The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A181926 Diagonal sums of Fibonomial triangle A010048. 4
 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: A346779 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 18 02:30 EDT 2024. Contains 373468 sequences. (Running on oeis4.)