斐波那契数列若干项的和

王守恩

数串(1)——0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418,

0=0,
1=1,
2=2,
3=3,
4=1+3,
5=5,
6=1+5,
7=2+5,
8=8,
9=1+8,
10=2+8,
11=3+8,
12=1+3+8,
13=13,
14=1+13,
15=2+13,
16=3+13,
17=1+3+13,
18=5+13,
19=1+5+13,
20=2+5+13,
21=21,
22=1+21,
23=2+21,
24=3+21,
25=1+3+21,
26=5+21,
27=1+5+21,
28=2+5+21,
29=8+21,
30=1+8+21,
31=2+8+21,
32=3+8+21,
33=1+3+8+21,
34=34,
35=1+34,
36=2+34,
37=3+34,
38=1+3+34,
39=5+34,
40=1+5+34,
41=2+5+34,
42=8+34,
43=1+8+34,
44=2+8+34,
45=3+8+34,
46=1+3+8+34,
47=13+34,
48=1+13+34,
49=2+13+34,
50=3+13+34,
51=1+3+13+34,
52=5+13+34,
53=1+5+13+34,
54=2+5+13+34,
55=1,
56=1+55,
57=2+55,
58=3+55,
59=1+3+55,
60=5+55,

得到数串(2)——1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 2, 3, 3, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 3, 4, 4, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 3, 4, 4, 2, 3, 3, 3, 4, 3, 4, 4, 3, 4, 4, 4, 5,

a(1)=0=0,
a(2)=4=1+3,
a(3)=12=1+3+8,
a(4)=33=1+3+8+21,
a(5)=88=1+3+8+21+55,
a(6)=232=1+3+8+21+55+144,
a(7)=609=1+3+8+21+55+144+377,
a(8)=1596,
a(9)=4180,

得到数串(3)——0, 4, 12, 33, 88, 232, 609, 1596, 4180, 10945, 28656, 75024, 196417, ——是这串数吗？

求助：是这串数吗？

northwolves

A027941
a(n) = Fibonacci(2*n + 1) - 1.

0, 1, 4, 12, 33, 88, 232, 609, 1596, 4180, 10945, 28656, 75024, 196417, 514228, 1346268, 3524577, 9227464, 24157816, 63245985, 165580140, 433494436, 1134903169, 2971215072, 7778742048, 20365011073, 53316291172, 139583862444, 365435296161, 956722026040

Also: smallest number not writeable as the sum of fewer than n positive Fibonacci numbers. E.g., a(5)=88 because it is the smallest number that needs at least 5 Fibonacci numbers: 88 = 55 + 21 + 8 + 3 + 1. - Johan Claes, Apr 19 2005 [corrected for offset and clarification by Mike Speciner, Sep 19 2023] In general, a(n) is the sum of n positive Fibonacci numbers as a(n) = Sum_{i=1..n} A000045(2*i). See A001076 when negative Fibonacci numbers can be included in the sum. - Mike Speciner, Sep 24 2023
Except for first term, numbers a(n) that set a new record in the number of Fibonacci numbers needed to sum up to n. Position of records in sequence A007895. - Ralf Stephan, May 15 2005

王守恩

northwolves 发表于 2025-7-28 06:22
A027941
a(n) = Fibonacci(2*n + 1) - 1.

简单！丢了！换一串。

Table[Solve[{(Sin[Pi/6] + Sin[Pi/3] + Sin[Pi/2])^(2 n - 1) ==A*Sin[Pi/6]^(2 n - 1) + B*Sin[Pi/3]^(2 n - 1) + Sin[Pi/2]^(2 n - 1)}, {A, B}, PositiveIntegers], {n, 9}]

n=1, A(1)=1, B(1)=1,
n=2, A(2)=46, B(2)=10,
n=3, A(3)=1156, B(3)=76,
n=4, A(4)=26440, B(4)=568,
n=5, A(5)=594352, B(5)=4240,
n=6, A(6)=13318240, B(6)=31648,
n=7, A(7)=298263616, B(7)=236224,
n=8, A(8)=6678960256, B(8)=1763200,
n=9, A(9)=149557916416, B(9)=13160704,

