\u6570\u5217B\u7684\u89c4\u5f8b\uff1a<\/p>\n\n\n\n
- \n
- b\u1d62 = a\u1d62 \u00d7 a\u1d62<\/li>\n\n\n\n
- b\u2081 = 6 \u00d7 6 = 36<\/li>\n\n\n\n
- b\u2082 = 66 \u00d7 66 = 4356<\/li>\n\n\n\n
- …<\/li>\n<\/ul>\n\n\n\n
\u8981\u6c42\uff1a \u8ba1\u7b97 b\u2081 + b\u2082 + … + b\u2099<\/p>\n\n\n\n
\u5173\u952e\u95ee\u9898<\/h2>\n\n\n\n
\u5f53 n = 10,000,000 \u65f6\uff1a<\/p>\n\n\n\n
- \n
- a\u2099 \u4f1a\u975e\u5e38\u5927\uff08\u7ea610\u2077\u4f4d\u6570\uff09<\/li>\n\n\n\n
- b\u2099 \u4f1a\u66f4\u5927<\/li>\n\n\n\n
- \u65e0\u6cd5\u7528\u5e38\u89c4\u6574\u6570\u5b58\u50a8\uff01<\/li>\n<\/ul>\n\n\n\n
\u9700\u8981\u7528\u5b57\u7b26\u4e32\u6216\u5927\u6570\u8fd0\u7b97\u5904\u7406\u3002<\/p>\n\n\n\n
\u4e8e\u662f\u6211\u7684\u7b2c\u4e00\u7248\u4ee3\u7801\u5c31\u662f<\/p>\n\n\n\n
n = int(input())\n\n# \u7528\u5b57\u7b26\u4e32\u6a21\u62df\u5927\u6570\u8fd0\u7b97\na = \"6\" # a1 = 6\ntotal = \"0\" # \u603b\u548c\n\ndef string_add(x, y):\n \"\"\"\u5b57\u7b26\u4e32\u5927\u6570\u52a0\u6cd5\"\"\"\n # \u5bf9\u9f50\u4f4d\u6570\n max_len = max(len(x), len(y))\n x = x.zfill(max_len)\n y = y.zfill(max_len)\n \n result = []\n carry = 0\n for i in range(max_len - 1, -1, -1):\n digit_sum = int(x[i]) + int(y[i]) + carry\n result.append(str(digit_sum % 10))\n carry = digit_sum \/\/ 10\n if carry:\n result.append(str(carry))\n \n return ''.join(reversed(result))\n\ndef string_multiply(x, y):\n \"\"\"\u5b57\u7b26\u4e32\u5927\u6570\u4e58\u6cd5\"\"\"\n if x == \"0\" or y == \"0\":\n return \"0\"\n \n result = [0] * (len(x) + len(y))\n \n for i in range(len(x) - 1, -1, -1):\n for j in range(len(y) - 1, -1, -1):\n mul = int(x[i]) * int(y[j])\n p1, p2 = i + j, i + j + 1\n total = mul + result[p2]\n result[p2] = total % 10\n result[p1] += total \/\/ 10\n \n # \u8f6c\u4e3a\u5b57\u7b26\u4e32\uff0c\u53bb\u6389\u524d\u5bfc\u96f6\n result_str = ''.join(map(str, result)).lstrip('0')\n return result_str if result_str else \"0\"\n\nfor i in range(1, n + 1):\n # \u8ba1\u7b97 b_i = a_i * a_i\n b = string_multiply(a, a)\n \n # \u7d2f\u52a0\u5230\u603b\u548c\n total = string_add(total, b)\n \n # \u8ba1\u7b97\u4e0b\u4e00\u4e2a a_{i+1} = a_i * 10 + 6\n if i < n:\n a = a + \"6\" # \u76f4\u63a5\u5728\u672b\u5c3e\u52a06\u5373\u53ef\n\nprint(total)\n\n\u4f46\u6709\u4e24\u4e2a\u7ec6\u8282\u4f1a\u8ba9\u7a0b\u5e8f\u5728 n = 10^7 \u65f6 \u8d85\u65f6 \/ \u8d85\u5185\u5b58\uff1a\n\u5b57\u7b26\u4e32\u8d8a\u6765\u8d8a\u957fai \u7684\u957f\u5ea6\u662f i \u4f4d\uff0c\u5e73\u65b9\u540e\u957f\u5ea6 \u2248 2i\u3002\u5f53 i = 10^7 \u65f6\uff0c\u5355\u6761\u5b57\u7b26\u4e32\u5c31\u6709 2\u00d710^7 \u5b57\u7b26\uff0c\u518d\u4e58\u4e0a Python \u5b57\u7b26\u4e32\u4e0d\u53ef\u53d8\uff0c\u6bcf\u6b21\u62fc\u63a5\u3001\u4e58\u6cd5\u90fd\u4f1a\u62f7\u8d1d O(len) \u5b57\u8282\uff0c\u603b\u64cd\u4f5c\u91cf\u7ea6\u4e3a\n\u03a3_{i=1}^{n} O(i) = O(n\u00b2) \u2248 5\u00d710^13 \u6b21\u5b57\u7b26\u642c\u8fd0\uff0c\u8fdc\u8fdc\u8d85\u51fa 3 s \u9650\u5236\u3002\n\n\u5927\u6570\u52a0\u6cd5\u4e5f\u8d8a\u52a0\u8d8a\u5927\n\u603b\u548c\u957f\u5ea6\u6700\u540e\u8fbe\u5230 2n = 2\u00d710^7 \u4f4d\uff0c\u6bcf\u52a0\u4e00\u6b21\u90fd\u8981\u904d\u5386\u8fd9\u4e48\u957f\uff0c\u603b\u65f6\u95f4\u53c8\u662f O(n\u00b2)\u3002\n\u56e0\u6b64\u201c\u7eaf\u5b57\u7b26\u4e32\u6a21\u62df\u201d\u53ea\u80fd\u62ff\u5230 50 \u5206\uff08n \u2264 10^5\uff09\u5de6\u53f3\uff0cn = 10^7 \u5fc5\u5b9a TLE\/MLE\u3002<\/code><\/pre>\n\n\n\n<\/p>\n\n\n\n
\u6240\u4ee5\u628a\u4ee3\u7801\u63a8\u7ffb\u91cd\u5199<\/h2>\n\n\n\n
\u6570\u5b66\u5206\u6790<\/h2>\n\n\n\n
\u6570\u5217A\u7684\u901a\u9879\u516c\u5f0f\uff1a<\/p>\n\n\n\n
<\/p>\n\n\n\n
a\u1d62 = 666\u20266 (i\u4e2a6)
= 6 \u00d7 (10^(i-1) + 10^(i-2) + \u2026 + 10^0)
= 6 \u00d7 (10^i - 1) \/ 9
= 2 \u00d7 (10^i - 1) \/ 3<\/pre>\n\n\n\n\u6570\u5217B\u7684\u901a\u9879\u516c\u5f0f\uff1a<\/p>\n\n\n\n
b\u1d62 = a\u1d62\u00b2 = [2 \u00d7 (10^i - 1) \/ 3]\u00b2
= 4 \u00d7 (10^i - 1)\u00b2 \/ 9
= 4 \u00d7 (10^(2i) - 2\u00d710^i + 1) \/ 9<\/pre>\n\n\n\n\u6c42\u548c\u516c\u5f0f\uff1a<\/p>\n\n\n\n
S\u2099 = \u03a3(i=1 to n) b\u1d62
= \u03a3(i=1 to n) [4\u00d7(10^(2i) - 2\u00d710^i + 1) \/ 9]
= 4\/9 \u00d7 [\u03a310^(2i) - 2\u00d7\u03a310^i + n]<\/pre>\n\n\n\n\u5229\u7528\u7b49\u6bd4\u6570\u5217\u6c42\u548c\uff1a<\/p>\n\n\n\n
\u03a310^i = (10^(n+1) - 10) \/ 9
\u03a310^(2i) = (10^(2n+2) - 100) \/ 99<\/pre>\n\n\n\nPython\uff08\u6570\u5b66\u516c\u5f0f\uff09<\/p>\n\n\n\n
n = int(input())
def fast_power_mod(base, exp, mod):
\"\"\"\u5feb\u901f\u5e42\u8ba1\u7b97 (base^exp) % mod\"\"\"
result = 1
base = base % mod
while exp > 0:
if exp & 1:
result = (result * base) % mod
base = (base * base) % mod
exp >>= 1
return result
