A069011 -id:A069011 - cp's OEIS frontend

A227481 Number of squares in row n of the triangle in A069011.

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 3, 1, 2, 1, 2, 3, 1, 1, 4, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 5, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 1, 1, 4, 1, 1, 1, 5, 1

Offset: 0

Reinhard Zumkeller, Oct 11 2013

nonn

a(A074235(n)) = 1; a(A009023(n)) > 1.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

a227481 = sum . map a010052 . a069011_row

a(n) = sum(A010052(A069011(n,k)): k=0..n).

A132124 a(n) = n(n+1)(8*n + 1)/6.

Original entry on oeis.org

0, 3, 17, 50, 110, 205, 343, 532, 780, 1095, 1485, 1958, 2522, 3185, 3955, 4840, 5848, 6987, 8265, 9690, 11270, 13013, 14927, 17020, 19300, 21775, 24453, 27342, 30450, 33785, 37355, 41168, 45232, 49555, 54145, 59010, 64158, 69597, 75335, 81380, 87740, 94423

Offset: 0

Reinhard Zumkeller, Aug 12 2007

Convolution of the sequences (0,3,5,0,0,0,...) and (binomial(n+3, 3)), n >= 0. - Emeric Deutsch, Aug 30 2007

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

Cf. A000330, A033994, A132112, A050409.
Cf. A014105, A016061; A002412, A006331; A007585, A002378.
Row sums of A069011.

seq((1/6)*n*(n+1)*(8*n+1),n=0..40); # Emeric Deutsch, Aug 30 2007

a[n_] := n*(n + 1)*(8*n + 1)/6; Array[a, 42, 0] (* Amiram Eldar, May 20 2023 *)

a(n) = A132121(n,2) for n > 1.
G.f.: x*(3+5*x)/(1-x)^4. - Emeric Deutsch, Aug 30 2007
From Bruno Berselli, Nov 25 2010: (Start)
a(n) = n*A014105(n) - A016061(n-1), since A016061(n-1) = Sum_{k=0..n-1} A014105(k) (n > 0).
Also a(n) = A002412(n) + A006331(n) = A007585(n) + A002378(n). (End)
Sum_{n>=1} 1/a(n) = 54 - 24*(sqrt(2)+1)*Pi/7 - 24*(sqrt(2)+8)*log(2)/7 + 48*sqrt(2)*log(2-sqrt(2))/7. - Amiram Eldar, May 20 2023
E.g.f.: exp(x)*x*(18 + 33*x + 8*x^2)/6. - Stefano Spezia, Feb 21 2024

A143166 a(n) = n(8n^2 + 1)/3.

Original entry on oeis.org

0, 3, 22, 73, 172, 335, 578, 917, 1368, 1947, 2670, 3553, 4612, 5863, 7322, 9005, 10928, 13107, 15558, 18297, 21340, 24703, 28402, 32453, 36872, 41675, 46878, 52497, 58548, 65047, 72010, 79453, 87392, 95843, 104822, 114345, 124428, 135087, 146338, 158197

Offset: 0

Wolfdieter Lang, Sep 15 2008

One fourth of the sum of p^2 + q^2 over the discrete square frame of length 2*n centered around the origin (called the 2n-frame). See the Wolfdieter Lang link below.
Because the summation over p*q becomes zero due to symmetry, this is also the sum over, e.g., (p+q)^2.
The total number of sites (vertices) s(n) of a square around (0,0) with length 2*n, is (2*n+1)^2. The 2n-frame borders 8*n = s(n) - s(n-1) sites, for n>=1. For n=0 only the site (0,0) is considered.
The author was led to consider such sums by a (much more difficult) question asked by R. Thomale.
Convolution of 4*j-1 with 4*j-3, j=1..n. For n=4: [1,5,9,13] convolved with [3,7,11,15] gives a(4) = 1*(15) + 5*(11) + 9*(7) + 13*(3) = 172. - J. M. Bergot, May 27 2017

