Introduction

It's been more than 20 minutes since the negligent waiter has taken your order for the house special prime tofu steak with a side of chili fries.

Out of boredom, you start fiddling around with the condiments tray. To be efficient, you want to be familiar with the choice of sauces and spices before your order is finally served.

You also examine the toothpick holder and try to analyze its inner workings when - yikes - the holder's lid falls off and all 23 picks lay scattered on the table.

Being a good and hygiene oriented citizen, you decide not to just put them back in the holder. Instead of letting all the good wood go to waste, you start playing around with the picks.

In the first "round", you lay down one toothpick vertically. You've used a total of one toothpick.

In the second "round", at each end of the first toothpick, you add a perpendicular toothpick at its center point. You added two additional toothpicks for a total of three toothpicks.

In the next rounds, you continue to add perpendicular toothpicks to each free end of toothpicks already on the table.

With your 23 toothpicks, you can complete a total of six rounds:


You wonder if you'd be able to implement this sequence in your favorite programming language. Because your food still hasn't arrived, you decide to take out your laptop and start implementing...

Challenge
Implement a script that returns the amount of toothpicks needed to complete n amount of rounds of the toothpick sequence.

```
0 <= n <= 5000
```

Hint
You can attempt this brute force or get some inspiration from the math department.

Source: codewars | Difficulty: EASY
Tags: Mathematics

Complete search

Starter code: def toothpick(n):
	

Solutions:
from math import log2

def t(n):
	if n == 0:
		return 0
	k = int(log2(n))
	i = n - 2 ** k
	if i == 0:
		return (2 ** (2 * k + 1) + 1) // 3
	else:
		return t(2 ** k) + 2 * t(i) + t(i + 1) - 1
toothpick = t


from functools import lru_cache
msb = lru_cache(maxsize=None)(lambda n: 1 << n.bit_length() - 1)

@lru_cache(maxsize=None)
def toothpick(n):
	if n == 0:
		return 0
	x = msb(n)
	res = ((x ** 2 << 1) + 1) // 3
	if n != x:
		res += (toothpick(n - x) << 1) + toothpick(n - x + 1) - 1
	return res


def toothpick(N):
	if N <= 0:
		return 0
	n = 1 << N.bit_length() - 1
	i = N - n
	return toothpick(n) + 2 * toothpick(i) + toothpick(i + 1) - 1 if i else (2 * n ** 2 + 1) / 3


import requests
r = requests.get('https://oeis.org/A139250/b139250.txt').text.splitlines()

def toothpick(n):
	return int(r[n].split()[1])


