{"id":5089,"date":"2022-10-25T00:07:31","date_gmt":"2022-10-24T22:07:31","guid":{"rendered":"https:\/\/blogsaverroes.juntadeandalucia.es\/recursosdematematicas\/?p=5089"},"modified":"2022-10-25T20:43:45","modified_gmt":"2022-10-25T18:43:45","slug":"sucesion-de-jacobsthal","status":"publish","type":"post","link":"https:\/\/blogsaverroes.juntadeandalucia.es\/recursosdematematicas\/sucesion-de-jacobsthal\/","title":{"rendered":"Sucesi\u00f3n de Jacobsthal"},"content":{"rendered":"

La sucesi\u00f3n de Jacobsthal<\/strong>, debida al matem\u00e1tico alem\u00e1n Ernst Jacobstal<\/strong> (1882-1965), es la siguiente sucesi\u00f3n de n\u00fameros:<\/span><\/p>\n

0, 1, 1 , 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, . . .<\/span><\/strong><\/p>\n

\u2022 Esta sucesi\u00f3n se puede obtener como una sucesi\u00f3n recurrente conociendo los dos t\u00e9rminos anteriores:<\/span><\/p>\n

a1 <\/sub>= 0   ,  a2 <\/sub>= 1   ,  an <\/sub>= 2\u00b7an\u20132 <\/sub>+ an\u20131<\/sub><\/span><\/strong><\/p>\n\n\n\n\n
a3<\/sub>=2a1<\/sub>+a2<\/sub>=0+1=1<\/strong><\/span><\/td>\na4<\/sub>=2a2<\/sub>+a3<\/sub>=2+1=3<\/strong><\/span><\/td>\na5<\/sub>=2a3<\/sub>+a4<\/sub>=2+3=5<\/strong><\/span><\/td>\na6<\/sub>=2a4<\/sub>+a5<\/sub>=6+5=11<\/strong><\/span><\/td>\n<\/tr>\n
a7<\/sub>=2a5<\/sub>+a6<\/sub>=10+11=21<\/strong><\/span><\/td>\na8<\/sub>=2a6<\/sub>+a7<\/sub>=22+21=43<\/strong><\/span><\/td>\na9<\/sub>=2a7<\/sub>+a8<\/sub>=42+43=85<\/strong><\/span><\/td>\na10<\/sub>=2a8<\/sub>+a9<\/sub>=86+85=171<\/strong><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u2022 Esta sucesi\u00f3n se puede obtener tambi\u00e9n como una sucesi\u00f3n recurrente conociendo el t\u00e9rmino anterior:<\/span><\/p>\n

a1 <\/sub>= 0   ,  an <\/sub>= 2\u00b7an\u20131 <\/sub>+(-1)n<\/sup><\/span><\/strong><\/p>\n\n\n\n\n
a2<\/sub>=2a1<\/sub>+<\/span>(-1)2<\/sup><\/span>=0+1=<\/span>1<\/strong><\/span><\/td>\na3<\/sub>=2a2<\/sub>+<\/span>(-1)3<\/sup><\/span>=2\u20131=<\/span>1<\/strong><\/span><\/td>\na4<\/sub>=2a3<\/sub>+(-1)4<\/sup><\/span>=2+1=3<\/strong><\/span><\/td>\na5<\/sub>=2a4<\/sub>+(-1)5<\/sup><\/span>=6\u2013<\/span>1=5<\/strong><\/span><\/td>\n<\/tr>\n
a6<\/sub>=2a5<\/sub>+(-1)6<\/sup><\/span>=10+1=11<\/strong><\/span><\/td>\na7<\/sub>=2a6<\/sub>+(-1)7<\/sup><\/span>=22\u2013<\/span>1=21<\/strong><\/span><\/td>\na8<\/sub>=2a7<\/sub>+(-1)8<\/sup><\/span>=42+1=43<\/strong><\/span><\/td>\na9<\/sub>=2a8<\/sub>+(-1)9<\/sup><\/span>=86\u2013<\/span>1=85<\/strong><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u2022 Tambi\u00e9n se puede obtener cualquier t\u00e9rmino de la sucesi\u00f3n sin necesidad de conocer ninguno de los t\u00e9rminos anteriores, mediante el t\u00e9rmino general:<\/span><\/p>\n

