A213037 -id:A213037 - cp's OEIS frontend

A247375 Numbers m such that floor(m/2) is a square.

0, 1, 2, 3, 8, 9, 18, 19, 32, 33, 50, 51, 72, 73, 98, 99, 128, 129, 162, 163, 200, 201, 242, 243, 288, 289, 338, 339, 392, 393, 450, 451, 512, 513, 578, 579, 648, 649, 722, 723, 800, 801, 882, 883, 968, 969, 1058, 1059, 1152, 1153, 1250, 1251, 1352, 1353

Offset: 0

Bruno Berselli, Sep 15 2014

Union of A001105 and A058331.

Squares of the sequence are listed in A055792.

Jens Kruse Andersen, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A001105, A055792, A058331, A213037.

Cf. A130404 (numbers m such that floor(m/2) is a triangular number).

[n: n in [0..1400] | IsSquare(Floor(n div 2))];

Select[Range[0, 1400], IntegerQ[Sqrt[Floor[#/2]]] &]
LinearRecurrence[{1,2,-2,-1,1},{0,1,2,3,8},70] (* Harvey P. Dale, Oct 21 2021 *)

[n for n in [0..1400] if is_square(floor(n/2))]

G.f.: x*( 1 + x - x^2 + 3*x^3 ) / ( (1 - x)^3*(1 + x)^2 ).
a(n) = 1 + ( 2*n*(n-1) + (2*n-3)*(-1)^n - 1 )/4.
a(n+1) = 1 + A213037(n).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n >= 5. - Wesley Ivan Hurt, Dec 18 2020
Sum_{n>=1} 1/a(n) = Pi^2/12 + coth(Pi/sqrt(2))*Pi/(2*sqrt(2)) + 1/2. - Amiram Eldar, Sep 24 2022

A322630 Arithmetic table similar to multiplication with different rules for odd and even products, read by antidiagonals. T(n,k) = (n*k + A319929(n,k))/2.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 4, 4, 4, 5, 4, 7, 4, 5, 6, 6, 8, 8, 6, 6, 7, 6, 11, 8, 11, 6, 7, 8, 8, 12, 12, 12, 12, 8, 8, 9, 8, 15, 12, 17, 12, 15, 8, 9, 10, 10, 16, 16, 18, 18, 16, 16, 10, 10, 11, 10, 19, 16, 23, 18, 23, 16, 19, 10, 11

Offset: 1

David Lovler, Dec 20 2018

This table is akin to multiplication in that it is associative, 1 is the identity and 0 takes any number to 0. Associativity is proved by checking eight cases of three ordered odd and even numbers. Distributivity works except if an even number is partitioned into a sum of two odd numbers.

Excluding the first row and the first column, every number in the table is of the form 2i*j or 2i*j - 1 where i and j > 0. Every positive even number appears in the table. Odd numbers that do not appear are of the form 2p - 1 where p is a prime number.

			Array T(n,k) begins:
   1   2   3   4   5   6   7   8   9  10
   2   2   4   4   6   6   8   8  10  10
   3   4   7   8  11  12  15  16  19  20
   4   4   8   8  12  12  16  16  20  20
   5   6  11  12  17  18  23  24  29  30
   6   6  12  12  18  18  24  24  30  30
   7   8  15  16  23  24  31  32  39  40
   8   8  16  16  24  24  32  32  40  40
   9  10  19  20  29  30  39  40  49  50
  10  10  20  20  30  30  40  40  50  50

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150, flattened).

Cf. A076274, A327263.

0 and diagonal is A213037.

Table[Function[n, Switch[FromDigits[Mod[{n, k}, 2], 2], 0, n k/2, 1, (n k + n)/2, 2, (n k + k)/2, , (n k + n + k - 1)/2]][m - k + 1], {m, 11}, {k, m}] // Flatten (* _Michael De Vlieger, Jan 14 2022 *)

T319929(n,k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
T(n,k) = (T319929(n,k) + n*k)/2;
matrix(6, 6, n, k, T(n,k)) \\ Michel Marcus, Dec 22 2018

T(n,k) = (n*k + n + k - 1)/2 if n is odd and k is odd;
T(n,k) = (n*k + n)/2 if n is even and k is odd;
T(n,k) = (n*k + k)/2 if n is odd and k is even;
T(n,k) = n*k/2 if n is even and k is even.