v = [0, 1, 3, 7, 11, 15, 23, 35, 43, 47, 55, 67, 79, 95, 123, 155, 171, 175, 183, 195, 207, 223, 251, 283, 303, 319, 347, 383, 423, 483, 571, 651, 683, 687, 695, 707, 719, 735, 763, 795, 815, 831, 859, 895, 935, 995, 1083, 1163, 1199, 1215, 1243, 1279, 1319, 1379, 1467, 1551, 1607, 1667, 1759, 1871, 2011, 2219, 2475, 2667, 2731, 2735, 2743, 2755, 2767, 2783, 2811, 2843, 2863, 2879, 2907, 2943, 2983, 3043, 3131, 3211, 3247, 3263, 3291, 3327, 3367, 3427, 3515, 3599, 3655, 3715, 3807, 3919, 4059, 4267, 4523, 4715, 4783, 4799, 4827, 4863, 4903, 4963, 5051, 5135, 5191, 5251, 5343, 5455, 5595, 5803, 6059, 6255, 6343, 6403, 6495, 6607, 6747, 6955, 7215, 7439, 7611, 7823, 8119, 8483, 8971, 9643, 10347, 10795, 10923, 10927, 10935, 10947, 10959, 10975, 11003, 11035, 11055, 11071, 11099, 11135, 11175, 11235, 11323, 11403, 11439, 11455, 11483, 11519, 11559, 11619, 11707, 11791, 11847, 11907, 11999, 12111, 12251, 12459, 12715, 12907, 12975, 12991, 13019, 13055, 13095, 13155, 13243, 13327, 13383, 13443, 13535, 13647, 13787, 13995, 14251, 14447, 14535, 14595, 14687, 14799, 14939, 15147, 15407, 15631, 15803, 16015, 16311, 16675, 17163, 17835, 18539, 18987, 19119, 19135, 19163, 19199, 19239, 19299, 19387, 19471, 19527, 19587, 19679, 19791, 19931, 20139, 20395, 20591, 20679, 20739, 20831, 20943, 21083, 21291, 21551, 21775, 21947, 22159, 22455, 22819, 23307, 23979, 24683, 25135, 25287, 25347, 25439, 25551, 25691, 25899, 26159, 26383, 26555, 26767, 27063, 27427, 27915, 28587, 29295, 29775, 30011, 30223, 30519, 30883, 31371, 32047, 32791, 33411, 33967, 34687, 35643, 36859, 38507, 40555, 42411, 43435, 43691, 43695, 43703, 43715, 43727, 43743, 43771, 43803, 43823, 43839, 43867, 43903, 43943, 44003, 44091, 44171, 44207, 44223, 44251, 44287, 44327, 44387, 44475, 44559, 44615, 44675, 44767, 44879, 45019, 45227, 45483, 45675, 45743, 45759, 45787, 45823, 45863, 45923, 46011, 46095, 46151, 46211, 46303, 46415, 46555, 46763, 47019, 47215, 47303, 47363, 47455, 47567, 47707, 47915, 48175, 48399, 48571, 48783, 49079, 49443, 49931, 50603, 51307, 51755, 51887, 51903, 51931, 51967, 52007, 52067, 52155, 52239, 52295, 52355, 52447, 52559, 52699, 52907, 53163, 53359, 53447, 53507, 53599, 53711, 53851, 54059, 54319, 54543, 54715, 54927, 55223, 55587, 56075, 56747, 57451, 57903, 58055, 58115, 58207, 58319, 58459, 58667, 58927, 59151, 59323, 59535, 59831, 60195, 60683, 61355, 62063, 62543, 62779, 62991, 63287, 63651, 64139, 64815, 65559, 66179, 66735, 67455, 68411, 69627, 71275, 73323, 75179, 76203, 76463, 76479, 76507, 76543, 76583, 76643, 76731, 76815, 76871, 76931, 77023, 77135, 77275, 77483, 77739, 77935, 78023, 78083, 78175, 78287, 78427, 78635, 78895, 79119, 79291, 79503, 79799, 80163, 80651, 81323, 82027, 82479, 82631, 82691, 82783, 82895, 83035, 83243, 83503, 83727, 83899, 84111, 84407, 84771, 85259, 85931, 86639, 87119, 87355, 87567, 87863, 88227, 88715, 89391, 90135, 90755, 91311, 92031, 92987, 94203, 95851, 97899, 99755, 100783, 101063, 101123, 101215, 101327, 101467, 101675, 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183039, 183079, 183139, 183227, 183311, 183367, 183427, 183519, 183631, 183771, 183979, 184235, 184431, 184519, 184579, 184671, 184783, 184923, 185131, 185391, 185615, 185787, 185999, 186295, 186659, 187147, 187819, 188523, 188975, 189127, 189187, 189279, 189391, 189531, 189739, 189999, 190223, 190395, 190607, 190903, 191267, 191755, 192427, 193135, 193615, 193851, 194063, 194359, 194723, 195211, 195887, 196631, 197251, 197807, 198527, 199483, 200699, 202347, 204395, 206251, 207275, 207535, 207551, 207579, 207615, 207655, 207715, 207803, 207887, 207943, 208003, 208095, 208207, 208347, 208555, 208811, 209007, 209095, 209155, 209247, 209359, 209499, 209707, 209967, 210191, 210363, 210575, 210871, 211235, 211723, 212395, 213099, 213551, 213703, 213763, 213855, 213967, 214107, 214315, 214575, 214799, 214971, 215183, 215479, 215843, 216331, 217003, 217711, 218191, 218427, 218639, 218935, 219299, 219787, 220463, 221207, 221827, 222383, 223103, 224059, 225275, 226923, 228971, 230827, 231855, 232135, 232195, 232287, 232399, 232539, 232747, 233007, 233231, 233403, 233615, 233911, 234275, 234763, 235435, 236143, 236623, 236859, 237071, 237367, 237731, 238219, 238895, 239639, 240259, 240815, 241535, 242491, 243707, 245355, 247403, 249263, 250319, 250683, 250895, 251191, 251555, 252043, 252719, 253463, 254083, 254639, 255359, 256315, 257531, 259179, 261231, 263127, 264323, 265007, 265727, 266683, 267899, 269551, 271647, 273755, 275551, 277383, 279779, 282907, 286987, 292331, 298283, 303019, 305323, 305839, 305855, 305883, 305919, 305959, 306019, 306107, 306191, 306247, 306307, 306399, 306511, 306651, 306859, 307115, 307311, 307399, 307459, 307551, 307663, 307803, 308011, 308271, 308495, 308667, 308879, 309175, 309539, 310027, 310699, 311403, 311855, 312007, 312067, 312159, 312271, 312411, 312619, 312879, 313103, 313275, 313487, 313783, 314147, 314635, 315307, 316015, 316495, 316731, 316943, 317239, 317603, 318091, 318767, 319511, 320131, 320687, 321407, 322363, 323579, 325227, 327275, 329131, 330159, 330439, 330499, 330591, 330703, 330843, 331051, 331311, 331535, 331707, 331919, 332215, 332579, 333067, 333739, 334447, 334927, 335163, 335375, 335671, 336035, 336523, 337199, 337943, 338563, 339119, 339839, 340795, 342011, 343659, 345707, 347567, 348623, 348987, 349199, 349495, 349859, 350347, 351023, 351767, 352387, 352943, 353663, 354619, 355835, 357483, 359535, 361431, 362627, 363311, 364031, 364987, 366203, 367855, 369951, 372059, 373855, 375687, 378083, 381211, 385291, 390635, 396587, 401323, 403631, 404167, 404227, 404319, 404431, 404571, 404779, 405039, 405263, 405435, 405647, 405943, 406307, 406795, 407467, 408175, 408655, 408891, 409103, 409399, 409763, 410251, 410927, 411671, 412291, 412847, 413567, 414523, 415739, 417387, 419435, 421295, 422351, 422715, 422927, 423223, 423587, 424075, 424751, 425495, 426115, 426671, 427391, 428347, 429563, 431211, 433263, 435159, 436355, 437039, 437759, 438715, 439931, 441583, 443679, 445787, 447583, 449415, 451811, 454939, 459019, 464363, 470315, 475055, 477391, 478011, 478223, 478519, 478883, 479371, 480047, 480791, 481411, 481967, 482687, 483643, 484859, 486507, 488559, 490455, 491651, 492335, 493055, 494011, 495227, 496879, 498975, 501083, 502879, 504711, 507107, 510235, 514315, 519659, 525615, 530391, 532867, 533807, 534527, 535483, 536699, 538351, 540447, 542555, 544351, 546183, 548579, 551707, 555787, 561135, 567135, 572123, 575199, 577287, 579683, 582811, 586895, 592295, 598595, 604607, 610031, 616091, 624011, 634347, 647851, 664491, 681131, 692907, 698027, 699051, 699055, 699063, 699075, 699087, 699103, 699131, 699163, 699183, 699199, 699227, 699263, 699303, 699363, 699451, 699531, 699567, 699583, 699611, 699647, 699687, 699747, 699835, 699919, 699975, 700035, 700127, 700239, 700379, 700587, 700843, 701035, 701103, 701119, 701147, 701183, 701223, 701283, 701371, 701455, 701511, 701571, 701663, 701775, 701915, 702123, 702379, 702575, 702663, 702723, 702815, 702927, 703067, 703275, 703535, 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def toothpick(n):
	return v[n]


