A060036 -id:A060036 - cp's OEIS frontend

A048152 Triangular array T read by rows: T(n,k) = k^2 mod n, for 1 <= k <= n, n >= 1.

0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 4, 4, 1, 0, 1, 4, 3, 4, 1, 0, 1, 4, 2, 2, 4, 1, 0, 1, 4, 1, 0, 1, 4, 1, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 5, 3, 3, 5, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 3, 12, 10, 10, 12, 3, 9, 4, 1, 0

Offset: 1

Clark Kimberling

			Rows:
  0;
  1, 0;
  1, 1, 0;
  1, 0, 1, 0;
  1, 4, 4, 1, 0;
  1, 4, 3, 4, 1, 0;

T. D. Noe, Rows n = 1..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Quadratic Residue

Cf. A060036.
Cf. A225126 (central terms).
Cf. A070430 (row 5), A070431 (row 6), A053879 (row 7), A070432 (row 8), A008959 (row 10), A070435 (row 12), A070438 (row 15), A070422 (row 20).
Cf. A046071 (in ascending order, without zeros and duplicates).
Cf. A063987 (for primes, in ascending order, without zeros and duplicates).

a048152 n k = a048152_tabl !! (n-1) !! (k-1)
a048152_row n = a048152_tabl !! (n-1)
a048152_tabl = zipWith (map . flip mod) [1..] a133819_tabl
-- Reinhard Zumkeller, Apr 29 2013

Flatten[Table[PowerMod[k,2,n],{n,15},{k,n}]] (* Harvey P. Dale, Jun 20 2011 *)

T(n,k) = A133819(n,k) mod n, k = 1..n. - Reinhard Zumkeller, Apr 29 2013

T(n,k) = (T(n,k-1) + (2k+1)) mod n. - Andrés Ventas, Apr 06 2021

A048153 a(n) = Sum_{k=1..n} (k^2 mod n).

Original entry on oeis.org

0, 1, 2, 2, 10, 13, 14, 12, 24, 45, 44, 38, 78, 77, 70, 56, 136, 129, 152, 130, 182, 209, 184, 148, 250, 325, 288, 294, 406, 365, 372, 304, 484, 561, 490, 402, 666, 665, 572, 540, 820, 805, 860, 726, 840, 897, 846, 680, 980, 1125, 1156, 1170, 1378, 1305, 1210

Offset: 1

Clark Kimberling

nonn

See A048152 for the array T[n,k] = k^2 mod n.
Starting with a(2)=1 each 4th term is odd: a(n=2+4*k) = 1, 13, 45, 77, 129, 209, 325, 365, ... - Zak Seidov, Apr 22 2009
Positions of squares in A048153: 1, 2, 33, 51, 69, 105, 195, 250, 294, 1250, 4913, 9583, 13778, 48778, 65603, 83521.
Corresponding values of squares are: {0, 1, 22, 34, 46, 70, 130, 175, 203, 875, 3468, 6734, 9711, 34481, 46308, 58956}^2 = {0, 1, 484, 1156, 2116, 4900, 16900, 30625, 41209, 765625, 12027024, 45346756, 94303521, 1188939361, 2144430864, 3475809936}. - Zak Seidov, Nov 02 2011
For n > 1 also row sums of A060036. - Reinhard Zumkeller, Apr 29 2013
Conjecture: a(n) <= (n^2-1)/2. - Aspen A.M. Meissner, Mar 06 2025

			a(5) = 1^2 + 2^2 + (3^2 mod 5) + (4^2 mod 5) + (5^2 mod 5) = 1 + 4 + 4 + 1 + 0 = 10. (It is easily seen that the last term, n^2 mod n, is always zero and would not need to be included.) - _M. F. Hasler_, Oct 21 2013

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A000330, A048152, A215573.

a048153 = sum . a048152_row  -- Reinhard Zumkeller, Apr 29 2013

Table[Sum[PowerMod[k,2,n], {k,n-1}], {n,1,10000}] (* Zak Seidov, Nov 02 2011 *)

a(n)=sum(k=1,n,k^2%n) \\ Charles R Greathouse IV, Oct 21 2013

def A048153(n): return sum(k**2%n for k in range(1,n)) # Chai Wah Wu, Jun 02 2024

a(n) == n*(n+1)*(2n+1)/6 (mod n). - Charles R Greathouse IV, Dec 28 2011
a(n) == n*(n-1)*(2n-1)/6 (mod n). - Chai Wah Wu, Jun 02 2024
a(n) mod n = A215573(n). - Alois P. Heinz, Jun 03 2024

Definition made more explicit by M. F. Hasler, Oct 21 2013

A133819 Triangle whose rows are sequences of increasing squares: 1; 1,4; 1,4,9; ... .

Original entry on oeis.org

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

Offset: 1

Peter Bala, Sep 25 2007

Reading the triangle by rows produces the sequence 1,1,4,1,4,9,1,4,9,16,..., analogous to A002260.

Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. A133819 is reluctant sequence of A000290. - Boris Putievskiy, Jan 11 2013

			The triangle T(n, k) starts:
1;
1, 4;
1, 4, 9;
1, 4, 9, 16;
1, 4, 9, 16, 25;

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
M. de Frenicle, Methode pour trouver la solutions des problemes par les exclusions, in: Divers ouvrages de mathematiques et de physique par messieurs de l'academie royale des sciences, (1693) pp 1-44, page 11. - _Paul Curtz_, Aug 18 2008
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A215630, A215631, A000290, A002260.