# \u8ba1\u7b97 S_n = \u03a3 b_i
# \u4f7f\u7528\u63a8\u5bfc\u7684\u516c\u5f0f
# S_n = 4\/9 * [(10^(2n+2)-100)\/99 - 2*(10^(n+1)-10)\/9 + n]
# \u4f46\u7531\u4e8e\u6d89\u53ca\u5206\u6570\uff0c\u6211\u4eec\u7528\u6a21\u8fd0\u7b97\u7684\u9006\u5143\u6216\u8005\u76f4\u63a5\u5927\u6570\u8ba1\u7b97
# Python 3.8+ \u53ef\u4ee5\u7528 int \u76f4\u63a5\u5904\u7406\uff0c\u4f46\u9700\u8981\u4f18\u5316
from decimal import Decimal, getcontext
# \u8bbe\u7f6e\u8db3\u591f\u7684\u7cbe\u5ea6
getcontext().prec = 50
def compute_sum(n):
# \u4f7f\u7528\u516c\u5f0f\u76f4\u63a5\u8ba1\u7b97\uff0c\u907f\u514d\u5faa\u73af
# S_n = 4\/9 * [((10^(2*n+2) - 100) \/ 99) - 2 * ((10^(n+1) - 10) \/ 9) + n]
two_pow = Decimal(10) ** (2 * n + 2)
one_pow = Decimal(10) ** (n + 1)
term1 = (two_pow - Decimal(100)) \/ Decimal(99)
term2 = Decimal(2) * (one_pow - Decimal(10)) \/ Decimal(9)
term3 = Decimal(n)
sum_b = Decimal(4) \/ Decimal(9) * (term1 - term2 + term3)
return int(sum_b)
result = compute_sum(n)
print(result)<\/pre>\n\n\n\n\u4f46\u662f\u6211\u53d1\u73b0\u8fd8\u662f\u4f1a\u65f6\u95f4\/\u5185\u5b58\u8d85\u9650<\/p>\n\n\n\n
![\"\"](\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\")
<\/div><\/figure>\n\n\n\n\u6240\u4ee5\u6700\u540e\u6211\u9009\u62e9\u4f7f\u7528C++\u6765\u5b9e\u73b0<\/p>\n\n\n\n
<\/h1>\n\n\n\n
#include <cstdio>\n\nlong long ans[29000000], n;\n\nint main(){\n scanf(\"%lld\", &n);\n \n ans[1] = n * 6;\n \n for(int i = 2; i <= n + 1; i++){\n ans[i] = 3 + (n-i+1) * 5 + (i-2) \/ 2 * 4;\n }\n \n for(int i = 2 * n, j = 2; i >= n + 2; i--, j++){\n ans[i] = j \/ 2 * 4;\n }\n \n \/\/ \u8fdb\u4f4d\n for(int i = 1; i < 2 * n; i++){\n ans[i+1] += ans[i] \/ 10;\n ans[i] %= 10;\n }\n \n \/\/ \u8f93\u51fa\uff08\u6ce8\u610f\u8f93\u51fa\u8303\u56f4\uff09\n for(int i = 2 * n; i >= 1; i--){\n if(ans[i] != 0 || i != 2*n) \/\/ \u8df3\u8fc7\u524d\u5bfc\u96f6\n putchar(ans[i] + '0');\n }\n putchar('\\n');\n \n return 0;\n}<\/code><\/pre>\n\n\n\n\u7b97\u6cd5\u5206\u6790<\/h2>\n\n\n\n
\u8fd9\u4e2a\u89e3\u6cd5\u4f7f\u7528\u4e86\u4e00\u4e2a\u6570\u5b66\u89c4\u5f8b\uff0c\u76f4\u63a5\u8ba1\u7b97\u6bcf\u4e00\u4f4d\u7684\u503c\uff0c\u800c\u4e0d\u662f\u9010\u4e2a\u8ba1\u7b97 a\u1d62 \u548c b\u1d62\u3002<\/p>\n\n\n\n
\u8ba9\u6211\u9a8c\u8bc1\u4e00\u4e0b\u8fd9\u4e2a\u89c4\u5f8b\uff1a<\/p>\n\n\n\n
\u5bf9\u4e8e n = 3\uff08\u6837\u4f8b\uff09\uff1a<\/p>\n\n\n\n
- \n
- \u7ed3\u679c\u5e94\u8be5\u662f\uff1a447948<\/li>\n\n\n\n
- \u6570\u7ec4\u957f\u5ea6\uff1a2 \u00d7 n = 6<\/li>\n<\/ul>\n\n\n\n
\u8ba1\u7b97\u8fc7\u7a0b\uff1a<\/p>\n\n\n\n