B(n)=1, 10, 76, 568, 4240, 31648, 236224, 1763200, 13160704, 98232832, 733219840, 5472827392, 40849739776, 304906608640, 2275853910016, ——A107903——Generalized NSW numbers.——条文没我们的有意义。

A(n)=1, 46, 1156, 26440, 594352, 13318240, 298263616, 6678960256, 149557916416, 3348948866560, 74990693573632, 1679214509639680, 37601483354976256,——OEIE就没有了。

固定n, 在A(n), B(n)有解的前提下,  C(n)表示可以取到的最大值。

Table[Solve[{(Sin[Pi/6] + Sin[Pi/3] + Sin[Pi/2])^(2 n - 1) ==A*Sin[Pi/6]^(2 n - 1) + B*Sin[Pi/3]^(2 n - 1) + C*Sin[Pi/2]^(2 n - 1)}, {A, B, C}, PositiveIntegers], {n, 3}]

C(n)=1, 6, 37, 207, 1161, 6504, 36410, 203826, 1141037, 6387614, 35758350,——电脑罢工了——好不容易搞了这么几个————连我也找不到通项公式。

northwolves

$A107903(n)=\lfloor 2^{n-2}\left(1+\sqrt{3}\right)  \left(2+\sqrt{3}\right)^{n-1}\rfloor$

northwolves

A(n)=1, 46, 1156, 26440, 594352, 13318240, 298263616, 6678960256, 149557916416, 3348948866560, 74990693573632, 1679214509639680, 37601483354976256,——OEIE就没有了。
-----------------------

$A(n)=\frac{6^n}{12}  \left(\left(\sqrt{3}+3\right) \left(2-\sqrt{3}\right)^n+\left(3-\sqrt{3}\right) \left(\sqrt{3}+2\right)^n\right)-2^{2 n-1}$

northwolves

Table[Solve[{(Sin[Pi/6] + Sin[Pi/3] + Sin[Pi/2])^(2 n - 1) ==A*Sin[Pi/6]^(2 n - 1) + B*Sin[Pi/3]^(2 n - 1) + C*Sin[Pi/2]^(2 n - 1)}, {A, B, C}, PositiveIntegers], {n, 3}]

C(n)=1, 6, 37, 207, 1161, 6504, 36410, 203826, 1141037, 6387614, 35758350,——电脑罢工了——好不容易搞了这么几个————连我也找不到通项公式。
---------------------------------------
$（\frac{3}{2}+\frac{\sqrt3}{2}）^{2n-1}=A*(frac{1}{2})^{2n-1}+B*(frac{\sqrt3}{2})^{2n-1}+C}$

显然B为定值，C为左式取整: $C_n=\lfloor\frac{ \left(3+\sqrt{3}\right)^{2 n-1}}{4^n}\rfloor$

王守恩

继续骚扰。

Table[FindInstance[{(Sin[Pi/k] + Sin[2 Pi/k] + Sin[3 Pi/k])^(2 n + 1) == A*Sin[Pi/k]^(2 n + 1) + B*Sin[2 Pi/k]^(2 n + 1) + C*Sin[3 Pi/k]^(2 n + 1)}, {A, B, C}, PositiveIntegers, 1], {n, 16}, {k, 3, 6}]

