Bodo Zinser - cp's OEIS frontend

A290187 Minimum number of moves to rearrange a Tower of Hanoi puzzle with 3 colors. There are 3 rods with n discs in descending sizes. On the first rod there are n red discs, on the 2nd n blue discs and on the 3rd n white ones.

5, 23, 68, 168, 377, 801, 1657, 3377, 6825, 13729, 27545, 55185

Offset: 1

Bodo Zinser, Jul 23 2017

Like the known Tower of Hanoi you have 3 rods with n discs in descending sizes. But here on the first rod there are n red discs, on the 2nd n blue discs and on the 3rd n white ones.
The task is to rearrange the discs so that red ones go to rod 2, the blue ones to rod 3 and the white ones to rod 1. Sequence lists the number of minimum moves needed to rearrange the discs.
I have written a program to find the minimum moves, it might be improved for higher values.

			For n=1 the moves are 23-12-32-31-23 (meaning to move a disc from rod 2 to 3, then 1 to 2, ...).
For n=2 the moves are 31-21-23-13-13-13-12-31-31-31-32-12-12-12-31-21-21-21-23-13-13-12-31.

Cut The Knot, 3-Colors Tower of Hanoi
Bodo Zinser, Tower of Hanoi with three colors (explanation of the algorithm)
Index entries for sequences related to Towers of Hanoi

Cf. A000225, A055622 (another 3-color variant of the Towers of Hanoi).

A182478 Numbers that can be truncated in base 10 such that the sum of the factorials of the truncations equals that number.

Original entry on oeis.org

1, 2, 145, 40585, 6402374184741226, 121645100891988866, 121666023198802103, 121666023198802144, 2432902008177819519, 2432902008217006118, 2432902008656812499, 4872206390059820318

Offset: 1

Bodo Zinser, May 01 2012

			a(5)=6402374184741226=6!+4!+(02)!+3!+7!+4!+18!+4!+7!+4!+12!+2!+6!
a(6): 2-digit-truncations are 12,10,19
a(7): 2-digit-truncs are 16,19
a(8): 2-digit-truncs are 16,19
a(9): 2-digit-trunc is 20
a(10): 2-digit-truncs are 20,11
a(11): 2-digit-truncs are 20,12
a(12): 2-digit-truncs are 20,20,18

A014080 is a subsequence.

A104113 Numbers which when chopped into one, two or more parts, added and squared result in the same number.

Original entry on oeis.org

0, 1, 81, 100, 1296, 2025, 3025, 6724, 8281, 9801, 10000, 55225, 88209, 136161, 136900, 143641, 171396, 431649, 455625, 494209, 571536, 627264, 826281, 842724, 893025, 929296, 980100, 982081, 998001, 1000000, 1679616, 2896804, 3175524, 4941729, 7441984

Offset: 1

Bodo Zinser, Mar 05 2005

Every term is congruent to 0 or 1 modulo 9. - Andrea Tarantini, Sep 27 2021

			1296 is a term since (1+29+6)^2 = 36^2 = 1296.

John Drake, Table of n, a(n) for n = 1..408 (terms 1..80 from Mehrad Mahmoudian, terms 81..225 from Giovanni Resta)
Mehrad Mahmoudian, R code to produce the sequence
Mehrad Mahmoudian, Decompositions for a(1)-a(80)
Project Euler, Problem 719: Number Splitting (2020)

Cf. A038206, A102766.

Join[{0},Select[Select[Range@3000^2,Mod[#,9]<2&],(n=#;MemberQ[(Total/@(FromDigits/@#&/@Union[DeleteCases[SplitBy[#,#==-1&],{-1}]&/@(Insert[IntegerDigits@n,-1,#]&/@(List/@#&/@Rest@Subsets[Range@IntegerLength@n]))]))^2,#])&]] (* Giorgos Kalogeropoulos, Oct 28 2021 *)

def expr(t, d): # can you express target t with digits d, only adding +'s
    if t < 0: return False
    if t == int(d): return True
    return any(expr(t-int(d[:i]), d[i:]) for i in range(1, len(d)))
def aupto(limit):
    alst, k, k2 = [], 0, 0
    while k2 <= limit:
        if expr(k, str(k2)):
            alst.append(k2)
        k, k2 = k+1, k2 + 2*k + 1
    return alst
print(aupto(7500000)) # Michael S. Branicky, Sep 27 2021

a(n) = A038206(n)^2. - Andrea Tarantini, Sep 27 2021

a(30) and beyond from Mehrad Mahmoudian, Dec 16 2019

A102762 Curvatures of (largest) kissing circles along the circumference, starting with curvature = -1 and 2.