			The total sums S(n) are [0, 12, 100, 392, 1080, 2420, 4732, 8400, 13872, 21660, 32340, ...].
The 2n-frame sums are 4*a(n) = [0, 12, 88, 292, 688, 1340, 2312, 3668, 5472, 7788, 10680, 14212, 18448, 23452, 29288, 36020, 43712, 52428, 62232, 73188, 85360]. The sum is over 8*n numbers.
For n=1 the 8 numbers of the 2-frame are 2,1,2; 1,1; 2,1,2, summing to 4*a(1)=12.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, The squares for n=0..4.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A069011, A118465.

[n*(8*n^2+1)/3: n in [0..40]]; // Vincenzo Librandi, Feb 05 2014

A143166:=n->n*(8*n^2+1)/3; seq(A143166(n), n=0..50); # Wesley Ivan Hurt, Feb 03 2014

Table[n (8 n^2 + 1)/3, {n, 0, 50}] (* Wesley Ivan Hurt, Feb 03 2014 *)
CoefficientList[Series[x (3 + 10 x + 3 x^2)/(1 - x)^4,{x, 0, 40}], x] (* Vincenzo Librandi, Feb 05 2014 *)

a(n) = (1/4)*(S(n) - S(n-1)), with a(0)=0 and S(n):=sum(sum(p^2+q^2,p=-n..+n),q=-n..+n) = 2*sum(sum((p^2,p=-n..+n), q=-n..+n) = 2*sum(p^2,p=-n..+n)*sum(1,q=-n..n) = 2*2*(n*(n+1)*(2*n+1))/6)*(2*n+1) = (2/3)*n*(n+1)*(2*n+1)^2.
a(n) = n*(8*n^2 + 1)/3.
G.f.: x*(3 + 10*x + 3*x^2)/(1-x)^4. - Vincenzo Librandi, Feb 05 2014
a(n) = T(n, 0) + 2*(Sum_{k=1..n-1}T(n,k)) + T(n,k), with the triangle T(n, k) = A069011(n, k) = n^2 + k^2. - Wolfdieter Lang, Aug 15 2019
From Elmo R. Oliveira, Aug 07 2025: (Start)
E.g.f.: exp(x)*x*(9 + 24*x + 8*x^2)/3.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = A118465(n)/3. (End)

A105125 Triangle read by rows: T(n,k) = n^3 + k^3, n >= 0, 0 <= k <= n.

Original entry on oeis.org

0, 1, 2, 8, 9, 16, 27, 28, 35, 54, 64, 65, 72, 91, 128, 125, 126, 133, 152, 189, 250, 216, 217, 224, 243, 280, 341, 432, 343, 344, 351, 370, 407, 468, 559, 686, 512, 513, 520, 539, 576, 637, 728, 855, 1024, 729, 730, 737, 756, 793, 854, 945, 1072, 1241, 1458, 1000, 1001, 1008, 1027

Offset: 0

Roger L. Bagula, Apr 09 2005

			Triangle begins (modulo 2 plot is a checkerboard):
  {0}
  {1, 2}
  {8, 9, 16}
  {27, 28, 35, 54}
  {64, 65, 72, 91, 128}
  {125, 126, 133, 152, 189, 250}
  ...
The identity for T(2, 1): 9 = 3*(3^2 + 3*1^2)/4 = 3*12/4 = 9. - _Wolfdieter Lang_, May 15 2015

Cf. A069011. Different from A004999. A257238, A025581, A051162.

seq(seq(n^3+k^3,k=0..n),n=0..10); # Robert Israel, May 15 2015

f[n_, m_, p_] := n^p + m^p p = 3 a = Table[Table[f[n, m, p], {n, 0, m}], {m, 0, 20}] aa = Flatten[a]