<\/summary>\n
<\/p>\n<\/details>\n\n\n\n
ans[1] = n \u00d7 6 = 3 \u00d7 6 = 18
ans[2] = 3 + (3-2+1) \u00d7 5 + (2-2)\/2 \u00d7 4 = 3 + 5 + 0 = 8
ans[3] = 3 + (3-3+1) \u00d7 5 + (3-2)\/2 \u00d7 4 = 3 + 5 + 4 = 12
ans[4] = 3 + (3-4+1) \u00d7 5 + (4-2)\/2 \u00d7 4 = 3 + 0 + 4 = 7
ans[6] = 2\/2 \u00d7 4 = 4 (j=2)
ans[5] = 3\/2 \u00d7 4 = 4 (j=3, \u6574\u9664)<\/pre>\n\n\n\n\u8fdb\u4f4d\u5904\u7406\u524d\uff1a
[18, 8, 12, 7, 4, 4]<\/code><\/p>\n\n\n\n\u8fdb\u4f4d\u5904\u7406\u540e\uff1a<\/p>\n\n\n\n
ans[1] = 18 % 10 = 8, carry = 1
ans[2] = 8 + 1 = 9, carry = 0
ans[3] = 12 % 10 = 2, carry = 1
ans[4] = 7 + 1 = 8, carry = 0
ans[5] = 4, carry = 0
ans[6] = 4, carry = 0<\/pre>\n\n\n\n\u4f46\u662f\u5982\u679c\u91cd\u65b0\u68c0\u67e5\u4ee3\u7801\u903b\u8f91<\/p>\n\n\n\n
ans[1] = n * 6; \/\/ ans[1] = 18<\/p>\n\n\n\n
for(int i = 2; i <= n + 1; i++){
ans[i] = 3 + (n-i+1) * 5 + (i-2) \/ 2 * 4;
}<\/p>\n\n\n\n\u5bf9\u4e8e n=3:<\/p>\n\n\n\n
- \n
- ans[2] = 3 + (3-2+1)\u00d75 + 0 = 3 + 5 = 8<\/li>\n\n\n\n
- ans[3] = 3 + (3-3+1)\u00d75 + 4 = 3 + 5 + 4 = 12<\/li>\n\n\n\n
- ans[4] = 3 + (3-4+1)\u00d75 + 4 = 3 + 0 + 4 = 7<\/li>\n<\/ul>\n\n\n\n
for(int i = 2 * n, j = 2; i >= n + 2; i--, j++){
ans[i] = j \/ 2 * 4;
}<\/pre>\n\n\n\n\u5bf9\u4e8e n=3, \u4ece i=6 \u5230 i=4:<\/p>\n\n\n\n
- \n
- i=6, j=2: ans[6] = 2\/2\u00d74 = 4<\/li>\n\n\n\n
- i=5, j=3: ans[5] = 3\/2\u00d74 = 4 (\u6574\u9664)<\/li>\n\n\n\n
- i=4, j=4: ans[4] = 4\/2\u00d74 = 8 (\u8986\u76d6\u4e86\u4e4b\u524d\u76847\uff01)<\/li>\n<\/ul>\n\n\n\n
\u6240\u4ee5\u5b9e\u9645\u6570\u7ec4\u662f\uff1a
[18, 8, 12, 8, 4, 4]<\/code><\/p>\n\n\n\n\u8fdb\u4f4d\u540e\uff1a<\/p>\n\n\n\n
- \n
- i=1: ans[1]=8, ans[2]+=1 \u2192 9<\/li>\n\n\n\n
- i=2: ans[2]=9, ans[3]+=1 \u2192 13<\/li>\n\n\n\n
- i=3: ans[3]=3, ans[4]+=1 \u2192 9<\/li>\n\n\n\n
- i=4: ans[4]=9<\/li>\n\n\n\n
- i=5: ans[5]=4<\/li>\n\n\n\n
- i=6: ans[6]=4<\/li>\n<\/ul>\n\n\n\n
\u7ed3\u679c\uff1a
[8, 9, 3, 9, 4, 4]<\/code> \u2192 449389<\/p>\n\n\n\n\u6700\u540e\u7684\u4ee3\u7801<\/p>\n\n\n\n
#include <cstdio>\n\nlong long ans[29000000], n;\n\nint main(){\n scanf(\"%lld\", &n);\n \n ans[1] = n * 6;\n \n for(int i = 2; i <= n + 1; i++){\n ans[i] = 3 + (n-i+1) * 5 + (i-2) \/ 2 * 4;\n }\n \n for(int i = 2 * n, j = 2; i >= n + 2; i--, j++){\n ans[i] = j \/ 2 * 4;\n }\n \n \/\/ \u8fdb\u4f4d\n for(int i = 1; i < 2 * n; i++){\n ans[i+1] += ans[i] \/ 10;\n ans[i] %= 10;\n }\n \n \/\/ \u8f93\u51fa\uff08\u6ce8\u610f\u8f93\u51fa\u8303\u56f4\uff09\n for(int i = 2 * n; i >= 1; i--){\n if(ans[i] != 0 || i != 2*n) \/\/ \u8df3\u8fc7\u524d\u5bfc\u96f6\n putchar(ans[i] + '0');\n }\n putchar('\\n');\n \n return 0;\n}<\/code><\/pre>\n\n\n\n<\/div><\/figure>\n\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"
