フィボナッチ数列の定義
フィボナッチ数列の定義
フィボナッチ数列は次の漸化式で定義される。
\begin{align*} F_{0} & =0\\ F_{1} & =1\\ F_{n+2} & =F_{n+1}+F_{n}\qquad\left(n\in\mathbb{N}_{0}\right) \end{align*} フィボナッチ数列に現れる数をフィボナッチ数といいます。
フィボナッチ数列は次の漸化式で定義される。
\begin{align*} F_{0} & =0\\ F_{1} & =1\\ F_{n+2} & =F_{n+1}+F_{n}\qquad\left(n\in\mathbb{N}_{0}\right) \end{align*} フィボナッチ数列に現れる数をフィボナッチ数といいます。
フィボナッチ数一覧
\[ \begin{array}{|c|c|} \hline n & F_{n}\\ \hline 0 & 0\\ \hline 1 & 1\\ \hline 2 & 1\\ \hline 3 & 2\\ \hline 4 & 3\\ \hline 5 & 5\\ \hline 6 & 8\\ \hline 7 & 13\\ \hline 8 & 21\\ \hline 9 & 34\\ \hline 10 & 55\\ \hline 11 & 89\\ \hline 12 & 144\\ \hline 13 & 233\\ \hline 14 & 377\\ \hline 15 & 610\\ \hline 16 & 987\\ \hline 17 & 1,597\\ \hline 18 & 2,584\\ \hline 19 & 4,181\\ \hline 20 & 6,765\\ \hline 21 & 10,946\\ \hline 22 & 17,711\\ \hline 23 & 28,657\\ \hline 24 & 46,368\\ \hline 25 & 75,025\\ \hline 26 & 121,393\\ \hline 27 & 196,418\\ \hline 28 & 317,811\\ \hline 29 & 514,229\\ \hline 30 & 832,040\\ \hline 31 & 1,346,269\\ \hline 32 & 2,178,309\\ \hline 33 & 3,524,578\\ \hline 34 & 5,702,887\\ \hline 35 & 9,227,465\\ \hline 36 & 14,930,352\\ \hline 37 & 24,157,817\\ \hline 38 & 39,088,169\\ \hline 39 & 63,245,986\\ \hline 40 & 102,334,155\\ \hline 41 & 165,580,141\\ \hline 42 & 267,914,296\\ \hline 43 & 433,494,437\\ \hline 44 & 701,408,733\\ \hline 45 & 1,134,903,170\\ \hline 46 & 1,836,311,903\\ \hline 47 & 2,971,215,073\\ \hline 48 & 4,807,526,976\\ \hline 49 & 7,778,742,049\\ \hline 50 & 12,586,269,025\\ \hline 51 & 20,365,011,074\\ \hline 52 & 32,951,280,099\\ \hline 53 & 53,316,291,173\\ \hline 54 & 86,267,571,272\\ \hline 55 & 139,583,862,445\\ \hline 56 & 225,851,433,717\\ \hline 57 & 365,435,296,162\\ \hline 58 & 591,286,729,879\\ \hline 59 & 956,722,026,041\\ \hline 60 & 1,548,008,755,920\\ \hline 61 & 2,504,730,781,961\\ \hline 62 & 4,052,739,537,881\\ \hline 63 & 6,557,470,319,842\\ \hline 64 & 10,610,209,857,723\\ \hline 65 & 17,167,680,177,565\\ \hline 66 & 27,777,890,035,288\\ \hline 67 & 44,945,570,212,853\\ \hline 68 & 72,723,460,248,141\\ \hline 69 & 117,669,030,460,994\\ \hline 70 & 190,392,490,709,135\\ \hline 71 & 308,061,521,170,129\\ \hline 72 & 498,454,011,879,264\\ \hline 73 & 806,515,533,049,393\\ \hline 74 & 1,304,969,544,928,657\\ \hline 75 & 2,111,485,077,978,050\\ \hline 76 & 3,416,454,622,906,707\\ \hline 77 & 5,527,939,700,884,757\\ \hline 78 & 8,944,394,323,791,464\\ \hline 79 & 14,472,334,024,676,221\\ \hline 80 & 23,416,728,348,467,685\\ \hline 81 & 37,889,062,373,143,906\\ \hline 82 & 61,305,790,721,611,591\\ \hline 83 & 99,194,853,094,755,497\\ \hline 84 & 160,500,643,816,367,088\\ \hline 85 & 259,695,496,911,122,585\\ \hline 86 & 420,196,140,727,489,673\\ \hline 87 & 679,891,637,638,612,258\\ \hline 88 & 1,100,087,778,366,101,931\\ \hline 89 & 1,779,979,416,004,714,189\\ \hline 90 & 2,880,067,194,370,816,120\\ \hline 91 & 4,660,046,610,375,530,309\\ \hline 92 & 7,540,113,804,746,346,429\\ \hline 93 & 12,200,160,415,121,876,738\\ \hline 94 & 19,740,274,219,868,223,167\\ \hline 95 & 31,940,434,634,990,099,905\\ \hline 96 & 51,680,708,854,858,323,072\\ \hline 97 & 83,621,143,489,848,422,977\\ \hline 98 & 135,301,852,344,706,746,049\\ \hline 99 & 218,922,995,834,555,169,026\\ \hline 100 & 354,224,848,179,261,915,075 \\\hline \end{array} \]
