A127647 -id:A127647 - cp's OEIS frontend

A143211 Triangle read by rows, T(n,k) = Fibonacci(n)*Fibonacci(k).

1, 1, 1, 2, 2, 4, 3, 3, 6, 9, 5, 5, 10, 15, 25, 8, 8, 16, 24, 40, 64, 13, 13, 26, 39, 65, 104, 169, 21, 21, 42, 63, 105, 168, 273, 441, 34, 34, 68, 102, 170, 272, 442, 714, 1156, 55, 55, 110, 165, 275, 440, 715, 1155, 1870, 3025, 89, 89, 178, 267, 445, 712, 1157, 1869

Offset: 1

Gary W. Adamson, Jul 30 2008

			First few rows of the triangle:
   1;
   1,  1;
   2,  2,  4;
   3,  3,  6,  9;
   5,  5, 10, 15,  25;
   8,  8, 16, 24,  40,  64;
  13, 13, 26, 39,  65, 104, 169;
  21, 21, 42, 63, 105, 168, 273, 441;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000045 (left border), A007598 (right border), A127647,

Cf. A024458 (diagonal row sums), A143212 (row sums).

F:=Fibonacci; [F(n)*F(k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 20 2024

With[{F=Fibonacci}, Table[F[k]*F[n], {n,12}, {k,n}]]//Flatten (* G. C. Greubel, Jul 20 2024 *)

def A143211(n,k): return fibonacci(n)*fibonacci(k)
flatten([[A143211(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Jul 20 2024

T(n, k) = Fibonacci(n)*Fibonacci(k).
T(n, k) = A127647 * A000012 * A127647, as infinite lower triangular matrices.
T(n, 1) = A000045(n).
T(n, n) = A007598(n).
Sum_{k=1..n} T(n, k) = A143212(n).
From G. C. Greubel, Jul 20 2024: (Start)
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*Fibonacci(n)*(Fibonacci(n-1) - (-1)^n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A024458(n). (End)

A132921 Triangle read by rows: T(n,k) = n + Fibonacci(k) - 1, 1 <= k <= n.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 5, 6, 5, 5, 6, 7, 9, 6, 6, 7, 8, 10, 13, 7, 7, 8, 9, 11, 14, 19, 8, 8, 9, 10, 12, 15, 20, 28, 9, 9, 10, 11, 13, 16, 21, 29, 42, 10, 10, 11, 12, 14, 17, 22, 30, 43, 64, 11, 11, 12, 13, 15, 18, 23, 31, 44, 65, 99, 12, 12, 13, 14, 16, 19, 24, 32, 45, 66, 100, 155

Offset: 1

Gary W. Adamson, Sep 05 2007

Right border = A081659, row sums = A132922: (1, 4, 10, 19, 32, ...).

			First few rows of the triangle are:
  1;
  2, 2;
  3, 3, 4;
  4, 4, 5, 6;
  5, 5, 6, 7, 9;
  ...
Column 3 = 4, 5, 6, 7, ...; since A081659(2) = 4.

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

Row sums are A132922.

Cf. A127648, A127647, A081659.

T[n_,k_]:=n+Fibonacci[k]-1;Table[T[n,k],{n,12},{k,n}]//Flatten (* James C. McMahon, Mar 09 2025 *)

T(n,k)=if(k<=n, n + fibonacci(k) - 1, 0) \\ Andrew Howroyd, Sep 01 2018

Equals (A127648 * A000012 + A000012 * A127647) - A000012 as infinite lower triangular matrices.

Name clarified and terms a(56) and beyond from Andrew Howroyd, Sep 01 2018

A131411 Triangle read by rows: T(n,k) = Fibonacci(n) + Fibonacci(k) - 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 9, 8, 8, 9, 10, 12, 15, 13, 13, 14, 15, 17, 20, 25, 21, 21, 22, 23, 25, 28, 33, 41, 34, 34, 35, 36, 38, 41, 46, 54, 67, 55, 55, 56, 57, 59, 62, 67, 75, 88, 109, 89, 89, 90, 91, 93, 96, 101, 109, 122, 143, 177, 144, 144, 145, 146, 148, 151, 156, 164, 177, 198, 232, 287

Offset: 1

Gary W. Adamson, Jul 08 2007

Left column = Fibonacci numbers. Right column = A001595: (1, 1, 3, 5, 9, 15, 25,...).

Row sums = A131412: (1, 2, 7, 15, 32, 62, 117, 214,...).

			First few rows of the triangle are:
   1;
   1,  1;
   2,  2,  3;
   3,  3,  4,  5;
   5,  5,  6,  7,  9;
   8,  8,  9, 10, 12, 15;
  13, 13, 14, 15, 17, 20, 25;
  21, 21, 22, 23, 25, 28, 33, 41;
  ...

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

Row sums are A131412.

Cf. A000012, A000045, A001595, A104763, A127647, A131410.