Original entry on oeis.org

-1, 2, 2, 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403, 2502, 2603, 2706, 2811, 2918, 3027, 3138, 3251, 3366

Offset: 0

Bodo Zinser, Feb 10 2005

A059100 has a totally different description but is the same sequence (omitting the first two numbers here)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Daniel Bach ("Dan"), Kissing circles.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A059100, A060790.

a(n) = if(n>1, n^2 - 4*n + 6, [-1,2][n+1]) \\ Andrew Howroyd, Feb 25 2018

a(n) = a(1) + a(2) + a(n-1) + 2*sqrt(a(1)*a(2) + a(1)*a(n-1) + a(2)*a(n-1)) = 1 + a(n-1) + 2*(sqrt(-2 + a(n-1))). (Descartes' curvature-theorem)
From Colin Barker, Jan 07 2013: (Start)
a(n) = n^2 - 4*n + 6 for n > 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.
G.f.: -(x^4 + 4*x^3 - 7*x^2 + 5*x - 1)/(x - 1)^3.
(End)

A102766 Numbers n that can be chopped into two parts, which when added and squared result in n.

Original entry on oeis.org

1, 81, 100, 2025, 3025, 9801, 10000, 88209, 494209, 998001, 1000000, 4941729, 7441984, 23804641, 24502500, 25502500, 28005264, 52881984, 60481729, 99980001, 100000000, 300814336, 493817284, 1518037444, 6049417284, 6832014336, 9048004641, 9999800001, 10000000000

Offset: 1

Bodo Zinser, Feb 10 2005

			a(7) = 88209 is a term as (88+209)^2 = 297^2 = 88209.

Michael S. Branicky, Table of n, a(n) for n = 1..131
Henry Ernest Dudeney, Amusements in Mathematics, 1917. See problem 113, "The torn number".

Supersequence of A238237.

from math import isqrt
from itertools import count, islice
def ok(n):
    if n == 1: return True
    r = isqrt(n)
    if r**2 != n: return False
    s = str(n)
    return any(int(s[:i])+int(s[i:])== r for i in range(1, len(s)))
def agen(): yield from (k**2 for k in count(1) if ok(k**2))
print(list(islice(agen(), 29))) # Michael S. Branicky, Dec 29 2024

a(n) = A248353(n)^2.

a(1)=1 prepended by Max Alekseyev, Aug 04 2017

a(27) and beyond from Michael S. Branicky, Dec 29 2024

A096915 Smallest prime which when appended to n produces a prime.

Original entry on oeis.org

3, 3, 7, 3, 3, 7, 3, 3, 7, 3, 3, 7, 7, 23, 7, 3, 3, 11, 3, 11, 11, 3, 3, 11, 7, 3, 7, 3, 3, 7, 3, 17, 7, 7, 3, 7, 3, 3, 7, 13, 11, 11, 3, 3, 7, 3, 23, 7, 19, 3, 13, 3, 23, 7, 7, 3, 7, 7, 3, 7, 3, 11, 11, 3, 3, 19, 3, 3, 11, 13, 29, 7, 3, 3, 7, 43, 3, 7, 7, 11, 11, 3, 11, 19, 3, 3, 7, 3, 23, 7, 37, 41

Offset: 1

Bodo Zinser, Aug 18 2004

			a(20)=11 because 11 is prime and 2011 is the smallest prime starting with 20 (2003 is not allowed).

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A088606, A089777 (the resulting primes). Records: A137177, A144593.

f[n_] := Block[{p = 2, a = IntegerDigits[n]}, While[ !PrimeQ[ FromDigits[ Join[a, IntegerDigits[ Prime[p]]] ]], p++ ]; Prime[p]]; Table[ f[n], {n, 92}] (* Robert G. Wilson v, Aug 20 2004 *)
sp[n_]:=Module[{p=3},While[CompositeQ[n*10^IntegerLength[p]+p],p= NextPrime[ p]];p]; Array[sp,100] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 26 2019 *)

A096915(n) = { local(p=1); until(isprime(eval(Str(n,p=nextprime(p+2)))),);p} \\ M. F. Hasler, Jan 05 2009

More terms from Robert G. Wilson v, Aug 20 2004

Cross-reference to indices of records corrected by M. F. Hasler, Jan 14 2009

A093090 Start with a(1)=1, a(2)=3, apply rule of A093089.

Original entry on oeis.org

1, 3, 4, 7, 11, 18, 29, 47, 76, 1, 23, 77, 24, 1, 0, 1, 1, 25, 1, 1, 2, 26, 26, 2, 3, 28, 52, 28, 5, 31, 80, 80, 33, 36, 1, 11, 1, 60, 1, 13, 69, 37, 12, 12, 61, 61, 14, 82, 1, 6, 49, 24, 73, 1, 22, 75, 96, 83, 7, 55, 73, 97, 74, 23, 97, 1, 71, 1, 79, 90, 62, 1, 28, 1, 70, 1, 71, 97, 1, 20

Offset: 1

Bodo Zinser, Mar 20 2004

Cf. A093086, A093087, A093088, A093089.