Cf. A048152, A060036.

a133819 n k = a133819_tabl !! (n-1) !! (k-1)
a133819_row n = a133819_tabl !! (n-1)
a133819_tabl = map (`take` (tail a000290_list)) [1..]
-- Reinhard Zumkeller, Nov 11 2012

With[{sqs=Range[12]^2},Flatten[Table[Take[sqs,n],{n,12}]]] (* Harvey P. Dale, Sep 09 2012 *)

T(n, k) = k^2, n >= k >= 1. - Wolfdieter Lang, Dec 02 2014
O.g.f.: (1+qx)/((1-x)(1-qx)^3) = 1 + x(1 + 4q) + x^2(1 + 4q + 9q^2) + ... .
a(n) = A000290(m+1), where m = n-t(t+1)/2, t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Jan 11 2013

Offset changed by Reinhard Zumkeller, Nov 11 2012

A225126 Central terms of the triangle in A048152.

Original entry on oeis.org

0, 1, 4, 2, 7, 3, 10, 4, 13, 5, 16, 6, 19, 7, 22, 8, 25, 9, 28, 10, 31, 11, 34, 12, 37, 13, 40, 14, 43, 15, 46, 16, 49, 17, 52, 18, 55, 19, 58, 20, 61, 21, 64, 22, 67, 23, 70, 24, 73, 25, 76, 26, 79, 27, 82, 28, 85, 29, 88, 30, 91, 31, 94, 32, 97, 33, 100

Offset: 1

Reinhard Zumkeller, Apr 29 2013

a(n) = A123684(n) for n > 1;
a(n) = A048152(2*n-1,n), central terms;
also a(n) = A060036(2*n-1,n-1) for n > 1.
a(n+1)=the remainder when n^2 is divided by 2n+1. - J. M. Bergot, Jun 25 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Haskell
```
a225126 n = a048152 (2 * n - 1)  n
```

a(n)=n--^2%(2*n+1) \\ Charles R Greathouse IV, Jun 25 2013

a(1) = 0, a(2*n) = n and a(2*n+1) = 3*n+1.

a(n) = 2*a(n-2)-a(n-4) for n>5. G.f.: -x^2*(x^3-4*x-1) / ((x-1)^2*(x+1)^2). - Colin Barker, May 01 2013

A060037 Triangular array T read by rows: T(n,k)=k^2 mod n, for k=1,2,...,[n/2], n=2,3,...

Original entry on oeis.org

1, 1, 1, 0, 1, 4, 1, 4, 3, 1, 4, 2, 1, 4, 1, 0, 1, 4, 0, 7, 1, 4, 9, 6, 5, 1, 4, 9, 5, 3, 1, 4, 9, 4, 1, 0, 1, 4, 9, 3, 12, 10, 1, 4, 9, 2, 11, 8, 7, 1, 4, 9, 1, 10, 6, 4, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 16, 8, 2, 15, 13, 1, 4, 9, 16, 7, 0, 13, 10, 9, 1, 4, 9, 16, 6, 17, 11, 7, 5, 1, 4, 9, 16, 5, 16

Offset: 2

N. J. A. Sloane, Mar 17 2001

			1; 1; 1,0; 1,4; 1,4,3; 1,4,2; ...

T. D. Noe, Rows n=2..100 of triangle, flattened

Cf. A048152, A060036.

Flatten[Table[PowerMod[k,2,n],{n,2,20},{k,Floor[n/2]}]] (* Harvey P. Dale, Mar 05 2012 *)

More terms from Larry Reeves (larryr(AT)acm.org), Mar 20 2001

A343720 Triangle read by rows: T(n,k) = k^2 mod n for k = 0..n-1, n >= 1.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 4, 4, 1, 0, 1, 4, 3, 4, 1, 0, 1, 4, 2, 2, 4, 1, 0, 1, 4, 1, 0, 1, 4, 1, 0, 1, 4, 0, 7, 7, 0, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 5, 3, 3, 5, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 3, 12, 10, 10, 12, 3, 9, 4, 1

Offset: 1

Jon E. Schoenfield, Apr 26 2021

Similar to A048152 and A060036, but each row in this sequence begins at k = 0 and ends at k = n-1 (the minimum and maximum residues modulo n, respectively).

			Triangle begins:
  n\k| 0  1  2  3  4  5  6  7  8  9 10 11
  ---+-----------------------------------
   1 | 0
   2 | 0, 1
   3 | 0, 1, 1
   4 | 0, 1, 0, 1
   5 | 0, 1, 4, 4, 1
   6 | 0, 1, 4, 3, 4, 1
   7 | 0, 1, 4, 2, 2, 4, 1
   8 | 0, 1, 4, 1, 0, 1, 4, 1
   9 | 0, 1, 4, 0, 7, 7, 0, 4, 1
  10 | 0, 1, 4, 9, 6, 5, 6, 9, 4, 1
  11 | 0, 1, 4, 9, 5, 3, 3, 5, 9, 4, 1
  12 | 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Eric Weisstein's World of Mathematics, Quadratic Residue

Cf. A048152, A060036.

T(n,k) = k^2 % n \\ Andrew Howroyd, Jan 05 2024

T(n,k) = k^2 mod n.

T(n,k) = T(n,n-k).

cp's OEIS Frontend

A048152 Triangular array T read by rows: T(n,k) = k^2 mod n, for 1 <= k <= n, n >= 1.

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A048153 a(n) = Sum_{k=1..n} (k^2 mod n).

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A133819 Triangle whose rows are sequences of increasing squares: 1; 1,4; 1,4,9; ... .

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A225126 Central terms of the triangle in A048152.

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A060037 Triangular array T read by rows: T(n,k)=k^2 mod n, for k=1,2,...,[n/2], n=2,3,...

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A343720 Triangle read by rows: T(n,k) = k^2 mod n for k = 0..n-1, n >= 1.

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