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

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

1,4

Comments

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,...).

Links

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

Formula

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

Example

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;
  ...

Mathematica

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

Prog

(PARI) T(n, k) = if(k<=n, fibonacci(n) + fibonacci(k) - 1, 0); \\ Andrew Howroyd, Aug 10 2018
(Magma) F:=Fibonacci; [F(n)+F(k)-1: k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 13 2019
(Sage) f=fibonacci; [[f(n)+f(k)-1 for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 13 2019
(GAP) F:=Fibonacci;; Flat(List([1..15], n-> List([1..n], k-> F(n) +F(k) -1 ))); # G. C. Greubel, Jul 13 2019

Crossrefs

Row sums are A131412.

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

Sequence in context: A226222 A140473 A115384 * A300068 A194202 A261224

Adjacent sequences: A131408 A131409 A131410 * A131412 A131413 A131414

Keyword

nonn,tabl

Author

Gary W. Adamson, Jul 08 2007

Extensions

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

Status

approved