A065941 -id:A065941 - cp's OEIS frontend

A131779 Triangle read by rows: T(n,k) = 2*A065941(n-1,k-1) - (-1)^(n+k).

1, 3, 1, 1, 3, 1, 3, 1, 5, 1, 1, 3, 5, 5, 1, 3, 1, 9, 5, 7, 1, 1, 3, 9, 9, 11, 7, 1, 3, 1, 13, 9, 21, 11, 9, 1, 1, 3, 13, 13, 29, 21, 19, 9, 1, 3, 1, 17, 13, 43, 29, 41, 19, 11, 1, 1, 3, 17, 17, 55, 43, 69, 41, 29, 11, 1, 3, 1, 21, 17, 73, 55, 113, 69, 71, 29, 13, 1

Offset: 1

Gary W. Adamson, Jul 14 2007

			First few rows of the triangle are:
  1;
  3, 1;
  1, 3, 1;
  3, 1, 5, 1;
  1, 3, 5, 5, 1;
  3, 1, 9, 5, 7, 1;
  1, 3, 9, 9, 11, 7, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Row sums are A131780.

Cf. A065941.

T(n,k) = {if(k<=n, 2*binomial(n-1-k\2, (k-1)\2) - (-1)^(n+k), 0)} \\ Andrew Howroyd, Sep 08 2018

T(n,k) = 2*binomial(n-1-floor(k/2), floor((k-1)/2)) - (-1)^(n+k). - Andrew Howroyd, Sep 08 2018

a(28)-a(29) corrected and terms a(56) and beyond from Andrew Howroyd, Sep 08 2018

A131376 Triangle read by rows: T(n,k) = A007318(n,k) + A065941(n,k) - A168561(n,k).

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 2, 5, 1, 1, 5, 6, 6, 1, 2, 3, 14, 9, 8, 1, 1, 7, 14, 24, 16, 9, 1, 2, 4, 27, 30, 45, 21, 11, 1, 1, 9, 25, 62, 70, 66, 31, 12, 1, 2, 5, 44, 71, 147, 120, 104, 38, 14, 1, 1, 11, 39, 128, 203, 273, 217, 140, 51, 15, 1, 2, 6, 65, 139, 366, 434, 518, 329, 200, 60, 17, 1

Offset: 0

Gary W. Adamson, Jul 04 2007

The old definition was: A007318 + A065941 - A049310. - N. J. A. Sloane, Aug 09 2019

Row sums = A117591: (1, 3, 5, 10, 19, 37, 72, ...).

			First few rows of the triangle are:
  1;
  2, 1;
  1, 3, 1;
  2, 2, 5, 1;
  1, 5, 6, 6, 1;
  2, 3, 14, 9, 8, 1;
  1, 7, 14, 24, 16, 9, 1;
  ...

Cf. A007318, A065941, A049310, A117591 (row sums), A131375, A168561, A309213.

Table[Binomial[n, k] + Binomial[n - Floor[(k+1)/2], Floor[k/2]] - If[EvenQ[n+k], Binomial[(n+k)/2, k], 0], {n, 0, 11}, {k, 0, n}] // Flatten  (* Amiram Eldar, May 31 2025 *)

The old definition of A131376 did not match its data, as Michel Marcus pointed out. The definition has been corrected here, keeping the data. The old definition with corrected data is now A309213. - N. J. A. Sloane, Aug 09 2019

More terms from Amiram Eldar, May 31 2025

A131913 Product of the square matrix in A065941 and the column vector (1, 2, 3, ...)'.

Original entry on oeis.org

1, 3, 6, 13, 25, 48, 89, 163, 294, 525, 929, 1632, 2849, 4947, 8550, 14717, 25241, 43152, 73561, 125075, 212166, 359133, 606721, 1023168, 1722625, 2895843, 4861254, 8149933, 13646809, 22825200, 38136089, 63653827, 106146534, 176849517, 294401825, 489706272

Offset: 0

Gary W. Adamson, Jul 27 2007

			a(4) = 25 = (1, 1, 3, 2, 1) dot (1, 2, 3, 4, 5) = (1 + 2 + 9 + 8 + 5), where (1, 1, 3, 2, 1) = row 4 of triangle A065941.
a(4) = 25 = A010049(4) + A001629(6) = 5 + 20.
a(5) = 48 = A055244(6) + A001629(4) = 43 + 5.

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045, A065941, A010049, A001629, A055244.

a(n) = A010049(n) + A001629(n+2) = A055244(n+1) + A001629(n-1).
From Philippe Deléham, Dec 28 2013: (Start)
G.f.: (1+x-x^2)/(1-x-x^2)^2.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4), a(0)=1, a(1)=3, a(2)=6, a(3)=13.
a(n) = a(n-1) + a(n-2) + 2*Fibonacci(n). (End)

A309213 A007318 + A065941 - A049310.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 2, 6, 5, 1, 1, 5, 12, 6, 1, 2, 3, 14, 17, 8, 1, 3, 7, 14, 24, 26, 9, 1, 2, 12, 27, 30, 45, 33, 11, 1, 1, 9, 45, 62, 70, 66, 45, 12, 1, 2, 5, 44, 111, 147, 120, 104, 54, 14, 1, 3, 11, 39, 128, 273, 273, 217, 140, 69, 15, 1, 2, 18, 65, 139, 366, 546, 518, 329, 200, 80, 17, 1

Offset: 0

Gary W. Adamson, Jul 04 2007

Row sums = 1, 3, 7, 14, 25, 45, 85, ... (This is probably a new sequence and should be added to the OEIS.) - N. J. A. Sloane, Aug 09 2019

			First few rows of the triangle are:
1,
2, 1,
3, 3, 1,
2, 6, 5, 1,
1, 5, 12, 6, 1,
2, 3, 14, 17, 8, 1,
3, 7, 14, 24, 26, 9, 1,
...

Jinyuan Wang, Table of n, a(n) for n = 0..5150

Cf. A007318, A065941, A049310, A117591, A131375, A168561.

T007318(n, k) = binomial(n, k);
T065941(n, k) = binomial(n - (k+1)\2, k\2);
T049310(n, k) = if ((n+k)%2, 0, (-1)^((n+k)/2 + k) * binomial((n+k)/2, k));
T(n, k) = T007318(n, k) + T065941(n, k) - T049310(n, k); \\ Michel Marcus, Apr 28 2014

A007318 + A065941 - A168561 as infinite lower triangular matrices.

The old definition of A131376 did not match the data, as Michel Marcus pointed out. The definition there has been corrected, keeping the old data. The present sequence uses the old definition with corrected data from Michel Marcus. - N. J. A. Sloane, Aug 09 2019

More terms from Jinyuan Wang, Aug 29 2019

A131344 A046854 * A065941.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 8, 7, 8, 3, 1, 13, 12, 18, 9, 4, 1, 21, 20, 38, 21, 14, 4, 1, 34, 33, 76, 47, 39, 15, 5, 1, 55, 54, 147, 97, 100, 43, 21, 5, 1, 89, 88, 277, 194, 236, 115, 69, 22, 6, 1

Offset: 0

Gary W. Adamson, Jun 30 2007

Left border = Fibonacci numbers starting with F(2). Row sums = A131246: (1, 3, 6, 13, 27, 57,...). A131345 = A065941 * A046854.

			First few rows of the triangle are:
1;
2, 1;
3, 2, 1;
5, 4, 3, 1;
8, 7, 8, 3, 1;
13, 12, 18, 9, 4, 1;
...

Cf. A046854, A065941, A131246, A131345.

A046854 * A065941 as infinite lower triangular matrices.

A131345 Triangle read by rows: A065941 * A046854 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 5, 3, 1, 8, 10, 8, 3, 1, 13, 20, 19, 10, 4, 1, 21, 38, 42, 26, 14, 4, 1, 34, 71, 89, 65, 41, 16, 5, 1, 55, 130, 182, 151, 110, 50, 21, 5, 1, 89, 235, 363, 338, 276, 146, 72, 23, 6, 1, 144, 420, 709, 730, 659, 392, 223, 83, 29, 6, 1

Offset: 0

Gary W. Adamson, Jun 30 2007

Left border = Fibonacci numbers starting with F(2). Row sums = A131244: (1, 3, 6, 14, 30, 67, 146,...). A131344 = A046854 * A065941.

			First few rows of the triangle are:
1;
2, 1;
3, 2, 1;
5, 5, 3, 1;
8, 10, 8, 3, 1;
13, 20, 19, 10, 4, 1;
21, 38, 42, 26, 14, 4, 1;
...

Cf. A046854, A065941, A131244, A131344.

A065941 * A046854 as infinite lower triangular matrics.

a(49) split and more terms from Georg Fischer, May 29 2023

A131400 A046854 + A065941 - I (Identity matrix).

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 2, 3, 3, 1, 2, 3, 6, 3, 1, 2, 4, 7, 7, 4, 1, 2, 4, 11, 8, 11, 4, 1, 2, 5, 12, 15, 15, 12, 5, 1, 2, 5, 17, 16, 30, 16, 17, 5, 1, 2, 6, 18, 27, 36, 36, 27, 18, 6, 1, 2, 6, 24, 28, 63, 42, 63, 28, 24, 6, 1, 2, 7, 25, 44, 71, 84, 84, 71, 44, 25, 7, 1

Offset: 0

Gary W. Adamson, Jul 06 2007

Row sums = A001595: (1, 3, 5, 9, 15, 25, 41, 67,...).

			First few rows of the triangle are:
  1;
  2, 1;
  2, 2,  1;
  2, 3,  3, 1;
  2, 3,  6, 3,  1;
  2, 4,  7, 7,  4, 1;
  2, 4, 11, 8, 11, 4, 1; ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A046854, A065941, A001595.