Name clarified by David Lovler, Jan 24 2022

A354594 a(n) = n^2 + 2*floor(n/2)^2.

Original entry on oeis.org

0, 1, 6, 11, 24, 33, 54, 67, 96, 113, 150, 171, 216, 241, 294, 323, 384, 417, 486, 523, 600, 641, 726, 771, 864, 913, 1014, 1067, 1176, 1233, 1350, 1411, 1536, 1601, 1734, 1803, 1944, 2017, 2166, 2243, 2400, 2481, 2646, 2731, 2904

Offset: 0

David Lovler, Jun 01 2022

The first bisection is A033581, the second bisection is A080859. - Bernard Schott, Jun 07 2022

David Lovler, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A008794, A033581, A080859, A247375.

Cf. A213037, A322744 (diagonal), A354595, A354596.

a[n_] := n^2 + 2 Floor[n/2]^2
Table[a[n], {n, 0, 90}]    (* A354594 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 6, 11, 24}, 60]

PARI
```
a(n) = n^2 + 2*(n\2)^2;
```

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n >= 5.
a(n) = A000290(n) + 2*A008794(n).
G.f.: x*(1 + 5*x + 3*x^2 + 3*x^3)/((1 - x)^3*(1 + x)^2).
E.g.f.: (x*(1 + 3*x)*cosh(x) + (1 + 3*x + 3*x^2)*sinh(x))/2. - Stefano Spezia, Jun 07 2022

A354595 a(n) = n^2 + 4*floor(n/2)^2.

Original entry on oeis.org

0, 1, 8, 13, 32, 41, 72, 85, 128, 145, 200, 221, 288, 313, 392, 421, 512, 545, 648, 685, 800, 841, 968, 1013, 1152, 1201, 1352, 1405, 1568, 1625, 1800, 1861, 2048, 2113, 2312, 2381, 2592, 2665, 2888, 2965, 3200, 3281, 3528, 3613, 3872

Offset: 0

David Lovler, Jun 01 2022

The first bisection is A139098, the second bisection is A102083.

David Lovler, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A008794, A102083, A139098, A247375.

Cf. A213037, A327259 (diagonal), A354594, A354596.

a[n_] := n^2 + 4 Floor[n/2]^2
Table[a[n], {n, 0, 90}]    (* A354595 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 8, 13, 32}, 60]

PARI
```
a(n) = n^2 + 4*(n\2)^2;
```

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n >= 5.
a(n) = A000290(n) + 4*A008794(n).
G.f.: x*(1 + 7*x + 3*x^2 + 5*x^3)/((1 - x)^3*(1 + x)^2).
E.g.f.: 2*x^2*cosh(x) + (1 + 2*x + 2*x^2)*sinh(x). - Stefano Spezia, Jun 07 2022

A354596 Array T(n,k) = k^2 + (2n-4)*floor(k/2)^2, n >= 0, k >= 0, read by descending antidiagonals.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 5, 2, 1, 0, 0, 7, 4, 1, 0, 9, 8, 9, 6, 1, 0, 0, 17, 16, 11, 8, 1, 0, 13, 18, 25, 24, 13, 10, 1, 0, 0, 31, 36, 33, 32, 15, 12, 1, 0, 17, 32, 49, 54, 41, 40, 17, 14, 1, 0, 0, 49, 64, 67, 72, 49, 48, 19, 16, 1, 0, 21, 50, 81, 96, 85, 90, 57, 56, 21, 18, 1, 0

Offset: 0

David Lovler, Jun 01 2022

Column k is an arithmetic progression with difference 2*A008794(k).
Odd rows of A133728 triangle are contained in row 0.
For i = 0 through 4, row i is 0 and the diagonal of A319929, A322630 = A213037, A003991, A322744, and A327259, respectively. In general, row i is 0 and the diagonal of array U(i;n,k) described in A327263.