{{{A -> 7, B -> 1, C -> 2}},  {{A -> 19, B -> 7, C -> 1}},   {{A -> 4,  B -> 2,  C -> 15}},    {{A -> 6,  B -> 10,  C -> 6}}},
{{{A -> 31, B -> 1, C -> 2}}, {{A -> 28, B -> 41, C -> 204}}, {{A -> 11, B -> 28, C -> 94}}, {{A -> 1156, B -> 76,  C -> 1}}},
{{{A -> 127, B -> 1, C -> 2}}, {{A -> 28, B -> 239, C -> 2676}}, {{A -> 29, B -> 28, C -> 814}}, {{A -> 3528, B -> 568, C -> 180}}},
{{{A -> 511, B -> 1, C -> 2}}, {{A -> 28, B -> 1393, C -> 31492}}, {{A -> 76, B -> 28, C -> 5749}}, {{A -> 14256, B -> 4240, C -> 1134}}},
{{{A -> 2047, B -> 1, C -> 2}}, {{A -> 28, B -> 8119, C -> 367396}}, {{A -> 199, B -> 28, C -> 39574}}, {{A -> 55392, B -> 31648, C -> 6477}}},
{{{A -> 8191, B -> 1, C -> 2}}, {{A -> 28, B -> 47321, C -> 4282980}}, {{A -> 521, B -> 28, C -> 271414}}, {{A -> 222272, B -> 236224, C -> 36383}}},
{{{A -> 32767, B -> 1, C -> 2}}, {{A -> 28, B -> 275807, C -> 49926372}}, {{A -> 1364, B -> 28, C -> 1860469}}, {{A -> 907392, B -> 1763200, C -> 203799}}},
{{{A -> 131071, B -> 1, C -> 2}}, {{A -> 28, B -> 1607521, C -> 581984740}}, {{A -> 3571, B -> 28, C -> 12752014}}, {{A -> 3584768, B -> 13160704, C -> 1141010}}},
{{{A -> 524287, B -> 1, C -> 2}}, {{A -> 28, B -> 9369319, C -> 6784111588}},  {{A -> 9349, B -> 28, C -> 87403774}}, {{A -> 14177792, B -> 98232832, C -> 6387587}}},
{{{A -> 2097151, B -> 1, C -> 2}}, {{A -> 28, B -> 54608393, C -> 79081400292}}, {{A -> 24476, B -> 28, C -> 599074549}}, {{A -> 57074688, B -> 733219840, C -> 35758323}}},
{{{A -> 8388607, B -> 1, C -> 2}}, {{A -> 28, B -> 318281039, C -> 921840357348}}, {{A -> 64079, B -> 28, C -> 4106118214}}, {{A -> 232343552, B -> 5472827392, C -> 200177942}}},
{{{A -> 33554431, B -> 1, C -> 2}}, {{A -> 28, B -> 1855077841, C -> 10745758687204}}, {{A -> 167761, B -> 28, C -> 28143753094}}, {{A -> 912699392, B -> 40849739776, C -> 1120611503}}},
{{{A -> 134217727, B -> 1, C -> 2}}, {{A -> 28, B -> 10812186007, C -> 125261742817252}}, {{A -> 439204, B -> 28, C -> 192900153589}}, {{A -> 3675414528, B -> 304906608640, C -> 6273268722}}},
{{{A -> 536870911, B -> 1, C -> 2}}, {{A -> 28, B -> 63018038201, C -> 1460157879058404}}, {{A -> 1149851, B -> 28, C -> 1322157322174}}, {{A -> 14684798976, B -> 2275853910016, C -> 35118236526}}},
{{{A -> 2147483647, B -> 1, C -> 2}}, {{A -> 28, B -> 367296043199, C -> 17020847577432036}}, {{A -> 3010349, B -> 28, C -> 9062201101774}}, {{A -> 57985236992, B -> 16987204845568, C -> ,196594564607}}},
{{{A -> 8589934591, B -> 1, C -> 2}}, {{A -> 28, B -> 2140758220993, C -> 198409539412951012}}, {{A -> 7881196, B -> 28, C -> 62113250390389}}, {{A -> 240222666752, B -> 126794223124480, C -> 1100551355531}}}}