\"\"<\/p>\n\n\n\n\n
a0<\/sub>=(1\u2013<\/span>1)\/3=0<\/span><\/td>\na1<\/sub>=(2+1)\/3<\/span>=<\/span>1<\/strong><\/span><\/td>\na2<\/sub>=(4\u2013<\/span>1)\/3=1<\/strong><\/span><\/td>\na3<\/sub>=(8+1)\/3=3<\/strong><\/span><\/td>\n<\/tr>\n
a4<\/sub>=(16\u2013<\/span>1)\/3=5<\/strong><\/span><\/td>\na5<\/sub>=(32+1)\/3=<\/span>11<\/strong><\/span><\/td>\na<\/span>6<\/sub>=(64\u2013<\/span>1)\/3=21<\/strong><\/span><\/td>\na7<\/sub>=(128+1)\/3=43<\/strong><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u2022 Y otra forma de obtener los t\u00e9rminos de esta sucesi\u00f3n, excepto el primero de ellos, es como sumas y diferencias alternas de potencias de 2:<\/span><\/span><\/p>\n\n\n\n\n\n\n\n\n\n
n=1  \u2192   a1 <\/sub>= 0<\/span><\/td>\n<\/tr>\n
n=2  \u2192   a2 <\/sub>= 20 <\/sup>= 1 <\/span><\/td>\n<\/tr>\n
n=3  \u2192   a3 <\/sub>= 21 <\/sup>\u2013 <\/span>20 <\/sup>= 2 \u2013 <\/span>1 = 1<\/span><\/td>\n<\/tr>\n
n=4  \u2192   a4 <\/sub>= 22 <\/sup>\u2013 <\/span>21 <\/sup>+ 20<\/sup> <\/sup>= 4 \u2013 2 <\/span>+ 1 = 3<\/span><\/td>\n<\/tr>\n
n=5  \u2192   a5 <\/sub>= 23 <\/sup>\u2013 <\/span>22 <\/sup>+ 21 <\/sup>\u2013 <\/span>20 <\/sup>= 8 \u2013 <\/span>4 + 2 \u2013 <\/span>1 = 5<\/span><\/td>\n<\/tr>\n
n=6  \u2192   a6 <\/sub>= 24 <\/sup>\u2013 <\/span>23 <\/sup>+ 22 <\/sup>\u2013 <\/span>21 <\/sup>+ 20<\/sup> <\/sup>= 16 \u2013 8 <\/span>+ 4 \u2013 2 <\/span>+ 1 = 11<\/span><\/td>\n<\/tr>\n
n=7  \u2192   a7 <\/sub>= 25 <\/sup>\u2013 <\/span>24 <\/sup>+ 23 <\/sup>\u2013 <\/span>22 <\/sup>+ 21 <\/sup>\u2013 <\/span>20 <\/sup>= 32 \u2013 <\/span>16 + 8 \u2013 <\/span>4 + 2 \u2013 <\/span>1 = 21<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Esta relaci\u00f3n se puede expresar de la forma:<\/span><\/p>\n

\"\"<\/p>\n","protected":false},"excerpt":{"rendered":"

La sucesi\u00f3n de Jacobsthal, debida al matem\u00e1tico alem\u00e1n Ernst Jacobstal (1882-1965), es la siguiente sucesi\u00f3n de n\u00fameros: 0, 1, 1 , 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, ....<\/p>\n","protected":false},"author":7927,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"footnotes":""},"categories":[2075204],"tags":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/blogsaverroes.juntadeandalucia.es\/recursosdematematicas\/wp-json\/wp\/v2\/posts\/5089"}],"collection":[{"href":"https:\/\/blogsaverroes.juntadeandalucia.es\/recursosdematematicas\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogsaverroes.juntadeandalucia.es\/recursosdematematicas\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogsaverroes.juntadeandalucia.es\/recursosdematematicas\/wp-json\/wp\/v2\/users\/7927"}],"replies":[{"embeddable":true,"href":"https:\/\/blogsaverroes.juntadeandalucia.es\/recursosdematematicas\/wp-json\/wp\/v2\/comments?post=5089"}],"version-history":[{"count":16,"href":"https:\/\/blogsaverroes.juntadeandalucia.es\/recursosdematematicas\/wp-json\/wp\/v2\/posts\/5089\/revisions"}],"predecessor-version":[{"id":5107,"href":"https:\/\/blogsaverroes.juntadeandalucia.es\/recursosdematematicas\/wp-json\/wp\/v2\/posts\/5089\/revisions\/5107"}],"wp:attachment":[{"href":"https:\/\/blogsaverroes.juntadeandalucia.es\/recursosdematematicas\/wp-json\/wp\/v2\/media?parent=5089"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogsaverroes.juntadeandalucia.es\/recursosdematematicas\/wp-json\/wp\/v2\/categories?post=5089"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogsaverroes.juntadeandalucia.es\/recursosdematematicas\/wp-json\/wp\/v2\/tags?post=5089"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}