T(n,k) = n^3 + k^3, n >= 0, 0 <= k <= n.
T(n, k) = A051162(n, k)*(A051162(n, k)^2 + 3* A025581(n, k)^2)/4. See the comment on A051162 for this identity. - Wolfdieter Lang, May 15 2015
G.f. for triangle: -(9*x^5*y^3 - 8*x^4*y^3 - x^4*y^2 + 7*x^3*y^3 - 36*x^3*y^2 - 2*x^2*y^3 + 5*x^3*y + 27*x^2*y^2 + 12*x^2*y - 8*x*y^2 - x^2 + 3*x*y - 4*x - 2*y - 1)*x/((x-1)^4*(x*y-1)^4). - Robert Israel, May 15 2015

A372667 Norm i^2+j^2+k^2 of (i,j,k) for 0 <= k <= j <= i.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 12, 9, 10, 11, 13, 14, 17, 18, 19, 22, 27, 16, 17, 18, 20, 21, 24, 25, 26, 29, 34, 32, 33, 36, 41, 48, 25, 26, 27, 29, 30, 33, 34, 35, 38, 43, 41, 42, 45, 50, 57, 50, 51, 54, 59, 66, 75, 36, 37, 38, 40, 41, 44, 45, 46, 49, 54, 52, 53

Offset: 0

A. Timothy Royappa, May 09 2024

In crystallography, these triples (i,j,k) can be interpreted as Miller indices, which can be sorted into a list: (0 0 0), (1 0 0), (1 1 0), (1 1 1), (2 0 0), (2 1 0), (2 1 1), (2 2 0), (2 2 1), (2 2 2), (3 0 0), (3 1 0), (3 1 1), (3 2 0), etc.

			The first few triples are:
   0, 0, 0
   1, 0, 0
   1, 1, 0
   1, 1, 1
   2, 0, 0
   2, 1, 0
   2, 1, 1
   2, 2, 0
   2, 2, 1
   2, 2, 2
   3, 0, 0
   ...

C. Suryanarayana and M. Grant Norton, X-Ray Diffraction - A Practical Approach, Springer Science + Business Media, 1998, p. 83.

Robert Israel, Table of n, a(n) for n = 0..10000
S. Phil Ahrenkiel, List of the first few Miller indices, in order, excluding (0 0 0)

The table of triples forms A331195.

Cf. A070770, A069011 (2-dimensional analog), A004215 (complement to this sequence)

a:=[];
for i from 0 to 10 do for j from 0 to i do for k from 0 to j do
a:=[op(a),i^2+j^2+k^2]; od: od: od: a; # N. J. A. Sloane, Jun 03 2024

print([i**2 + j**2 + k**2 for i in range(7) for j in range(i+1) for k in range(j+1)]) # Andrey Zabolotskiy, May 09 2024

More terms from Andrey Zabolotskiy, May 09 2024

A386823 Triangle read by rows: T(n,k) = numerator((n^2 - k^2)/(n^2 + k^2)), where 0 <= k < n.

Original entry on oeis.org

1, 1, 3, 1, 4, 5, 1, 15, 3, 7, 1, 12, 21, 8, 9, 1, 35, 4, 3, 5, 11, 1, 24, 45, 20, 33, 12, 13, 1, 63, 15, 55, 3, 39, 7, 15, 1, 40, 77, 4, 65, 28, 5, 16, 17, 1, 99, 12, 91, 21, 3, 8, 51, 9, 19, 1, 60, 117, 56, 105, 48, 85, 36, 57, 20, 21, 1, 143, 35, 15, 4, 119, 3, 95, 5, 7, 11, 23

Offset: 1

Stefano Spezia, Aug 04 2025

			The triangle of the fractions begins as:
  1/1;
  1/1,   3/5;
  1/1,   4/5,  5/13;
  1/1, 15/17,   3/5,  7/25;
  1/1, 12/13, 21/29,  8/17,  9/41;
  1/1, 35/37,   4/5,   3/5,  5/13, 11/61;
  1/1, 24/25, 45/53, 20/29, 33/65, 12/37, 13/85;
  ...

