author:guy keyword:nice - cp's OEIS frontend

A169678 The second of a pair of sequences A and B with property that all the differences |a_i - b_j| are distinct - for precise definition see Comments lines in A169677.

0, 3, 12, 26, 45, 72, 105, 149, 199, 255, 316, 392, 401, 502, 596, 733, 865, 891, 1086, 1119, 1311, 1330, 1646, 1773, 2011, 2324, 2371, 2554, 2692, 3055, 3258, 3820, 3960, 4063, 4606, 5126, 5515, 5535, 6228, 6233, 7134, 7515, 7861, 8619

Offset: 1

R. K. Guy and N. J. A. Sloane, Mar 27 2010

Computed by Alois P. Heinz and Wouter Meeussen, Mar 27 2010

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A169677, A169679, A169680, A169690-A169693.

A001042 a(n) = a(n-1)^2 - a(n-2)^2.

Original entry on oeis.org

1, 2, 3, 5, 16, 231, 53105, 2820087664, 7952894429824835871, 63248529811938901240357985099443351745, 4000376523371723941902615329287219027543200136435757892789536976747706216384

Offset: 0

N. J. A. Sloane, R. K. Guy

The next term has 152 digits. - Franklin T. Adams-Watters, Jun 11 2009

Archimedeans Problems Drive, Eureka, 27 (1964), 6.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..13
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968
R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A001146, A000278, A000283, A058182, A007018.

Cf. A064236 (numbers of digits).

a001042 n = a001042_list !! n
a001042_list = 1 : 2 : zipWith (-) (tail xs) xs
               where xs = map (^ 2) a001042_list
-- Reinhard Zumkeller, Dec 16 2013

RecurrenceTable[{a[0]==1,a[1]==2,a[n]==a[n-1]^2-a[n-2]^2},a,{n,0,12}] (* Harvey P. Dale, Jan 11 2013 *)

a(n) ~ c^(2^n), where c = 1.1853051643868354640833201434870139866230288004895868726506278977814490371... . - Vaclav Kotesovec, Dec 17 2014

More terms from James Sellers, Sep 19 2000.

A001056 a(n) = a(n-1)*a(n-2) + 1, a(0) = 1, a(1) = 3.

Original entry on oeis.org

1, 3, 4, 13, 53, 690, 36571, 25233991, 922832284862, 23286741570717144243, 21489756930695820973683319349467, 500426416062641238759467086706254193219790764168482, 10754042042885415070816603338436200915110904821126871858491675028294447933424899095

Offset: 0

N. J. A. Sloane, R. K. Guy

Archimedeans Problems Drive, Eureka, 19 (1957), 13.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..17
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A001622 (phi), A258112.

a:=[1,3];; for n in [3..13] do a[n]:=a[n-1]*a[n-2]+1; od; a; # G. C. Greubel, Sep 19 2019

a001056 n = a001056_list !! n
a001056_list = 1 : 3 : (map (+ 1 ) $
               zipWith (*) a001056_list $ tail a001056_list)
-- Reinhard Zumkeller, Aug 15 2012

I:=[1,3]; [n le 2 select I[n] else Self(n-1)*Self(n-2) + 1: n in [1..13]]; // G. C. Greubel, Sep 19 2019

a:= proc (n) option remember;
if n=0 then 1
elif n=1 then 3
else a(n-1)*a(n-2) + 1
end if
end proc;
seq(a(n), n = 0..13); # G. C. Greubel, Sep 19 2019

RecurrenceTable[{a[0]==1,a[1]==3,a[n]==a[n-1]*a[n-2]+1},a,{n,0,14}] (* Harvey P. Dale, Jul 17 2011 *)
t = {1, 3}; Do[AppendTo[t, t[[-1]] * t[[-2]] + 1], {n, 2, 14}] (* T. D. Noe, Jun 25 2012 *)

m=13; v=concat([1,3], vector(m-2)); for(n=3, m, v[n]=v[n-1]*v[n-2] +1 ); v \\ G. C. Greubel, Sep 19 2019

def a(n):
    if (n==0): return 1
    elif (n==1): return 3
    else: return a(n-1)*a(n-2) + 1
[a(n) for n in (0..13)] # G. C. Greubel, Sep 19 2019

a(n) ~ c^(phi^n), where c = A258112 = 1.7978784900091604813559508837..., phi = (1+sqrt(5))/2 = A001622. - Vaclav Kotesovec, Dec 17 2014

A036688 Number of distinct n-digit suffixes of base-10 squares not containing the digit 0.

