A129371 -id:A129371 - cp's OEIS frontend

A050409 Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n} k^2.

0, 5, 29, 86, 190, 355, 595, 924, 1356, 1905, 2585, 3410, 4394, 5551, 6895, 8440, 10200, 12189, 14421, 16910, 19670, 22715, 26059, 29716, 33700, 38025, 42705, 47754, 53186, 59015, 65255, 71920, 79024, 86581, 94605, 103110, 112110, 121619

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A033994, A129371, A132112, A132121, A132124.
Cf. A225144. - Bruno Berselli, Jun 06 2013
Cf. A045943: Sum_{k = n..2*n} k.
Cf. A304993: Sum_{k = n..2*n} k*(k+1)/2.

List([0..40], n-> n*(n+1)*(14*n+1)/6); # G. C. Greubel, Oct 30 2019

[&+[k^2: k in [n..2*n]]: n in [0..40]]; // Bruno Berselli, Feb 11 2011

I:=[0, 5, 29, 86]; [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, Jun 22 2012

seq(add((n+k)^2,k=0..n),n=0..40); # Zerinvary Lajos, Dec 01 2006

LinearRecurrence[{4,-6,4,-1},{0,5,29,86},40] (* Vincenzo Librandi, Jun 22 2012 *)
Table[(n(n+1)(14n+1))/6,{n,0,40}] (* Harvey P. Dale, Mar 08 2020 *)

PARI
```
a(n)=sum(k=n,n+n,k^2)
```

vector(40, n, n*(n-1)*(14*n-13)/6) \\ G. C. Greubel, Oct 30 2019

[n*(n+1)*(14*n+1)/6 for n in (0..40)] # G. C. Greubel, Oct 30 2019

a(n) = n*(n+1)*(14*n+1)/6.
a(n) = A132121(n,4) for n>3. - Reinhard Zumkeller, Aug 12 2007
From Bruno Berselli, Feb 11 2011: (Start)
G.f.: x*(5+9*x)/(1-x)^4.
a(n) = A129371(2*n). (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 22 2012
E.g.f.: x*(30 + 57*x + 14*x^2)*exp(x)/6. - G. C. Greubel, Oct 30 2019

A129370 a(n) = n^2 - (n-1)^2*(1 - (-1)^n)/8.

Original entry on oeis.org

0, 1, 4, 8, 16, 21, 36, 40, 64, 65, 100, 96, 144, 133, 196, 176, 256, 225, 324, 280, 400, 341, 484, 408, 576, 481, 676, 560, 784, 645, 900, 736, 1024, 833, 1156, 936, 1296, 1045, 1444, 1160, 1600, 1281, 1764, 1408

Offset: 0

Paul Barry, Apr 11 2007

Partial sums are A129371.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A000567 (odd bisection), A016742 (even bisection), A129371.

[n^2 -(n-1)^2*(n mod 2)/4: n in [0..60]]; // G. C. Greubel, Jan 31 2024

Table[n^2-(n-1)^2 (1-(-1)^n)/8,{n,0,50}] (* Harvey P. Dale, Oct 22 2011 *)

a(n)=n^2-(n-1)^2*(1-(-1)^n)/8 \\ Charles R Greathouse IV, Sep 28 2015

[n^2 -(n-1)^2*(n%2)/4 for n in range(61)] # G. C. Greubel, Jan 31 2024

a(n) = (1/8)*( (7*n^2 + 2*n - 1) + (-1)^n*(n-1)^2 ).
G.f.: x*(1 + 4*x + 5*x^2 + 4*x^3)/(1-x^2)^3.
E.g.f.: (1/4)*( x*(5+4*x)*cosh(x) - (1-4*x-3*x^2)*sinh(x) ). - G. C. Greubel, Jan 31 2024

A055461 Square decrescendo subsequences: triangle T(n,k) = (n-k)^2, n >= 1, 0 <= k < n.

Original entry on oeis.org

1, 4, 1, 9, 4, 1, 16, 9, 4, 1, 25, 16, 9, 4, 1, 36, 25, 16, 9, 4, 1, 49, 36, 25, 16, 9, 4, 1, 64, 49, 36, 25, 16, 9, 4, 1, 81, 64, 49, 36, 25, 16, 9, 4, 1, 100, 81, 64, 49, 36, 25, 16, 9, 4, 1, 121, 100, 81, 64, 49, 36, 25, 16, 9, 4, 1, 144, 121, 100, 81, 64, 49, 36, 25, 16, 9, 4, 1

Offset: 1

Henry Bottomley, Jun 26 2000

			From _Omar E. Pol_, Jan 26 2014: (Start)
Triangle begins:
    1;
    4,  1;
    9,  4,  1;
   16,  9,  4,  1;
   25, 16,  9,  4,  1;
   36, 25, 16,  9,  4,  1;
   49, 36, 25, 16,  9,  4,  1;
   64, 49, 36, 25, 16,  9,  4,  1;
   81, 64, 49, 36, 25, 16,  9,  4,  1;
  100, 81, 64, 49, 36, 25, 16,  9,  4,  1;
  ...
For n = 7 the row sum is 49 + 36 + 25 + 16 + 9 + 4 + 1 = A000330(7) = 140.
The alternating row sum is 49 - 36 + 25 - 16 + 9 - 4 + 1 = A000217(7) = 28.
(End)

Robert Israel, Rows n = 1..141, flattened

Cf. A000290, A000292, A004736, A129371, A194274.

Cf. A000217 (alternating row sums), A000330 (row sums).

[(n-k)^2: k in [0..n-1], n in [1..15]]; // G. C. Greubel, Jan 31 2024

for n from 1 to 10 do
  seq((n-k)^2, k=0..n-1)
od; # Robert Israel, Jan 18 2018

Table[Range[n,1,-1]^2,{n,20}]//Flatten (* Harvey P. Dale, Apr 17 2020 *)

flatten([[(n-k)^2 for k in range(n)] for n in range(1,16)]) # G. C. Greubel, Jan 31 2024

a(n) = A004736(n)^2.
Sum_{k=0..n-1} T(n, k) = A000330(n) (row sums). - Michel Marcus, Dec 31 2012
G.f. as triangle: x*(1+x)/((1-x*y)*(1-x)^3). - Robert Israel, Jan 18 2018
Sum_{k=0..n-1} (-1)^k*T(n, k) = A000217(n) (alternating row sums). - Omar E. Pol, Jan 24 2014
From G. C. Greubel, Jan 31 2024: (Start)
T(2*n-1, n-1) = A000290(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000292(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A194274(n).
Sum_{k=0..floor(n/2)} T(n, k) = A129371(n). (End)

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A050409 Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n} k^2.

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A129370 a(n) = n^2 - (n-1)^2*(1 - (-1)^n)/8.

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A055461 Square decrescendo subsequences: triangle T(n,k) = (n-k)^2, n >= 1, 0 <= k < n.

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