nonzero pentagonal pyramidal numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A169801 a(n) = ((n-1)^2*n^2*(n+1)^2)/6 - 2*Sum_{l=2..n}Sum_{k=2..n}(n-k+1)*(n-l+1)*(k-1)*(l-1).

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

A169801 a(n) = ((n-1)^2n^2(n+1)^2)/6 - 2Sum_{l=2..n}Sum_{k=2..n}(n-k+1)(n-l+1)(k-1)(l-1).