B:=Binomial;;
T:= function(n,k)
    if k=n then return 1;
    else return B(Int((n+k)/2), k) + B(n - Int((k+1)/2), Int(k/2));
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Jul 13 2019

B:=Binomial; [k eq n select 1 else B(Floor((n+k)/2), k) + B(n - Floor((k+1)/2), Floor(k/2)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 13 2019

With[{B = Binomial}, Table[If[k==n, 1, B[Floor[(n+k)/2], k] + B[n - Floor[(k+1)/2], Floor[k/2]]], {n,0,12}, {k,0,n}]]//Flatten (* G. C. Greubel, Jul 13 2019 *)

b=binomial; T(n,k) = if(k==n, 1, b((n+k)\2, k) + b(n - (k+1)\2, k\2));
for(n=0,12, for(k=0,n, print1(T(n,k), ", ", ))) \\ G. C. Greubel, Jul 13 2019

def T(n, k):
    b=binomial;
    if (k==n): return 1
    else: return b(floor((n+k)/2), k) + b(n - floor((k+1)/2), floor(k/2))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 13 2019

More terms added by G. C. Greubel, Jul 13 2019

A131767 2*A007318 - A065941.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 7, 9, 6, 1, 1, 9, 16, 17, 7, 1, 1, 11, 25, 36, 24, 9, 1, 1, 13, 36, 65, 60, 36, 10, 1, 1, 15, 49, 106, 125, 102, 46, 12, 1, 1, 17, 64, 161, 231, 237, 148, 62, 13, 1

Offset: 0

Gary W. Adamson, Jul 13 2007

			First few rows of the triangle are:
1;
1, 1;
1, 3, 1;
1, 5, 4, 1;
1, 7, 9, 6, 1;
1, 9, 16, 17, 7, 1;
1, 11, 25, 36, 24, 9, 1;
...

Cf. A007318, A065941, A027934 (row sums).

2*A007318 - A065941 as infinite lower triangular matrices.

A153341 Triangle read by rows, A065941 * A007318.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 5, 8, 5, 1, 8, 17, 15, 6, 1, 13, 35, 41, 25, 8, 1, 21, 68, 98, 78, 36, 9, 1, 34, 129, 222, 220, 135, 51, 11, 1, 55, 239, 477, 562, 425, 210, 66, 12, 1, 89, 436, 991, 1355, 1222, 751, 314, 86, 14, 1, 144, 785, 2001, 3110, 3248, 2373, 1225, 440, 105, 15, 1, 233, 1399, 3953, 6883, 8171, 6923, 4263, 1905, 605, 130, 17, 1

Offset: 0

Gary W. Adamson, Dec 24 2008

Row sums = A026581: (1, 3, 7, 19, 47, 123,...)

			First few rows of the triangle =
1;
2, 1;
3, 3, 1;
5, 8, 5, 1;
8, 17, 15, 6, 1;
13, 35, 41, 25, 8, 1;
21, 68, 98, 78, 36, 9, 1;
34, 129, 222, 220, 135, 51, 11, 1;
55, 239, 477, 562, 425, 210, 66, 12, 1;
89, 436, 991, 1355, 1222, 751, 314, 86, 14, 1;
...

Cf. A007318, A065941.

Triangle read by rows, A065941 * Pascal's triangle.

a(37) = 239 corrected and more terms from Georg Fischer, May 29 2023

A131373 A046854 + A065941 - A000012.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 5, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 10, 7, 10, 3, 1, 1, 4, 11, 14, 14, 11, 4, 1, 1, 4, 16, 15, 29, 15, 16, 4, 1, 1, 5, 17, 26, 35, 35, 26, 17, 5, 1

Offset: 0

Gary W. Adamson, Jul 03 2007

Row sums = A131269: (1, 2, 3, 6, 11, 20, 35, 60,...).

			First few rows of the triangle are:
1;
1, 1;
1, 1, 1;
1, 2, 2, 1;
1, 2, 5, 2, 1;
1, 3, 6, 6, 3, 1;
1, 3, 10, 7, 10, 3, 1;
1, 4, 11, 14, 14, 11, 4, 1;
1, 4, 16, 15, 29, 15, 16, 4, 1;
...

Cf. A056854, A065941, A000012, A131269.

A046854 + A065941 - A000012 as infinite lower triangular matrices.

cp's OEIS Frontend

A131779 Triangle read by rows: T(n,k) = 2*A065941(n-1,k-1) - (-1)^(n+k).

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A131376 Triangle read by rows: T(n,k) = A007318(n,k) + A065941(n,k) - A168561(n,k).

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A131913 Product of the square matrix in A065941 and the column vector (1, 2, 3, ...)'.

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A309213 A007318 + A065941 - A049310.

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A131344 A046854 * A065941.

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A131345 Triangle read by rows: A065941 * A046854 as infinite lower triangular matrices.

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A131400 A046854 + A065941 - I (Identity matrix).

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A131767 2*A007318 - A065941.

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