F:=Fibonacci;; Flat(List([1..15], n-> List([1..n], k-> F(n) +F(k) -1 ))); # G. C. Greubel, Jul 13 2019

F:=Fibonacci; [F(n)+F(k)-1: k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 13 2019

With[{F=Fibonacci}, Table[F[n]+F[k]-1, {n,15}, {k,n}]//Flatten] (* G. C. Greubel, Jul 13 2019 *)

T(n,k) = if(k<=n, fibonacci(n) + fibonacci(k) - 1, 0); \\ Andrew Howroyd, Aug 10 2018

f=fibonacci; [[f(n)+f(k)-1 for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 13 2019

Equals A131410 + A104763 - A000012 as infinite lower triangular matrices.

Name changed and terms a(56) and beyond from Andrew Howroyd, Aug 10 2018

A132919 Triangle read by rows: T(n,k) = Fibonacci(n) + k - 1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 13, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 34, 35, 36, 37, 38, 39, 40, 41, 42, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

Gary W. Adamson, Sep 05 2007

Left border = Fibonacci numbers, right border = A081659.

Infinite lower triangular matrix by rows: n-th row = n terms of: F(n) followed by (F(n) + 1), (F(n) + 2), (F(n) + 3), ...

			First few rows of the triangle:
  1;
  1,  2;
  2,  3,  4;
  3,  4,  5,  6;
  5,  6,  7,  8,  9;
  8,  9, 10, 11, 12, 13;
  ...

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

Row sums are A132920.
Cf. A127647, A127648, A081659, A000045.
Cf. A132921, A132923.

T[n_,k_]:=Fibonacci[n]+k-1;Table[T[n,k],{n,11},{k,n}]//Flatten (* James C. McMahon, Mar 09 2025 *)

T(n,k) = if(k<=n, fibonacci(n) + k - 1, 0); \\ Andrew Howroyd, Aug 10 2018

Name changed and terms a(56) and beyond from Andrew Howroyd, Aug 10 2018

A144154 A Fibonacci triangle, row sums = A023610.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 5, 3, 4, 5, 8, 5, 6, 6, 5, 13, 8, 10, 9, 10, 8, 21, 13, 16, 15, 15, 16, 13, 34, 21, 26, 24, 25, 24, 26, 21, 55, 34, 42, 39, 40, 40, 39, 42, 34

Offset: 1

Gary W. Adamson, Sep 12 2008

Row sums = A023610: (1, 3, 7, 15, 30, 58,...).

			First few rows of the triangle =
1;
2, 1;
3, 2, 2;
5, 3, 4, 3;
8, 5, 6, 6, 5;
13, 8, 10, 9, 10, 8;
21, 13, 16, 15, 15, 16, 13;
34, 21, 26, 24, 25, 24, 26, 21;
... Row 4 = (5, 3, 4, 3) = termwise products of (5, 3, 2, 1) and (1, 1, 2, 3).

A023610

The triangle as an infinite lower triangular matrix = A * B. A = a Fibonacci subsequences decrescendo triangle: (1; 2,1; 3,2,1; 5,3,2,1;...) and B = A127647, an infinite lower triangular matrix with the Fibonacci sequence as the main diagonal and the rest zeros.

A152203 Triangle T(n,k) = (2n+1-2k)*fibonacci(k), read by rows.

Original entry on oeis.org

1, 3, 1, 5, 3, 2, 7, 5, 6, 3, 9, 7, 10, 9, 5, 11, 9, 14, 15, 15, 8, 13, 11, 18, 21, 25, 24, 13, 15, 13, 22, 27, 35, 40, 39, 21, 17, 15, 26, 33, 45, 56, 65, 63, 34, 19, 17, 30, 39, 55, 72, 91, 105, 102, 55, 21, 19, 34, 45, 65, 88, 117, 147, 170, 165

Offset: 1

Gary W. Adamson, Nov 29 2008

			First few rows of the triangle =
1;
3, 1;
5, 3, 2;
7, 5, 6, 3;
9, 7, 10, 9, 5;
11, 9, 14, 15, 15, 8;
13, 11, 18, 21, 25, 24, 13;
15, 13, 22, 27, 35, 40, 39, 21;
17, 15, 26, 33, 45, 56, 65, 63, 34;
...
Row 4 = (7, 5, 6, 3) = termwise products of (7, 5, 3, 1) and (1, 1, 2, 3).

Cf. A099375, A127647, A001891 (row sums).

Flatten[Table[(2n+1-2k)Fibonacci[k],{n,15},{k,n}]] (* Harvey P. Dale, Mar 15 2015 *)

Triangle read by rows, A099375 * A127647 = termwise products of odd numbers and the Fibonacci series.

cp's OEIS Frontend

A143211 Triangle read by rows, T(n,k) = Fibonacci(n)*Fibonacci(k).

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A132921 Triangle read by rows: T(n,k) = n + Fibonacci(k) - 1, 1 <= k <= n.

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A131411 Triangle read by rows: T(n,k) = Fibonacci(n) + Fibonacci(k) - 1.

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A132919 Triangle read by rows: T(n,k) = Fibonacci(n) + k - 1.

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A144154 A Fibonacci triangle, row sums = A023610.

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A152203 Triangle T(n,k) = (2n+1-2k)*fibonacci(k), read by rows.

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