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A000012 (k=0), A000290, A005408, A066830 (k=1), A069011, A094728, A386824 (denominators).

T[n_,k_]:=Numerator[(n^2-k^2)/(n^2+k^2)]; Table[T[n,k],{n,12},{k,0,n-1}]//Flatten

T(n,n-1) = A005804(n-1).

A386824 Triangle read by rows: T(n,k) = denominator((n^2 - k^2)/(n^2 + k^2)), where 0 <= k < n.

Original entry on oeis.org

1, 1, 5, 1, 5, 13, 1, 17, 5, 25, 1, 13, 29, 17, 41, 1, 37, 5, 5, 13, 61, 1, 25, 53, 29, 65, 37, 85, 1, 65, 17, 73, 5, 89, 25, 113, 1, 41, 85, 5, 97, 53, 13, 65, 145, 1, 101, 13, 109, 29, 5, 17, 149, 41, 181, 1, 61, 125, 65, 137, 73, 157, 85, 185, 101, 221

Offset: 1

Stefano Spezia, Aug 04 2025

			The triangle of the fractions begins as:
  1/1;
  1/1,   3/5;
  1/1,   4/5,  5/13;
  1/1, 15/17,   3/5,  7/25;
  1/1, 12/13, 21/29,  8/17,  9/41;
  1/1, 35/37,   4/5,   3/5,  5/13, 11/61;
  1/1, 24/25, 45/53, 20/29, 33/65, 12/37, 13/85;
  ...

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A000012 (k=0), A000290, A001844, A069011, A094728, A228564 (k=1), A243883 (k=2), A386823 (numerators).

T[n_,k_]:=Denominator[(n^2-k^2)/(n^2+k^2)]; Table[T[n,k],{n,11},{k,0,n-1}]//Flatten

T(n,n-1) = A001844(n-1).

A162868 Least common multiple of all squares and all sums of two squares up to n^2 + n^2.

Original entry on oeis.org

1, 2, 40, 4680, 1591200, 1891936800, 4270101357600, 11089453225687200, 32565776278494961756800, 28429922691126101613686400, 42204464874461454985621846571472000

Offset: 0

Carl R. White, Jul 15 2009

Also the least common multiple of all rows of triangle A069011 up to the n-th row.

lcm(0) is taken to be 1, which follows from 0! = 1.

			a(3) = lcm(0^2+0^2; 0^2+1^2, 1^2+1^2; 0^2+2^2, 1^2+2^2, 2^2+2^2; 0^2+3^2, 1^2+3^2, 2^2+3^2, 3^2+3^2) = lcm(0; 1, 2; 4, 5, 8; 9, 10, 13, 18) = 4680.

Cf. A069011, A001481.

a(n) = {mcl = 1; for (x = 0, n, for (y = 0, n, if (v = x^2+y^2, mcl = lcm(mcl, v)););); mcl;} \\ Michel Marcus, Sep 03 2013

a(n) = lcm({ x,y:N | 0 <= x <= y <= n; x^2+y^2 })

cp's OEIS Frontend

A227481 Number of squares in row n of the triangle in A069011.

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A132124 a(n) = n*(n+1)*(8*n + 1)/6.

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A143166 a(n) = n*(8*n^2 + 1)/3.

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A105125 Triangle read by rows: T(n,k) = n^3 + k^3, n >= 0, 0 <= k <= n.

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A372667 Norm i^2+j^2+k^2 of (i,j,k) for 0 <= k <= j <= i.

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A386823 Triangle read by rows: T(n,k) = numerator((n^2 - k^2)/(n^2 + k^2)), where 0 <= k < n.

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A386824 Triangle read by rows: T(n,k) = denominator((n^2 - k^2)/(n^2 + k^2)), where 0 <= k < n.

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A162868 Least common multiple of all squares and all sums of two squares up to n^2 + n^2.

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A132124 a(n) = n(n+1)(8*n + 1)/6.

A143166 a(n) = n(8n^2 + 1)/3.