Original entry on oeis.org

5, 18, 119, 698, 5449, 41735, 359207, 3085197, 27434602, 243921771, 2188569304, 19636586858

Offset: 1

R. K. Guy

			Any square ends with one of [ 0 ], 1, 4, 5, 6, 9, so a(1) = 5.
a(3) = A000993(3) - a(2) - #{100, 104, 201, 204, 209, 304, 400, 401, 404, 409, 500, 504, 600, 601, 604, 609, 704, 801, 804, 809, 900, 904} = 159 - 18 - 22 = 119, cf. A122986. - _Reinhard Zumkeller_, Mar 21 2010

Josiah H. Drummond, Problem 57, Amer. Math. Monthly, Vol. 5 (1898), p. 26; reprinted on p. 906 of Vol. 105 (1998).

Cf. A036788.

(* A partly empirical script *) a[n_] := (Clear[qr]; qr[] = False; For[k = 1, k <= 10^n/4, k++, m = PowerMod[k, 2, 10^n]; If[m > 10^(n-1) && FreeQ[IntegerDigits[m], 0], qr[m] = True]]; For[cnt = 0; k = 10^(n-1)+1, k <= 10^n-1, k++, If[qr[k], cnt++]]; cnt); a[1] = 5; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 10}] (* _Jean-François Alcover, Jul 31 2015 *)

from math import isqrt
def a(n):
    suffixes = set()
    for k in range(isqrt(10 ** (n - 1)) + 1, 10 ** n):
        kk = k * k
        s = str(kk)[-n:]
        if "0" not in s and len(s) >= n:
            suffixes.add(s)
    return len(suffixes)
print([a(n) for n in range(1, 8)])  # Michael S. Branicky, May 18 2021

Explanation and more terms from David W. Wilson

a(11)-a(12) from Bert Dobbelaere, Mar 10 2021

A049343 Numbers m such that 2m and m^2 have same digit sum.

Original entry on oeis.org

0, 2, 9, 11, 18, 20, 29, 38, 45, 47, 90, 99, 101, 110, 119, 144, 146, 180, 182, 189, 198, 200, 245, 290, 299, 335, 344, 351, 362, 369, 380, 398, 450, 452, 459, 461, 468, 470, 479, 488, 495, 497, 639, 729, 794, 839, 848, 900, 929, 954, 990, 999

Offset: 1

R. K. Guy

An easy way to prove that this sequence is infinite is to observe that it contains all numbers of the form 10^k+1. - Stefan Steinerberger, Mar 31 2006

For n>0: digital root (A010888) of 2n or n^2 is either 4 or 9. - Reinhard Zumkeller, Oct 01 2007

Problem 117 in Loren Larson's translation of Paul Vaderlind's book.

Reinhard Zumkeller and Harvey P. Dale, Table of n, a(n) for n = 1..1000 (first 101 terms from Zumkeller)

Cf. A058369, A077436 (binary). - Reinhard Zumkeller, Apr 03 2011

import Data.List (elemIndices)
import Data.Function (on)
a049343 n = a049343_list !! (n-1)
a049343_list = elemIndices 0
   $ zipWith ((-) `on` a007953) a005843_list a000290_list
-- Reinhard Zumkeller, Apr 03 2011

[n: n in [0..1000] | &+Intseq(2*n) eq &+Intseq(n^2)]; // Vincenzo Librandi, Nov 17 2015

Select[Range[0, 1000], Sum[DigitCount[2# ][[i]]*i, {i, 1, 9}] == Sum[DigitCount[ #^2][[i]]*i, {i, 1, 9}] &] (* Stefan Steinerberger, Mar 31 2006 *)
Select[Range[0,1000],Total[IntegerDigits[2#]]==Total[ IntegerDigits[ #^2]]&] (* Harvey P. Dale, Sep 25 2012 *)

A007953(A005843(a(n))) = A007953(A000290(a(n))). - Reinhard Zumkeller, Oct 01 2007

A005530 Number of Boolean functions of n variables from Post class F(8,inf); number of degenerate Boolean functions of n variables.

