A176228 Triangle read by rows: T(n,k) = binomial(n,k) + Fibonacci(n) + 1.

2, 3, 3, 3, 4, 3, 4, 6, 6, 4, 5, 8, 10, 8, 5, 7, 11, 16, 16, 11, 7, 10, 15, 24, 29, 24, 15, 10, 15, 21, 35, 49, 49, 35, 21, 15, 23, 30, 50, 78, 92, 78, 50, 30, 23, 36, 44, 71, 119, 161, 161, 119, 71, 44, 36, 57, 66, 101, 176, 266, 308, 266, 176, 101, 66, 57

Offset

0,1

Comments

Row sums are: {2, 6, 10, 20, 36, 68, 127, 240, 454, 862, 1640, ...}.

Links

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

Formula

T(n, k) = binomial(n, k) + Fibonacci(n) + 1.

G.f.: 1/((-1 + x)*(-1 + y)) + x/((-1 + x + x^2)*(-1 + y)) + 1/(1 - x*(1 + y)). - Stefano Spezia, Nov 22 2019

Example

Triangle begins as:
   2;
   3,  3;
   3,  4,   3;
   4,  6,   6,   4;
   5,  8,  10,   8,   5;
   7, 11,  16,  16,  11,   7;
  10, 15,  24,  29,  24,  15,  10;
  15, 21,  35,  49,  49,  35,  21,  15;
  23, 30,  50,  78,  92,  78,  50,  30,  23;
  36, 44,  71, 119, 161, 161, 119,  71,  44, 36;
  57, 66, 101, 176, 266, 308, 266, 176, 101, 66, 57;

Maple

with(combinat); seq(seq(binomial(n, k) +fibonacci(n) +1, k=0..n), n=0..12); # G. C. Greubel, Nov 22 2019

Mathematica

T[n_, k_]:= Binomial[n, k] + Fibonacci[n] + 1; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(PARI) T(n, k) = binomial(n, k)+fibonacci(n)+1; \\ G. C. Greubel, Nov 22 2019
(Magma) [Binomial(n, k)+Fibonacci(n)+1: k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 22 2019
(Sage) [[binomial(n, k)+fibonacci(n)+1 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 22 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n, k)+Fibonacci(n) +1))); # G. C. Greubel, Nov 22 2019

Crossrefs

Cf. A000045, A007318.

Sequence in context: A098201 A341883 A175239 * A322822 A305232 A322974

Adjacent sequences: A176225 A176226 A176227 * A176229 A176230 A176231

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 12 2010

Extensions

Edited by G. C. Greubel, Nov 22 2019

Status

approved