More terms from Bodo Zinser, Mar 21 2004

A093089 "Fibonacci in pairs": start with a(1)=1, a(2)=1; repeatedly adjoin sum of previous two terms but chopped from the right into pairs of 2 digits.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 1, 44, 90, 45, 1, 34, 1, 35, 46, 35, 35, 36, 81, 81, 70, 71, 1, 17, 1, 62, 1, 51, 1, 41, 72, 18, 18, 63, 63, 52, 52, 42, 1, 13, 90, 36, 81, 1, 26, 1, 15, 1, 4, 94, 43, 14, 1, 3, 1, 26, 1, 17, 82, 27, 27, 16, 16, 5, 98, 1, 37, 57, 15, 4, 4, 27, 27, 18

Offset: 1

Bodo Zinser, Mar 20 2004

Do all pairs of digits appear infinitely often? The sequence is not periodic.

			... a(11)=a(9)+a(10), a(12)=left pair of (a(10)+a(11)=55+89=1 44), a(13)=right pair of (a(10)+a(11)=55+89=1 44), a(14)=a(11)+a(12) ...

Cf. A093086, A093087, A093088.

A093091 "Fibonacci in pairs from left": start with a(1)=1, a(2)=1; repeatedly adjoin sum of previous two terms but chopped from the left into pairs of 2 digits.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 14, 4, 10, 3, 18, 14, 13, 21, 32, 27, 34, 53, 59, 61, 87, 11, 2, 12, 0, 14, 8, 98, 13, 14, 12, 14, 22, 10, 6, 11, 1, 27, 26, 26, 36, 32, 16, 17, 12, 28, 53, 52, 62, 68, 48, 33, 29, 40, 81, 10, 5, 11, 4, 13, 0, 11, 6, 81, 62, 69, 12, 1, 91, 15, 16

Offset: 1

Bodo Zinser, Mar 20 2004

Do all pairs of digits appear infinitely often? The sequence is not periodic.

			... a(11)=a(9)+a(10), a(12)=left pair of (a(10)+a(11)=55+89=14 4),
a(13)=right pair of (a(10)+a(11)=55+89=14 4),
a(14)=left pair of (a(11)+a(12)=89+14=10 3),
a(15)=right pair of (a(11)+a(12)=89+14=10 3), a(16)=a(12)+a(13) ...

Cf. A093086, A093087, A093088, A093089, A093090.

A093093 "Fibonacci in digits - up and down": start with a(1)=1, a(2)=1; repeatedly adjoin either the sum of the two previous terms (if that sum happens to be odd) or else adjoin digits of the sum of previous two terms (if that sum happens to be even).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 3, 4, 2, 4, 7, 6, 6, 11, 13, 1, 2, 17, 2, 4, 1, 4, 3, 19, 19, 6, 5, 5, 7, 2, 2, 3, 8, 25, 11, 1, 0, 1, 2, 9, 4, 5, 11, 33, 3, 6, 1, 2, 1, 1, 3, 11, 13, 9, 1, 6, 4, 4, 3, 6, 9, 7, 3, 3, 2, 4, 1, 4, 2, 4, 2, 2, 1, 0, 7, 1, 0, 8, 7, 9, 15, 1, 6, 1, 0, 6, 5, 6, 5, 5, 6, 6, 6, 4, 3, 1, 7

Offset: 1

Bodo Zinser, Mar 20 2004

			... a(8)=a(6)+a(7), a(9)=left digit of (a(7)+a(8)=13+21=3 4) as 34 is even, a(10)=right digit of (a(7)+a(8)=13+21=3 4) as 34 is even, a(13)=a(9)+a(10) as odd, ...

Cf. A093086 to A093092.

cp's OEIS Frontend

User: Bodo Zinser

A290187 Minimum number of moves to rearrange a Tower of Hanoi puzzle with 3 colors. There are 3 rods with n discs in descending sizes. On the first rod there are n red discs, on the 2nd n blue discs and on the 3rd n white ones.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A182478 Numbers that can be truncated in base 10 such that the sum of the factorials of the truncations equals that number.

Views

Author

Keywords

Examples

Crossrefs

A104113 Numbers which when chopped into one, two or more parts, added and squared result in the same number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A102762 Curvatures of (largest) kissing circles along the circumference, starting with curvature = -1 and 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A102766 Numbers n that can be chopped into two parts, which when added and squared result in n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Formula

Extensions

A096915 Smallest prime which when appended to n produces a prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A093090 Start with a(1)=1, a(2)=3, apply rule of A093089.

Views

Author

Keywords

Crossrefs

Extensions

A093089 "Fibonacci in pairs": start with a(1)=1, a(2)=1; repeatedly adjoin sum of previous two terms but chopped from the right into pairs of 2 digits.

Views

Author

Keywords

Comments

Examples

Crossrefs

A093091 "Fibonacci in pairs from left": start with a(1)=1, a(2)=1; repeatedly adjoin sum of previous two terms but chopped from the left into pairs of 2 digits.

Views

Author

Keywords

Comments

Examples