			T(n,k) begins:
  0,   1,   0,   5,   0,   9,   0,  13, ...
  0,   1,   2,   7,   8,  17,  18,  31, ...
  0,   1,   4,   9,  16,  25,  36,  49, ...
  0,   1,   6,  11,  24,  33,  54,  67, ...
  0,   1,   8,  13,  32,  41,  72,  85, ...
  0,   1,  10,  15,  40,  49,  90, 103, ...
  0,   1,  12,  17,  48,  57, 108, 121, ...
  ...

David Lovler, Table of n, a(n) for n = 0..5150

Cf. A000290, A008794, A133728, A213037, A247375.
Cf. A266222, A266439.
Cf. A319929, A322630, A322744, A327259, A327263, A354594, A354595.

T[n_, k_] := k^2 + (2*n - 4)*Floor[k/2]^2; Table[T[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Jun 20 2022 *)

PARI
```
T(n,k) = k^2 + (2*n-4)*(k\2)^2;
```

T(n,k) = U(n;k,k) (see A327263).
For each row, T(n,k) = T(n,k-1) + 2*T(n,k-2) - 2*T(n,k-3) - T(n,k-4) + T(n,k-5), k >= 5.
G.f. for row n: x*(1 + (2*n-1)*x + 3*x^2 + (2*n-3)*x^3)/((1 - x)^3*(1 + x)^2). When n = 2, this reduces to x*(1 + x)/(1 - x)^3.
E.g.f. for row n: (((4-n)*x + n*x^2)*cosh(x) + (n-2 + n*x + n*x^2)*sinh(x))/2. When n = 2, this reduces to (x + x^2)*cosh(x) + (x + x^2)*sinh(x) = (x + x^2)*exp(x).

A349108 a(n) is the permanent of the n X n matrix A(n) that is defined as A[i,j,n] = (n mod 2) + abs((n + 1)/2 - i) + abs((n + 1)/2 - j).

Original entry on oeis.org

1, 1, 2, 66, 292, 41100, 314736, 108446352, 1267665984, 829171609920, 13696865136000, 14718069991152000, 325942368613966080, 524455030610743115520, 14983681934750599526400, 33855616071967479729408000, 1211736134642288777186918400, 3668200144503587527675580006400

Offset: 0

Stefano Spezia, Nov 08 2021

nonn

A(n) is an n X n matrix whose elements start from 1 at the center and get higher, the more they are close to the corners (see the examples).

det(A(1)) = 1 and det(A(n)) = 0 for n > 1.

			For n = 5 the matrix A(5) is
   5, 4, 3, 4, 5
   4, 3, 2, 3, 4
   3, 2, 1, 2, 3
   4, 3, 2, 3, 4
   5, 4, 3, 4, 5
with permanent a(5) = 41100.
For n = 6 the matrix A(6) is
   5, 4, 3, 3, 4, 5
   4, 3, 2, 2, 3, 4
   3, 2, 1, 1, 2, 3
   3, 2, 1, 1, 2, 3
   4, 3, 2, 2, 3, 4
   5, 4, 3, 3, 4, 5
with permanent a(6) = 314736.

Vaclav Kotesovec, Table of n, a(n) for n = 0..36

Cf. A213037 (trace of matrix A(n)), A349107.

A[i_, j_, n_] := Mod[n,2]+ Abs[(n + 1)/2 - j] +Abs[(n + 1)/2 - i]; a[n_]:=Permanent[Table[A[i,j,n],{i,n},{j,n}]]; Join[{1},Array[a,17]]

a(n) = matpermanent(matrix(n, n, i, j, (n%2) + abs((n + 1)/2 - i) + abs((n + 1)/2 - j))); \\ Michel Marcus, Nov 08 2021

a(2*n) = A349107(2*n).

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A247375 Numbers m such that floor(m/2) is a square.

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A322630 Arithmetic table similar to multiplication with different rules for odd and even products, read by antidiagonals. T(n,k) = (n*k + A319929(n,k))/2.

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A354594 a(n) = n^2 + 2*floor(n/2)^2.

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A354595 a(n) = n^2 + 4*floor(n/2)^2.

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A354596 Array T(n,k) = k^2 + (2n-4)*floor(k/2)^2, n >= 0, k >= 0, read by descending antidiagonals.

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A349108 a(n) is the permanent of the n X n matrix A(n) that is defined as A[i,j,n] = (n mod 2) + abs((n + 1)/2 - i) + abs((n + 1)/2 - j).

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