from math import floor, log

def toothpick(n):
	if n < 2:
		return n
	msb_n = 2 ** floor(log(n, 2))
	j = n - msb_n
	return (2 * msb_n ** 2 + 1) / 3 + 2 * toothpick(j) + toothpick(j + 1) - 1


import math

def isPowerOfTwo(n):
	return math.ceil(math.log2(n)) == math.floor(math.log2(n))

def toothpick(n):
	if n == 0:
		return 0
	elif n == 1:
		return 1
	elif isPowerOfTwo(n):
		return int((2 ** (2 * math.log2(n) + 1) + 1) / 3)
	k = int(math.log2(n))
	i = n - 2 ** k
	return toothpick(2 ** k) + 2 * toothpick(i) + toothpick(i + 1) - 1


from math import log
from functools import lru_cache

@lru_cache(5000)
def toothpick(n):
	if n == 0:
		return 0
	else:
		k = int(log(n, 2))
		i = n - 2 ** k
		if i == 0:
			return (2 ** (2 * k + 1) + 1) / 3
		else:
			return toothpick(2 ** k) + 2 * toothpick(i) + toothpick(i + 1) - 1


def msb(n):
	t = 0
	while n >> t > 0:
		t += 1
	return 2 ** (t - 1)

def toothpick(n):
	k = (2 * msb(n) ** 2 + 1) // 3
	if n == 0:
		return 0
	elif n == msb(n):
		return k
	else:
		return k + 2 * toothpick(n - msb(n)) + toothpick(n - msb(n) + 1) - 1