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A176259 Triangle, T(n, k) = Fibonacci(k+1) + Fibonacci(n-k+1) - Fibonacci(n+1), read by rows. 1
1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, -1, -1, -1, 1, 1, -2, -3, -3, -2, 1, 1, -4, -6, -7, -6, -4, 1, 1, -7, -11, -13, -13, -11, -7, 1, 1, -12, -19, -23, -24, -23, -19, -12, 1, 1, -20, -32, -39, -42, -42, -39, -32, -20, 1, 1, -33, -53, -65, -71, -73, -71, -65, -53, -33, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,17

COMMENTS

Row sums are: {1, 2, 2, 2, -1, -8, -25, -60, -130, -264, -515,...}.

This sequence is a particular case of a symmetrical triangular sequence dependent upon a recurrence. The triangle is given by T(n, k, q) = b(k+1, q) + b(n-k+1, q) - b(n+1, q) where b(n, q) satisfies the recurrence b(n, q) = b(n-1, q) + q*b(n-2, 1). This sequence is for q=1. - G. C. Greubel, Nov 23 2019

LINKS

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

FORMULA

From G. C. Greubel, Nov 23 2019: (Start)

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

Sum_{k=0..n} T(n,k) = Fibonacci(n+4) - n*Fibonacci(n+1) - 2 (row sums). (End)

EXAMPLE

Triangle begins as:

  1;

  1,   1;

  1,   0,   1;

  1,   0,   0,   1;

  1,  -1,  -1,  -1,   1;

  1,  -2,  -3,  -3,  -2,   1;

  1,  -4,  -6,  -7,  -6,  -4,   1;

  1,  -7, -11, -13, -13, -11,  -7,   1;

  1, -12, -19, -23, -24, -23, -19, -12,   1;

  1, -20, -32, -39, -42, -42, -39, -32, -20,   1;

  1, -33, -53, -65, -71, -73, -71, -65, -53, -33, 1;

MAPLE

with(combinat); f:=fibonacci; seq(seq( f(k+1) + f(n-k+1) - f(n+1), k=0..n), n = 0..12); # G. C. Greubel, Nov 23 2019

MATHEMATICA

(* q = 0..10 *)

b[n_, q_]:= b[n, q]= If[n<2, n, b[n-1, q] + q*b[n-2, q]];

T[n_, k_, q_]:= T[n, k, q]= b[k+1, q] + b[n-k+1, q] - b[n+1, q];

Table[Flatten[Table[T[n, k, q], {n, 0, 12}, {k, 0, n}]], {q, 0, 10}] (* modified by G. C. Greubel, Nov 23 2019 *)

(* Second program *)T[n_, k_]= Fibonacci[k+1] +Fibonacci[n-k+1] -Fibonacci[n+1]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 23 2019 *)

PROG

(PARI) T(n, k) = my(f=fibonacci); f(k+1) + f(n-k+1) - f(n+1); \\ G. C. Greubel, Nov 23 2019

(MAGMA) F:=Fibonacci; [F(k+1) +F(n-k+1) -F(n+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 23 2019

(Sage) f=fibonacci; [[f(k+1) + f(n-k+1) - f(n+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 23 2019

(GAP) F:=Fibonacci;; Flat(List([0..12], n-> List([0..n], k-> F(k+1) + F(n-k+1) - F(n+1) ))); # G. C. Greubel, Nov 23 2019

CROSSREFS

Cf. A000045.

Sequence in context: A171414 A270921 A038529 * A132312 A090431 A107336

Adjacent sequences:  A176256 A176257 A176258 * A176260 A176261 A176262

KEYWORD

sign,tabl

AUTHOR

Roger L. Bagula, Apr 13 2010

EXTENSIONS

Edited by G. C. Greubel, Nov 23 2019

STATUS

approved

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Last modified May 29 05:47 EDT 2020. Contains 334697 sequences. (Running on oeis4.)