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A181926 Diagonal sums of Fibonomial triangle A010048. 3
1, 1, 2, 2, 4, 6, 13, 27, 70, 191, 609, 2130, 8526, 38156, 194000, 1109673, 7176149, 52238676, 429004471, 3970438003, 41454181730, 488046132076, 6482590679282, 97134793638750, 1641654359781521, 31285014253070731, 672372121341768918, 16299021330860540657 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Cf. A000045 (Fibonacci) as diagonal sums of A007318 (Pascal's Triangle). For Fibonacci numbers, the ratio A000045(i+1)/A000045(i) approaches the golden ratio (1+sqrt(5))/2 as i increases. For this sequence, it appears that (a(i+5)/a(i+4))/(a(i+1)/a(i)) approaches the golden ratio. - Dale Gerdemann, Apr 23 2015

This sequence can be interpreted as counting colored, square-domino tilings of a 1xn board, where the number of colors available for a domino with k squares to the left is Fib(k+1) and the number of colors available for a square with k dominoes to the left is Fib(k-1). "Fib(n)" here is A000045 (Fibonacci), extended so that Fib(-1) = 1, Fib(0) = 0,... . As an example, let d be a domino, s be a square an consider the uncolored tilings of length 5: sssss, sssd, ssds, sdss, dsss, sdd, dsd, dds. Then, after each 's' or 'd', write the number of colors available: s1s1s1s1s1, s1s1s1d3, s1s1d2s0, s1d1s0s0, d1s0s0s0, s1d1d1, d1s0d1, d1d1s1. So the number of colorings for these tilings is: 1,3,0,0,0,1,0,1 and the total number of colored tilings is 6 (= a(5)). - Dale Gerdemann, Apr 30 2015

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..195

Vaclav Kotesovec, Graph - The asymptotic ratio

FORMULA

a(n) = sum(fibonomial(k,n-k),k=ceiling(n/2)..n).

From Vaclav Kotesovec, Apr 29 2015: (Start)

a(n) ~ c * ((1+sqrt(5))/2)^(n^2/8), where

c = 1.472885929099569314607134281503815932269629515265... if mod(n,4)=0,

c = 1.472782295338429619549807628338486893461428897618... if mod(n,4)=1 or 3,

c = 1.472678661577289942545896597162143392952724631588... if mod(n,4)=2.

Or c = Sum_{j} ((1+sqrt(5))/2)^(-2*(j+(1-cos(Pi*n/2))/4)^2) / A062073, where A062073 = 1.2267420107203532444176302... is the Fibonacci factorial constant.

(End)

a(n) = Sum_{k=ceiling(n/2)..n} A003266(k) / (A003266(2*k-n) * A003266(n-k)). - Vaclav Kotesovec, Apr 30 2015

MATHEMATICA

Table[Sum[Product[Fibonacci[k-j+1]/Fibonacci[j], {j, 1, n-k}], {k, Ceiling[n/2], n}], {n, 0, 30}] (* Vaclav Kotesovec, Apr 29 2015 *)

(* Or, since version 10 *) Table[Sum[Fibonorial[k]/Fibonorial[2k-n]/Fibonorial[n-k], {k, Ceiling[n/2], n}], {n, 0, 30}] (* Vaclav Kotesovec, Apr 30 2015 *)

(* List of the multiplicative constants from an asymptotic formula: *) {N[EllipticTheta[3, 0, GoldenRatio^(-2)]/QPochhammer[-(GoldenRatio^2)^(-1)], 80], N[Sum[GoldenRatio^(-2*(j + 1/4)^2), {j, -Infinity, Infinity}]/QPochhammer[-(GoldenRatio^2)^(-1)], 80], N[EllipticTheta[2, 0, GoldenRatio^(-2)]/QPochhammer[-(GoldenRatio^2)^(-1)], 80]} (* Vaclav Kotesovec, Apr 30 2015 *)

PROG

(Maxima) ffib(n):=prod(fib(k), k, 1, n);

fibonomial(n, k):=ffib(n)/(ffib(k)*ffib(n-k));

makelist(sum(fibonomial(k, n-k), k, ceiling(n/2), n), n, 0, 30);

CROSSREFS

Cf. A003266, A010048, A056569, A062073.

Sequence in context: A209025 A153955 A074028 * A061894 A116684 A276057

Adjacent sequences:  A181923 A181924 A181925 * A181927 A181928 A181929

KEYWORD

nonn,easy

AUTHOR

Emanuele Munarini, Apr 02 2012

EXTENSIONS

a(14) corrected by Vaclav Kotesovec, Apr 29 2015

STATUS

approved

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Last modified January 21 10:26 EST 2020. Contains 331105 sequences. (Running on oeis4.)