Original entry on oeis.org

2, 6, 38, 942, 325262, 25768825638, 129127208425774833206, 2722258935367507707190488025630791841374

Offset: 1

N. J. A. Sloane, R. K. Guy

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Tomescu, Introducere in Combinatorica. Editura Tehnica, Bucharest, 1972, p. 129.

Alois P. Heinz, Table of n, a(n) for n = 1..12
T. E. Allen, J. Goldsmith, N. Mattei, Counting, Ranking, and Randomly Generating CP-nets, 2014.
R. K. Guy, Letter to N. J. A. Sloane, Mar 1974
Y. Raekow and K. Ziegler, A taxonomy of non-cooperatively computable functions, Presented at WEWoRC 2011 (link to conference record).
I. Tomescu, Excerpts from "Introducese in Combinatorica" (1972), pp. 230-1, 44-5, 128-9. (Annotated scanned copy)
Index entries for sequences related to Boolean functions

a(n) = 2^(2^n) - A000371(n). Cf. A036239, A036240.

Sum[(-1)^(j + 1) Binomial[n, j] 2^2^(n - j), {j, 1, n}]

for(n=1,10, print1(sum(j=1,n, (-1)^(j+1)*binomial(n,j)*2^(2^(n-j))), ", ")) \\ G. C. Greubel, Oct 06 2017

a(n) = Sum_{j=1..n} (-1)^(j+1)*binomial(n,j)*2^(2^(n-j)).

More terms from Vladeta Jovovic, Goran Kilibarda

A038122 Start with {1,2,...,n}, replace any two numbers a,b with |a^2-b^2|, repeat until single number k remains; a(n) = minimal value of k.

Original entry on oeis.org

1, 3, 0, 16, 15, 63, 8, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0, 3, 1, 0, 0, 1, 3, 0, 4, 3, 3, 4, 0

Offset: 1

R. K. Guy

Due mostly to the efforts of Dean Hickerson, supported by David W. Wilson and Michael Kleber, it is now known that this has period 12 beginning at n=8.

			a(2) = 3 from (1,2); a(3) = 0 from ((1,2),3); a(4) = 16 from (((1,2),3),4); a(5) = 15 from ((((2,3),5),1),4)
a(6) = 63 from (((1,4),(3,5)),(2,6)) [ _Michael Kleber_ ]
a(7) = 8 from (((((4,5),6),(2,7)),1),3) [ Kleber ]
a(8) = 0 from ((((4,5),7)(2,6))((1,3),8)) [ Guy ]
a(9) = 3 from (2,(1,(((6,7),((3,4),8)),(5,9)))) [ Kleber ]
a(10)= 1 from ((((((((4,5),9),6),(8,10)),2),3),7),1) [ This and the following are due to _Dean Hickerson_ ]
a(11)= 0 from ((((((3,7),(9,11)),6),(8,10)),(1,2)),(4,5))
a(12)= 0 from ((((((1,3),7),(8,10)),(((5,6),9),(11,12))),2),4)
a(13)= 1 from (((((((((3,7),(9,11)),6),(8,10)),5),(12,13)),2),4),1) ...

Dean Hickerson and Michael Kleber, Reducing a Set by Subtracting Squares, J. Integer Sequences, Vol. 2, 1999, #4.
Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,-1,0,0,1).

LinearRecurrence[{0,0,1,0,0,-1,0,0,1},{1,3,0,16,15,63,8,0,3,1,0,0,1,3,0,4},120] (* Harvey P. Dale, Jul 29 2015 *)

a(n)=if(n<4||n>7, n*(n+1)/2%6, [16, 15, 63, 8][n-3]) \\ Charles R Greathouse IV, Feb 10 2017

def A038122(n): return (16,15,63,8)[n-4] if 3>1)%6 # Chai Wah Wu, Apr 17 2025

For n<4 and n>7, a(n) = n*(n+1)/2 mod 6 = A010875(A000217(n)). - Henry Bottomley, Feb 24 2003
a(n) = a(n-3)-a(n-6)+a(n-9) for n>16. - Colin Barker, Oct 01 2014
G.f.: x*(4*x^15 +60*x^14 +12*x^13 +8*x^12 -60*x^11 -12*x^10 -8*x^9 +60*x^8 +12*x^7 +7*x^6 -63*x^5 -12*x^4 -15*x^3 -3*x -1) / ((x -1)*(x^2 +1)*(x^2 +x +1)*(x^4 -x^2 +1)). - Colin Barker, Oct 01 2014

A110081 Integers n such that the digit set D = (0, 1, -n) used in base 3 expansions of the form Sum_{ -N < j < oo} d_j 3^{-j}, all d_j in D, can represent all real numbers.