这里出现了12个数字串——其中数串(B4)=A107903——可惜数串(A4), (C4)就这么也找不到通项公式了。

数串(A1)=7, 31, 127, 511, 2047, 8191, 32767, 131071, 524287,——a(n) = 5*a(n-1) - 4*a(n-2)。
数串(B1)
数串(C1)
数串(A2)
数串(B2)=7, 41, 239, 1393, 8119, 47321, 275807, 1607521, 9369319,——a(n) = 6*a(n-1) - a(n-2)。
数串(C2)=1, 204, 2676, 31492, 367396, 4282980, 49926372, 581984740,——a(n) = 13*a(n-1) - 16*a(n-1) + 4*a(n-2)。
数串(A3)=4, 11, 29, 76, 199, 521, 1364, 3571, 9349, 24476, 64079,167761,——a(n) = 3*a(n-1) - a(n-2)。
数串(B3)
数串(C3)=15, 94, 814, 5749, 39574, 271414, 1860469, 12752014, 87403774,——a(n) = 8*a(n-1) - 8*a(n-1) + a(n-2)。
数串(A4)=6, 1156, 3528, 14256, 55392, 222272, 907392, 3584768, 14177792, 57074688, 232343552, 912699392, 3675414528, 14684798976, 57985236992, 240222666752,——？
数串(B4)=10, 76, 568, 4240, 31648, 236224, 1763200, 13160704, 98232832,——a(n) = 8*a(n-1) - 4*a(n-2)——A107903——也就是我们的4#。
数串(C4)=180, 1134, 6477, 36383, 203799, 1141010, 6387587, 35758323, 200177942, 1120611503, 6273268722, 35118236526, 196594564607, 1100551355531,——？

王守恩

2025中国科技大学强基计划数学试题 ——第7题——扩散。

\(将1, 2, 3, ..., i, ..., n,  这 n 个数重新排列, 得到新序列a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}。 约定 i-1≤a_{i}≤i+1。求满足条件的排列数量。\)
{1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352}
LinearRecurrence[{1, 1}, {0, 1}, 40]

\(将1, 2, 3, ..., i, ..., n,  这 n 个数重新排列, 得到新序列a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}。 约定 i-1≤a_{i}≤i+2。求满足条件的排列数量。\)
{1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757, 4700770, 8646064, 15902591, 29249425, 53798080, 98950096, 181997601}
LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 40]

\(将1, 2, 3, ..., i, ..., n,  这 n 个数重新排列, 得到新序列a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}。 约定 i-1≤a_{i}≤i+3。求满足条件的排列数量。\)
{1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, 1055026, 2033628, 3919944, 7555935, 14564533, 28074040, 54114452, 104308960, 201061985, 387559437}
LinearRecurrence[{1, 1, 1, 1}, {0, 0, 0, 1}, 40]

\(将1, 2, 3, ..., i, ..., n,  这 n 个数重新排列, 得到新序列a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}。 约定 i-1≤a_{i}≤i+4。求满足条件的排列数量。\)
{1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513, 400096, 786568, 1546352, 3040048, 5976577, 11749641, 23099186, 45411804, 89277256, 175514464, 345052351, 678355061}
LinearRecurrence[{1, 1, 1, 1, 1}, {0, 0, 0, 0, 1}, 40]

\(将1, 2, 3, ..., i, ..., n,  这 n 个数重新排列, 得到新序列a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}。 约定 i-1≤a_{i}≤i+5。求满足条件的排列数量。\)
{1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, 29970, 59448, 117920, 233904, 463968, 920319, 1825529, 3621088, 7182728, 14247536, 28261168, 56058368, 111196417, 220567305, 437513522, 867844316}
LinearRecurrence[{1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 1}, 40]

\(将1, 2, 3, ..., i, ..., n,  这 n 个数重新排列, 得到新序列a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}。 约定 i-1≤a_{i}≤i+6。求满足条件的排列数量。\)
{1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, 31489, 62725, 124946, 248888, 495776, 987568, 1967200, 3918592, 7805695, 15548665, 30972384, 61695880, 122895984, 244804400, 487641600, 971364608}
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 1}, 40]

\(将1, 2, 3, ..., i, ..., n,  这 n 个数重新排列, 得到新序列a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}。 约定 i-1≤a_{i}≤i+7。求满足条件的排列数量。\)
{1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128, 32192, 64256, 128257, 256005, 510994, 1019960, 2035872, 4063664, 8111200, 16190208, 32316160, 64504063, 128752121, 256993248, 512966536, 1023897200}
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 1}, 40]

\(将1, 2, 3, ..., i, ..., n,  这 n 个数重新排列, 得到新序列a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}。 约定 i-1≤a_{i}≤i+8。求满足条件的排列数量。\)
{1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, 518145, 1035269, 2068498, 4132920, 8257696, 16499120, 32965728, 65866496, 131603200, 262947072, 525375999, 1049716729}
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1}, 40]