\[ \begin{array}{|c|c|} \hline n & F_{n}\\ \hline 0 & 0\\ \hline 1 & 1\\ \hline 2 & 1\\ \hline 3 & 2\\ \hline 4 & 3\\ \hline 5 & 5\\ \hline 6 & 8\\ \hline 7 & 13\\ \hline 8 & 21\\ \hline 9 & 34\\ \hline 10 & 55\\ \hline 11 & 89\\ \hline 12 & 144\\ \hline 13 & 233\\ \hline 14 & 377\\ \hline 15 & 610\\ \hline 16 & 987\\ \hline 17 & 1,597\\ \hline 18 & 2,584\\ \hline 19 & 4,181\\ \hline 20 & 6,765\\ \hline 21 & 10,946\\ \hline 22 & 17,711\\ \hline 23 & 28,657\\ \hline 24 & 46,368\\ \hline 25 & 75,025\\ \hline 26 & 121,393\\ \hline 27 & 196,418\\ \hline 28 & 317,811\\ \hline 29 & 514,229\\ \hline 30 & 832,040\\ \hline 31 & 1,346,269\\ \hline 32 & 2,178,309\\ \hline 33 & 3,524,578\\ \hline 34 & 5,702,887\\ \hline 35 & 9,227,465\\ \hline 36 & 14,930,352\\ \hline 37 & 24,157,817\\ \hline 38 & 39,088,169\\ \hline 39 & 63,245,986\\ \hline 40 & 102,334,155\\ \hline 41 & 165,580,141\\ \hline 42 & 267,914,296\\ \hline 43 & 433,494,437\\ \hline 44 & 701,408,733\\ \hline 45 & 1,134,903,170\\ \hline 46 & 1,836,311,903\\ \hline 47 & 2,971,215,073\\ \hline 48 & 4,807,526,976\\ \hline 49 & 7,778,742,049\\ \hline 50 & 12,586,269,025\\ \hline 51 & 20,365,011,074\\ \hline 52 & 32,951,280,099\\ \hline 53 & 53,316,291,173\\ \hline 54 & 86,267,571,272\\ \hline 55 & 139,583,862,445\\ \hline 56 & 225,851,433,717\\ \hline 57 & 365,435,296,162\\ \hline 58 & 591,286,729,879\\ \hline 59 & 956,722,026,041\\ \hline 60 & 1,548,008,755,920\\ \hline 61 & 2,504,730,781,961\\ \hline 62 & 4,052,739,537,881\\ \hline 63 & 6,557,470,319,842\\ \hline 64 & 10,610,209,857,723\\ \hline 65 & 17,167,680,177,565\\ \hline 66 & 27,777,890,035,288\\ \hline 67 & 44,945,570,212,853\\ \hline 68 & 72,723,460,248,141\\ \hline 69 & 117,669,030,460,994\\ \hline 70 & 190,392,490,709,135\\ \hline 71 & 308,061,521,170,129\\ \hline 72 & 498,454,011,879,264\\ \hline 73 & 806,515,533,049,393\\ \hline 74 & 1,304,969,544,928,657\\ \hline 75 & 2,111,485,077,978,050\\ \hline 76 & 3,416,454,622,906,707\\ \hline 77 & 5,527,939,700,884,757\\ \hline 78 & 8,944,394,323,791,464\\ \hline 79 & 14,472,334,024,676,221\\ \hline 80 & 23,416,728,348,467,685\\ \hline 81 & 37,889,062,373,143,906\\ \hline 82 & 61,305,790,721,611,591\\ \hline 83 & 99,194,853,094,755,497\\ \hline 84 & 160,500,643,816,367,088\\ \hline 85 & 259,695,496,911,122,585\\ \hline 86 & 420,196,140,727,489,673\\ \hline 87 & 679,891,637,638,612,258\\ \hline 88 & 1,100,087,778,366,101,931\\ \hline 89 & 1,779,979,416,004,714,189\\ \hline 90 & 2,880,067,194,370,816,120\\ \hline 91 & 4,660,046,610,375,530,309\\ \hline 92 & 7,540,113,804,746,346,429\\ \hline 93 & 12,200,160,415,121,876,738\\ \hline 94 & 19,740,274,219,868,223,167\\ \hline 95 & 31,940,434,634,990,099,905\\ \hline 96 & 51,680,708,854,858,323,072\\ \hline 97 & 83,621,143,489,848,422,977\\ \hline 98 & 135,301,852,344,706,746,049\\ \hline 99 & 218,922,995,834,555,169,026\\ \hline 100 & 354,224,848,179,261,915,075 \\\hline \end{array} \]
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フィボナッチ数列の総和
\[
\sum_{k=0}^{n}F_{k}=F_{n+2}-1
\]
フィボナッチ数の負整数での値
\[
F_{-n}=\left(-1\right)^{n+1}F_{n}
\]
フィボナッチ数列の商の極限
\[
\lim_{n\rightarrow\infty}\frac{F_{n+1}}{F_{n}}=\phi
\]
フィボナッチ数列同士の最大公約数
\[
\gcd\left(F_{m},F_{n}\right)=F_{\gcd\left(m,n\right)}
\]