Original entry on oeis.org

1, 7, 25, 31, 37, 73, 79, 85, 97, 103, 193, 241, 253, 271, 313, 319, 337, 343, 361, 517, 553, 661, 703, 721, 727, 733, 745, 751, 781, 799, 805, 865, 925, 943, 967, 1015, 1039, 1081, 1087, 1633, 1687, 1705, 1837, 1981, 2125, 2137, 2143, 2185, 2191, 2233, 2257, 2263, 2341, 2581, 2593, 2605, 2641, 2719, 2761, 2797, 2815, 2833, 2857, 2887, 2893, 2911, 3127

Offset: 1

N. J. A. Sloane, based on correspondence from R. K. Guy and Jeff Lagarias, Aug 31 2005

All nonnegative reals can be represented with ternary digits 0, 1, 2. If you're not allowed to use 2, then you only get something like the Cantor set. But you're back in business if you're allowed to use 0, 1, -1 - this gives the "balanced" ternary representation (so 1 is in the sequence).

The sequence is known to be infinite and irregular and is conjectured to have density zero.

			13/18 = 0.122111111111... in ternary which can't be represented without the 2's. But it is 10.x0111111111... if x = -7: 3 + 0 + (-7)/3 + 1/3^3 + 1/3^4 + 1/3^5 + ... = 3 - 7/3 + (1/27)/(1-(1/3)) = 13/18.

J. C. Lagarias, Crystals, Tilings and Packings, Hedrick Lectures, Math. Assoc. America MathFest, 2005.

Joerg Arndt, Table of n, a(n) for n = 1..1000
David W. Matula, Basic digit sets for radix representation, J. Assoc. Comput. Mach. 29 (1982), 1131-1143.
Don Reble, Python Program

More terms using Don Reble's program from Joerg Arndt, Sep 17 2017

A039958 Class number of maximal order in real quadratic field of radicand n.

Original entry on oeis.org

1, -1, 1, 2, -1, 1, 1, 2, 3, -2, 1, 2, -1, 1, 2, 4, -1, 3, 1, 2, 1, 1, 1, 2, 5, -2, 3, 2, -1, 2, 1, 4, 1, 2, 2, 6, -1, 1, 2, 2, -1, 2, 1, 2, 3, 1, 1, 4, 7, 5, 2, 2, -1, 3, 2, 2, 1, -2, 1, 2, -1, 1, 3, 8, -2, 2, 1, 2, 1, 2, 1, 6, -1, -2, 5, 2, 1, 2, 3, 4, 9, -4, 1, 2, -2, 1, 2, 2, -1, 3, 2, 2, 1, 1, 2, 4, -1, 7, 3, 10

Offset: 1

R. K. Guy, N. J. A. Sloane

Mollin gives a table for n < 10000.

R. A. Mollin, Quadratics, CRC Press, 1996, Table C1.

cp's OEIS Frontend

A169678 The second of a pair of sequences A and B with property that all the differences |a_i - b_j| are distinct - for precise definition see Comments lines in A169677.

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A001042 a(n) = a(n-1)^2 - a(n-2)^2.

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A001056 a(n) = a(n-1)*a(n-2) + 1, a(0) = 1, a(1) = 3.

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A036688 Number of distinct n-digit suffixes of base-10 squares not containing the digit 0.

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A049343 Numbers m such that 2m and m^2 have same digit sum.

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Haskell

Magma

Mathematica

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A005530 Number of Boolean functions of n variables from Post class F(8,inf); number of degenerate Boolean functions of n variables.

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Mathematica

PARI

Formula

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A038122 Start with {1,2,...,n}, replace any two numbers a,b with |a^2-b^2|, repeat until single number k remains; a(n) = minimal value of k.

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