\u7a81\u7136\u60f3\u5230\u597d\u4e45\u90fd\u6ca1\u6709\u505a\u7b97\u6cd5\u7684\u8bb2\u89e3\u4e86\uff0c\u8fd9\u6b21\u66f4\u65b0\u6253\u7b97\u7528\u4e00\u9053\u901a\u8fc7\u7387\u8f83\u4f4e\u7684\u9898\u76ee\u6765\u8bb2\u89e3\u4e00\u4e0b\u7b97\u6cd5 \u9898\u76ee\u5206\u6790 \u6570\u5217A\u7684\u89c4\u5f8b\uff1a […]<\/p>\n","protected":false},"author":1,"featured_media":151,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[10,1,40],"tags":[39,8,12],"class_list":["post-173","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-10","category-leam","category-40","tag-39","tag-8","tag-12"],"_links":{"self":[{"href":"https:\/\/www.thebugmuxi.xyz\/index.php\/wp-json\/wp\/v2\/posts\/173","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.thebugmuxi.xyz\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.thebugmuxi.xyz\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.thebugmuxi.xyz\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.thebugmuxi.xyz\/index.php\/wp-json\/wp\/v2\/comments?post=173"}],"version-history":[{"count":1,"href":"https:\/\/www.thebugmuxi.xyz\/index.php\/wp-json\/wp\/v2\/posts\/173\/revisions"}],"predecessor-version":[{"id":179,"href":"https:\/\/www.thebugmuxi.xyz\/index.php\/wp-json\/wp\/v2\/posts\/173\/revisions\/179"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.thebugmuxi.xyz\/index.php\/wp-json\/wp\/v2\/media\/151"}],"wp:attachment":[{"href":"https:\/\/www.thebugmuxi.xyz\/index.php\/wp-json\/wp\/v2\/media?parent=173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.thebugmuxi.xyz\/index.php\/wp-json\/wp\/v2\/categories?post=173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.thebugmuxi.xyz\/index.php\/wp-json\/wp\/v2\/tags?post=173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
![\"\"](\"https:\/\/www.https:\/\/www.thebugmuxi.xyz\/wp-content\/uploads\/2025\/12\/8f36820a-0805-4a13-8dc6-3c2de3091c4d-3-1024x687.png\")
<\/div><\/figure>\n\n\n\n![\"\"](\"https:\/\/www.https:\/\/www.thebugmuxi.xyz\/wp-content\/uploads\/2025\/12\/image-1024x692.png\")
<\/div><\/figure>\n\n\n\n