A132921 -id:A132921 - cp's OEIS frontend

A132922 Row sums of triangle A132921.

1, 4, 10, 19, 32, 50, 75, 110, 160, 233, 342, 508, 765, 1168, 1806, 2823, 4452, 7070, 11287, 18090, 29076, 46829, 75530, 121944, 197017, 318460, 514930, 832795, 1347080, 2179178, 3525507, 5703878, 9228520, 14931473, 24159006, 39089428, 63247317, 102335560

Offset: 1

Gary W. Adamson, Sep 05 2007

nonn

			a(4) = 19 = sum of row 4 terms of triangle A132921: (4 + 4 + 5 + 6).
a(5) = 32 = (1, 4, 6, 4, 1) dot (1, 3, 3, 0, 1) = (1 + 12 + 18 + 0 + 1).

Andrew Howroyd, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A132922.

LinearRecurrence[{4, -5, 1, 2, -1},{1, 4, 10, 19, 32},50] (* or *)
a[n_]:=n*(n - 1) + Fibonacci[n + 2] - 1; Array[a, 50] (* Stefano Spezia, Sep 01 2018 *)

a(n) = n*(n-1) + fibonacci(n+2) - 1; \\ Andrew Howroyd, Aug 28 2018

Vec((1 - x^2 - 2*x^3)/((1 - x)^3*(1 - x - x^2)) + O(x^40)) \\ Andrew Howroyd, Aug 28 2018

Binomial transform of [1, 3, 3, 0, 1, -1, 2, -3, 5, -8, ...].
From Andrew Howroyd, Aug 28 2018: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5) for n > 5.
a(n) = n*(n-1) + Fibonacci(n+2) - 1.
G.f.: x*(1 - x^2 - 2*x^3)/((1 - x)^3*(1 - x - x^2)).
(End)

Terms a(11) and beyond from Andrew Howroyd, Aug 28 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

cp's OEIS Frontend

A132922 Row sums of triangle A132921.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions