A079978 -id:A079978 - cp's OEIS frontend

A121262 The characteristic function of the multiples of four.

1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Aug 23 2006, Aug 30 2007

Period 4: repeat [1, 0, 0, 0].
a(n) is also the number of partitions of n where each part is four (Since the empty partition has no parts, a(0) = 1). Hence a(n) is also the number of 2-regular graphs on n vertices such that each component has girth exactly four. - Jason Kimberley, Oct 01 2011
This sequence is the Euler transformation of A185014. - Jason Kimberley, Oct 01 2011
Number of permutations satisfying -k <= p(i) - i <= r and p(i)-i not in I, i = 1..n, with k = 1, r = 3, I = {0, 1, 2}. - Vladimir Baltic, Mar 07 2012

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 82.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics 4 (2010), 119-135
Steve Chow, 0,0,0,1,0,0,0,1 (deriving an explicit formula for the sequence) :YouTube Video, 2017.
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

A011765 is another version of the same sequence.
Characteristic function of multiples of g: A000007 (g=0), A000012 (g=1), A059841 (g=2), A079978 (g=3), this sequence (g=4), A079998 (g=5), A079979 (g=6), A082784 (g=7). - Jason Kimberley, Oct 14 2011
Cf. A010873, A166486, A185014.

a121262 = (0 ^) . flip mod 4  -- Reinhard Zumkeller, Mar 04 2015
a121262_list = cycle [1,0,0,0]  -- Reinhard Zumkeller, Jan 06 2012

&cat [[1, 0, 0, 0]^^30]; // Wesley Ivan Hurt, Jul 07 2016

seq(op([1, 0, 0, 0]), n=0..50); # Wesley Ivan Hurt, Jul 07 2016

Table[Boole[IntegerQ[n/4]], {n,0,127}] (* Alonso del Arte, Jul 14 2013 *)

a(n)=!(n%4) \\ Charles R Greathouse IV, Oct 25 2012

a(n) = (1/4)*(2*cos(n*Pi/2) + 1 + (-1)^n).
Additive with a(p^e) = 1 if p = 2 and e > 1, 0 otherwise.
Sequence shifted right by 2 is additive with a(p^e) = 1 if p = 2 and e = 1, 0 otherwise.
a(n) = 1 - (C(n + 1, n + (-1)^(n+1)) mod 2).
a(n) = 0^(n mod 4). - Reinhard Zumkeller, Sep 30 2008
a(n) = !(n%4). - Jaume Oliver Lafont, Mar 01 2009
a(n) = (1/4)*(1 + I^n + (-1)^n + (-I)^n). - Paolo P. Lava, May 04 2010
a(n) = ((n-1)^k mod 4 - (n-1)^(k-1) mod 4)/2, k > 2. - Gary Detlefs, Feb 21 2011
a(n) = floor(1/2*cos(n*Pi/2) + 1/2). - Gary Detlefs, May 16 2011
G.f.: 1/(1 - x^4); a(n) = (1 + (-1)^n)*(1 + i^((n-1)*n))/4, where i = sqrt(-1). - Bruno Berselli, Sep 28 2011
a(n) = floor(((n+3) mod 4)/3). - Gary Detlefs, Dec 29 2011
a(n) = floor(n/4) - floor((n-1)/4). - Tani Akinari, Oct 25 2012
a(n) = ceiling( (1/2)*cos(Pi*n/2) ). - Wesley Ivan Hurt, May 31 2013
a(n) = ((1+(-1)^(n/2))*(1+(-1)^n))/4. - Bogart B. Strauss, Jul 14 2013
a(n) = C(n-1,3) mod 2. - Wesley Ivan Hurt, Oct 07 2014
a(n) = (((n+1) mod 4) mod 3) mod 2. - Ctibor O. Zizka, Dec 11 2014
a(n) = (sin(Pi*(n+1)/2)^2)/2 + sin(Pi*(n+1)/2)/2. - Mikael Aaltonen, Jan 02 2015
E.g.f.: (cos(x) + cosh(x))/2. - Vaclav Kotesovec, Feb 15 2015
a(n) = a(n-4) for n>3. - Wesley Ivan Hurt, Jul 07 2016
a(n) = (1-sqrt(2)*cos(n*Pi/2-3*Pi/4))/2 * cos(n*Pi/2). - (found by Steve Chow) Iain Fox, Nov 16 2017
a(n) = 1-A166486(n). - Antti Karttunen, Jul 29 2018
a(n) = (1-(-1)^A142150(n+1))/2. - Adriano Caroli, Sep 28 2019

More terms from Antti Karttunen, Jul 29 2018

A077267 Number of zeros in base-3 expansion of n.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 3, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 3, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 0, 4, 3, 3, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 1, 1, 2, 1, 1, 3, 2, 2, 2, 1, 1, 2

Offset: 0

Henry Bottomley, Nov 01 2002

			a(8)=0 since 8 written in base 3 is 22 with 0 zeros;
a(9)=2 since 9 written in base 3 is 100 with 2 zeros;
a(10)=1 since 10 written in base 3 is 101 with 1 zero.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters, F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009) 09.5.6
Eric Weisstein's World of Mathematics, Ternary.
Wikipedia, Ternary numeral system

Cf. A023416, A077266, A062756, A081603.

Cf. A007089, A081605, A032924, A081607, A081608, A077267, A134023. - Reinhard Zumkeller, Mar 23 2003

a077267 n = a079978 n + if n < 3 then 0 else a077267 (n `div` 3)
-- Reinhard Zumkeller, Feb 21 2013

Table[Count[IntegerDigits[n,3],0],{n,0,6!}] (* Vladimir Joseph Stephan Orlovsky, Jul 25 2009 *)
DigitCount[Range[0,110],3,0] (* Harvey P. Dale, Jul 04 2021 *)

a(1)=a(2)=0; a(3n)=a(n)+1; a(3n+1)=a(3n+2)=a(n). a(3^n-2)=a(3^n-1)=0; a(3^n)=n. a(n)=A077266(n, 3).
a(n) + A062756(n) + A081603(n) = A081604(n). - Reinhard Zumkeller, Mar 23 2003
G.f.: (Sum_{k>=0} x^(3^(k+1))/(1 + x^(3^k) + x^(2*3^k)))/(1-x). - Franklin T. Adams-Watters, Nov 03 2005
a(n) = A079978(n) if n < 3, A079978(n) + a(floor(n/3)) otherwise. - Reinhard Zumkeller, Feb 21 2013

a(0)=1 added, offset changed to 0 and b-file adjusted by Reinhard Zumkeller, Feb 21 2013

Wrong formula deleted by Reinhard Zumkeller, Feb 21 2013

A103221 Number of partitions of n into parts 2 and 3.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 6, 5, 6, 6, 6, 6, 7, 6, 7, 7, 7, 7, 8, 7, 8, 8, 8, 8, 9, 8, 9, 9, 9, 9, 10, 9, 10, 10, 10, 10, 11, 10, 11, 11, 11, 11, 12, 11, 12, 12, 12, 12, 13, 12, 13, 13, 13, 13, 14, 13, 14, 14, 14, 14, 15, 14, 15, 15

Offset: 0

Michael Somos, Jan 25 2005

Essentially the same as A008615.
Poincaré series [or Poincare series] for modular forms of weight w for the full modular group. As generators one may take the Eisenstein series E_4 (A004009) and E_6 (A013973).
Dimension of the space of weight 2n+12 cusp forms for Gamma_0( 1 ).
Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 5 ).
a(n) is the number of partitions of n into two nonnegative parts congruent modulo 3. - Andrew Baxter, Jun 28 2006
Also number of equivalence classes of period 2n billiards on an equilateral triangle. - Andrew Baxter, Jun 06 2008
a(n) is also the number of 2-regular multigraphs on n vertices, where each component is either a pair of parallel edges, or a triangle. - Jason Kimberley, Oct 14 2011
For n>1, a(n) is the number of partitions of 2n into positive parts x,y, and z such that x>=y and y=z. This sequence is used in calculating the probability of the need for a run-off election when n voters randomly cast ballots for two of three candidates running for two empty slots on a county commission. - Dennis P. Walsh, Apr 25 2013
Also, Molien series for invariants of finite Coxeter group A_2. The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1-x^i). Note that this is the root system A_k, not the alternating group Alt_k. - N. J. A. Sloane, Jan 11 2016
The coefficient of x^(2*n+1) in the power series expansion of the Weierstrass sigma function is a polynomial in the invariants g2 and g3 with a(n) terms. - Michael Somos, Jun 14 2016
a(n) is also the dimension of the complex vector space of modular forms M_{2*n} of weight 2*n and level 1 (full modular group). See Apostol p. 119, eq. (9) for k=2*n, and Ash and Gross, p. 178, Table 13.1. For a(6*k+1) = a(6*k+j)-1 for j = 0,2,3,4,5 and k >= 0 see A016921 (so-called dips, cf. Ash and Gross, p. 178.). - Wolfdieter Lang, Sep 16 2016
In an hexagonal tiling of the plane where the base tile is (0,0)--(2,1)--(3,3)--(1,4)--(-1,3)--(-2,1)--(0,0), a(n) is the number of vertices on the (n,0)--(n,n) closed line segment. - Luc Rousseau, Mar 22 2018

			For n=8, a(n)=2 since there are two partitions of 16 into 3 positive parts x, y, and z such that x >= y and y=z, namely, 16 = 8+4+4 and 16 = 6+5+5. - _Dennis P. Walsh_, Apr 25 2013
G.f. = 1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + 2*x^11 + ...

T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 119.
Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 178.
D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
E. Freitag, Siegelsche Modulfunktionen, Springer-Verlag, Berlin, 1983; p. 141, Th. 1.1.
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962.
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
J.-M. Kantor, Ou en sont les mathématiques, La formule de Molien-Weyl, SMF, Vuibert, p. 79
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 26. - N. J. A. Sloane, Aug 28 2010.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Max A. Alekseyev and Allan Bickle, Forbidden Subgraphs of Single Graphs, (2024). See p. 5.
Andrew M. Baxter and Ron Umble, Periodic Orbits of Billiards on an Equilateral Triangle, Amer. Math. Monthly, 115 (No. 6, 2008), 479-491.
Ansgar Freyer, Monika Ludwig, and Martin Rubey, Unimodular Valuations beyond Ehrhart, arXiv:2407.07691 [math.MG], 2024. See p. 8.
Jun-Ichi Igusa, On Siegel modular forms of genus 2 (II), Amer. J. Math., 86 (1964), 392-412, esp. p. 402.
Mohammed L. Nadji, Moussa Ahmia, Daniel F. Checa, and José L. Ramírez, Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants, J. Int. Seq. (2025) Vol. 28, Issue 3, Article 25.3.6. See p. 15.
Luc Rousseau, a(n) in an hexagonal tiling
Tetsuji Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89, 1022-1046, 1967.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N))
William A. Stein, The modular forms database
Index entries for Molien series
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Cf. A008615, A001399 (partial sums), A128115, A171386, A081753.
Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776, A266777, A266778, A266779, A266780, A266781.
Cf. A004009, A013973, A016921, A079978.

[Floor((n+2)/2)-Floor((n+2)/3): n in [0..100]]; // Vincenzo Librandi, Sep 18 2016

A103221:=n->floor((n+2)/2)-floor((n+2)/3): # Andrew Baxter, Jun 06 2008

a=b=c=d=0;Table[e=a+b-d+1;a=b;b=c;c=d;d=e,{n,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011 *)
LinearRecurrence[{0, 1, 1, 0, -1},{1, 0, 1, 1, 1},88] (* Ray Chandler, Sep 23 2015 *)
a[ n_] := With[{m = Max[-5 - n, n]}, (-1)^Boole[n < 0] SeriesCoefficient[ 1 / ((1 - x^2) (1 - x^3)), {x, 0, m}]]; (* Michael Somos, Jun 02 2019 *)

{a(n) = if( n<-4, -a(-5-n), polcoeff( 1 / ((1 - x^2) * (1 - x^3)) + x * O(x^n), n))};

a(n)=n+=2; n\2 - n\3 \\ Charles R Greathouse IV, Jul 31 2017

def A103221(n): return (n>>1)+1-(n+2)//3 # Chai Wah Wu, Apr 15 2025

def a(n) : return( len( CuspForms( Gamma0( 1), 2*n + 12, prec=1). basis())); # Michael Somos, May 29 2013

Euler transform of finite sequence [0, 1, 1] with offset 1, which is A171386.
a(n) = A008615(n+2). First differences of A001399.
a(n) = a(n-6) + 1 = a(n-2) + a(n-3) - a(n-5). - Henry Bottomley, Sep 02 2000
G.f.: 1/((1-x^2)*(1-x^3)).
a(n) = floor((n+2)/2) - floor((n+2)/3). - Andrew Baxter, Jun 06 2008
For odd n, a(n)=floor((n+3)/6). For even n, a(n)=floor((n+6)/6). - Dennis P. Walsh, Apr 25 2013
a(n) = floor(n/6)+1 unless n == 1 (mod 6); if n == 1 (mod 6), a(n) = floor(n/6). - Bob Selcoe, Sep 27 2014
a(n) = A081753(2*n); see the Dennis P. Walsh formula. - Wolfdieter Lang, Sep 16 2016
a(n)-a(n-2) = A079978(n). - R. J. Mathar, Jun 23 2021
E.g.f.: (3*(4 + x)*cosh(x) + exp(-x/2)*(6*cos(sqrt(3)*x/2) - 2*sqrt(3)*sin(sqrt(3)*x/2)) + 3*(1 + x)*sinh(x))/18. - Stefano Spezia, Mar 05 2023
a(n) = A008615(n-1)+A059841(n). - R. J. Mathar, May 03 2023

Name changed by Wolfdieter Lang, Sep 16 2016

A005063 Sum of squares of primes dividing n.

Original entry on oeis.org

0, 4, 9, 4, 25, 13, 49, 4, 9, 29, 121, 13, 169, 53, 34, 4, 289, 13, 361, 29, 58, 125, 529, 13, 25, 173, 9, 53, 841, 38, 961, 4, 130, 293, 74, 13, 1369, 365, 178, 29, 1681, 62, 1849, 125, 34, 533, 2209, 13, 49, 29, 298, 173, 2809, 13, 146, 53, 370, 845, 3481, 38, 3721

Offset: 1

N. J. A. Sloane

nonn

The set of these terms apart from 0 is A048261. - Bernard Schott, Feb 10 2022

Inverse Möbius transform of n^2 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000290, A005066, A005071, A005075, A005079, A005083, A059841, A079978.
Cf. A067666, A081403, A048261. - Franklin T. Adams-Watters, May 03 2009
Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), this sequence (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).
Cf. A010051.

A005063 := proc(n)
        add(d^2, d= numtheory[factorset](n)) ;
end proc;
seq(A005063(n),n=1..40) ; # R. J. Mathar, Nov 08 2011

a[n_] := Total[FactorInteger[n][[All, 1]]^2]; a[1]=0; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 20 2017 *)
Array[DivisorSum[#, #^2 &, PrimeQ] &, 61] (* Michael De Vlieger, Jul 11 2017 *)

a(n)=local(fm,t);fm=factor(n);t=0;for(k=1,matsize(fm)[1],t+=fm[k,1]^2);t \\ Franklin T. Adams-Watters, May 03 2009

a(n) = vecsum(apply(x->x^2, factor(n)[, 1])); \\ Michel Marcus, Sep 19 2020

from sympy import primefactors
def a(n): return sum(p**2 for p in primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017

(define (A005063 n) (if (= 1 n) 0 (+ (A000290 (A020639 n)) (A005063 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = p^2.
G.f.: Sum_{k>=1} prime(k)^2*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Dec 24 2016
From Antti Karttunen, Jul 11 2017: (Start)
a(n) = A005066(n) + 4*A059841(n).
a(n) = A005079(n) + A005083(n) + 4*A059841(n).
a(n) = A005071(n) + A005075(n) + 9*A079978(n).
(End)
Dirichlet g.f.: primezeta(s-2)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018
a(n) = Sum_{p|n, p prime} p^2. - Wesley Ivan Hurt, Feb 04 2022
a(n) = Sum_{d|n} d^2 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

More terms from Franklin T. Adams-Watters, May 03 2009

A000086 Number of solutions to x^2 - x + 1 == 0 (mod n).

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0

Offset: 1

N. J. A. Sloane

Number of elliptic points of order 3 for Gamma_0(n).
Equivalently, number of fixed points of Gamma_0(n) of type rho.
Values are 0 or a power of 2.
Shadow transform of central polygonal numbers A002061. - Michel Marcus, Jun 06 2013
Empirical: a(n) == A001615(n) (mod 3) for all natural numbers n. - John M. Campbell, Apr 01 2018
From Jianing Song, Jul 03 2018: (Start)
The comment above is true. Since both a(n) and A001615(n) are multiplicative we just have to verify that for prime powers. Note that A001615(p^e) = (p+1)*p^(e-1). For p == 1 (mod 3), p+1 == 2 (mod 3) so (p+1)*p^(e-1) == 2 (mod 3); for p == 2 (mod 3), p+1 is a multiple of 3 so (p+1)*p^(e-1) == 0 (mod 3). For p = 3, if e = 1 then p+1 == 1 (mod 3); if e > 1 then (p+1)*p^(e-1) == 0 (mod 3).
Equivalently, number of solutions to x^2 + x + 1 == 0 (mod n). (End)

			G.f. = x + x^3 + 2*x^7 + 2*x^13 + 2*x^19 + 2*x^21 + 2*x^31 + 2*x^37 + 2*x^39 + ...

Bruno Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 101.
Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (3).

Christian G. Bower, Table of n, a(n) for n = 1..2000
Harriet Fell, Morris Newman, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B (1963), 61-68.
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150; see Definition 7 for the shadow transform.
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. A65 (2009), 156-163. [See Table 4.]
N. J. A. Sloane, Transforms.

Cf. A000089, A000091, A001616, A014683.
Cf. A027748, A079978, A038502, A007949, A165952.
Cf. A341422 (without zeros).

a000086 n = if n `mod` 9 == 0 then 0
  else product $ map ((* 2) . a079978 . (+ 2)) $ a027748_row $ a038502 n
-- Reinhard Zumkeller, Jun 23 2013

with(numtheory); A000086 := proc (n) local d, s; if modp(n,9) = 0 then RETURN(0) fi; s := 1; for d in divisors(n) do if isprime(d) then s := s*(1+eval(legendre(-3,d))) fi od; s end: # Gene Ward Smith, May 22 2006

Array[ Function[ n, If[ EvenQ[ n ] || Mod[ n, 9 ]==0, 0, Count[ Array[ Mod[ #^2-#+1, n ]&, n, 0 ], 0 ] ] ], 84 ]
a[ n_] := If[ n < 1, 0, Length[ Select[ (#^2 - # + 1)/n & /@ Range[n], IntegerQ]]]; (* Michael Somos, Aug 14 2015 *)
a[n_] := a[n] = Product[{p, e} = pe; Which[p==1 || p==3 && e==1, 1, p==3 && e>1, 0, Mod[p, 3]==1, 2, Mod[p, 3]==2, 0, True, a[p^e]], {pe, FactorInteger[n]}]; Array[a, 105] (* Jean-François Alcover, Oct 18 2018 *)

{a(n) = if( n<1, 0, sum( x=0, n-1, (x^2 - x + 1)%n==0))}; \\ Nov 15 2002

{a(n) = if( n<1, 0, direuler( p=2, n, if( p==3, 1 + X, if( p%3==2, 1, (1 + X) / (1 - X)))) [n])}; \\ Nov 15 2002

Multiplicative with a(p^e) = 1 if p = 3 and e = 1; 0 if p = 3 and e > 1; 2 if p == 1 (mod 3); 0 if p == 2 (mod 3). - David W. Wilson, Aug 01 2001
a(A226946(n)) = 0; a(A034017(n)) > 0. - Reinhard Zumkeller, Jun 23 2013
a(2*n) = a(3*n + 2) = a(9*n) = a(9*n + 6) = 0. - Michael Somos, Aug 14 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*sqrt(3)/(3*Pi) = 0.367552... (A165952). - Amiram Eldar, Oct 11 2022

A024493 a(n) = C(n,0) + C(n,3) + ... + C(n,3[n/3]).

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 22, 43, 85, 170, 341, 683, 1366, 2731, 5461, 10922, 21845, 43691, 87382, 174763, 349525, 699050, 1398101, 2796203, 5592406, 11184811, 22369621, 44739242, 89478485, 178956971, 357913942, 715827883, 1431655765, 2863311530

Offset: 0

Clark Kimberling

First differences of A131708. First differences give A024495. - Paul Curtz, Nov 18 2007
a(n) = upper left term of X^n, where X = the 4 X 4 matrix [1,0,1,0; 1,1,0,0; 0,1,1,1; 0,0,0,1]. - Gary W. Adamson, Mar 01 2008
M^n * [1,0,0] = [a(n), A024495(n), A024494(n)], where M = a 3 X 3 matrix [1,1,0; 0,1,1; 1,0,1]. Sum of terms = 2^n. Example: M^5 * [1,0,0] = [11, 11, 10], sum = 2^5 = 32. - Gary W. Adamson, Mar 13 2009
Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1+M)^n = a(n) + A024494(n)*M + A024495(n)*M^2. - Stanislav Sykora, Jun 10 2012
Counts closed walks of length (n) at the vertices of a unidirectional triangle, containing a loop at each vertex. - David Neil McGrath, Sep 15 2014
{A024493, A131708, A024495} is the difference analog of the hyperbolic functions of order 3, {h_1(x), h_2(x), h_3(x)}. For a definition see the reference "Higher Transcendental Functions" and the Shevelev link. - Vladimir Shevelev, Jun 08 2017

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, 2nd. ed., Problem 38, p. 70.
Higher Transcendental Functions, Bateman Manuscript Project, Vol. 3, ed. A. Erdelyi, 1983 (chapter XVIII).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
P. H. Daus, Note on Sums Involving Binomial Coefficients, National Mathematics Magazine, volume 10, number 5, February 1936, pages 165-166.
John B. Dobson, A matrix variation on Ramus's identity for lacunary sums of binomial coefficients, arXiv preprint arXiv:1610.09361 [math.NT], 2016.
Arnold T. Saunders, Jr., Random Recursive Tree Evolution Algorithms: Identification and Characterization of Classes of Deletion Rules, Ph. D. thesis, The George Washington University, ProQuest Dissertations Publishing (2020) 27830773.
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Row sums of A098172.

Cf. A024494, A094715, A094717, A079978 (inverse binomial transform).

I:=[1,1,1]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+2*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jun 12 2017

A024493_list := proc(n) local i; series((exp(2*z)+2*cos(z*sqrt(3/4))*exp(z/2)) /3,z,n+2): seq(i!*coeff(%,z,i),i=0..n) end: A024493_list(33); # Peter Luschny, Jul 10 2012
seq((3*(-1)^(floor((n+1)/3))+(-1)^n+2^(n+1))/6, n=0..33); # Peter Luschny, Jun 14 2017

nn = 18; a = Sum[x^(3 i)/(3 i)!, {i, 0, nn}]; b = Exp[x];Range[0, nn]! CoefficientList[Series[a b , {x, 0, nn}], x]  (* Geoffrey Critzer, Dec 27 2011 *)
Differences[LinearRecurrence[{3,-3,2},{0,1,2},40]] (* Harvey P. Dale, Nov 27 2013 *)

a(n)=sum(i=0,n,sum(j=0,n,if(n-i-3*j,0,n!/(i)!/(3*j)!)))

a(n)=sum(k=0,n\3,binomial(n,3*k)) /* Michael Somos, Feb 14 2006 */

a(n)=if(n<0, 0, ([1,0,1;1,1,0;0,1,1]^n)[1,1]) /* Michael Somos, Feb 14 2006 */

a(n) = (1/3)*(2^n+2*cos( n*Pi/3 )).
G.f.: (1-x)^2/((1-2*x)*(1-x+x^2)) = (1-2*x+x^2)/(1-3*x+3*x^2-2*x^3). - Paul Barry, Feb 11 2004
a(n) = (1/3)*(2^n+b(n)) where b(n) is the 6-periodic sequence {2, 1, -1, -2, -1, 1}. - Benoit Cloitre, May 23 2004
Binomial transform of 1/(1-x^3). G.f.: (1-x)^2/((1-x)^3-x^3) = x/(1-x-2*x^2)+1/(1+x^3); a(n) = Sum_{k=0..floor(n/3)} binomial(n, 3*k); a(n) = Sum_{k=0..n} binomial(n,k)*(cos(2*Pi*k/3+Pi/3)/3+sin(2*Pi*k/3+Pi/3)/sqrt(3)+1/3); a(n) = A001045(n)+sqrt(3)*cos(Pi*n/3+Pi/6)/3+sin(Pi*n/3+Pi*/6)/3+(-1)^n/3. - Paul Barry, Jul 25 2004
a(n) = Sum_{k=0..n} binomial(n, 3*(n-k)). - Paul Barry, Aug 30 2004
G.f.: ((1-x)*(1-x^2)*(1-x^3))/((1-x^6)*(1-2*x)). - Michael Somos, Feb 14 2006
a(n+1)-2a(n) = -A010892(n). - Michael Somos, Feb 14 2006
E.g.f.: exp(x)*A(x) where A(x) is the e.g.f. for A079978. - Geoffrey Critzer, Dec 27 2011
Start with x(0)=1, y(0)=0, z(0)=0 and set x(n+1) = x(n) + z(n), y(n+1) = y(n) + x(n), z(n+1) = z(n) + y(n). Then a(n) = x(n). - Stanislav Sykora, Jun 10 2012
E.g.f.: (exp(2*z)+2*cos(z*sqrt(3/4))*exp(z/2))/3. - Peter Luschny, Jul 10 2012
Recurrence: a(0) = 1, a(1) = 1, a(2) = 1, a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3). - Christopher Hunt Gribble, Mar 25 2014
a(m+k) = a(m)*a(k) + A131708(m)*A024495(k) + A024495(m)*A131708(k). - Vladimir Shevelev, Jun 08 2017

A079998 The characteristic function of the multiples of five.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1

Offset: 0

Vladimir Baltic, Feb 10 2003

Number of permutations satisfying -k <= p(i) - i <= r and p(i) - i not in I, i = 1..n, with k = 2, r = 3, I = {-1, 0, 1, 2}.
a(n) = 1 if n = 5k, a(n) = 0 otherwise. Also, number of permutations satisfying -k <= p(i) - i <= r and p(i) - i not in I, i = 1..n, with k = 1, r = 4, I = {0, 1, 2, 3}.
a(n) is also the number of partitions of n with each part being five (a(0) = 1 because the empty partition has no parts to test equality with five). Hence a(n) is also the number of 2-regular graphs on n vertices with each component having girth exactly five. - Jason Kimberley, Oct 02 2011
This sequence is the Euler transformation of A185015. - Jason Kimberley, Oct 02 2011

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

Antti Karttunen, Table of n, a(n) for n = 0..16385
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135.
Benoit Cloitre, A study of a family of self-referential sequences, arXiv:2506.18103 [math.GM], 2025. See p. 10.
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A011558, A008587, A002524-A002529, A072827, A072850-A072856, A079955-A080014.

Characteristic function of multiples of g: A000007 (g = 0), A000012 (g = 1), A059841 (g = 2), A079978 (g = 3), A121262 (g = 4), this sequence (g = 5), A079979 (g = 6), A082784 (g = 7). - Jason Kimberley, Oct 14 2011

[Binomial(n-1,4) mod 5 : n in [0..100]]; // Wesley Ivan Hurt, Oct 06 2014

A079998:=n->binomial(n-1,4) mod 5: seq(A079998(n), n=0..100); # Wesley Ivan Hurt, Oct 06 2014

Table[Mod[Binomial[n - 1, 4], 5], {n, 0, 100}] (* Wesley Ivan Hurt, Oct 06 2014 *)
Table[Boole[Divisible[n, 5]], {n, 0, 99}] (* Alonso del Arte, Nov 29 2014 *)
PadRight[{},120,{1,0,0,0,0}] (* Harvey P. Dale, Jul 11 2023 *)

a(n)=!(n%5) \\ Charles R Greathouse IV, Mar 07 2012

(define (A079998 n) (if (zero? (modulo n 5)) 1 0)) ;; Antti Karttunen, Dec 21 2017

Recurrence: a(n) = a(n-5). G.f.: -1/(x^5 - 1).
a(n) = 1 - A011558(n); a(A008587(n)) = 1; a(A047201(n)) = 0. - Reinhard Zumkeller, Nov 30 2009
a(n) = floor(1/2*cos(2*n*Pi/5) + 1/2). - Gary Detlefs, May 16 2011
a(n) = floor(n/5) - floor((n-1)/5). - Tani Akinari, Oct 21 2012
a(n) = binomial(n - 1, 4) mod 5. - Wesley Ivan Hurt, Oct 06 2014

More terms from Antti Karttunen, Dec 21 2017

A005064 Sum of cubes of primes dividing n.

Original entry on oeis.org

0, 8, 27, 8, 125, 35, 343, 8, 27, 133, 1331, 35, 2197, 351, 152, 8, 4913, 35, 6859, 133, 370, 1339, 12167, 35, 125, 2205, 27, 351, 24389, 160, 29791, 8, 1358, 4921, 468, 35, 50653, 6867, 2224, 133, 68921, 378, 79507, 1339, 152, 12175, 103823, 35, 343, 133, 4940, 2205, 148877, 35, 1456, 351, 6886, 24397, 205379, 160

Offset: 1

N. J. A. Sloane

nonn

The set of these terms is A213519. - Bernard Schott, Feb 11 2022

Inverse Möbius transform of n^3 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000578, A005067, A005072, A005076, A005080, A005084, A059841, A079978.
Cf. A010051, A213519.
Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), A005063 (k=2), this sequence (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).

Array[DivisorSum[#, #^3 &, PrimeQ] &, 60] (* Michael De Vlieger, Jul 11 2017 *)
f[p_, e_] := p^3; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^3); \\ Michel Marcus, Jul 11 2017

from sympy import primefactors
def a(n): return sum(p**3 for p in primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017

(define (A005064 n) (if (= 1 n) 0 (+ (A000578 (A020639 n)) (A005064 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = p^3.
G.f.: Sum_{k>=1} prime(k)^3*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Dec 24 2016
From Antti Karttunen, Jul 11 2017: (Start)
a(n) = A005067(n) + 8*A059841(n).
a(n) = A005080(n) + A005084(n) + 8*A059841(n).
a(n) = A005072(n) + A005076(n) + 27*A079978(n).
(End)
Dirichlet g.f.: primezeta(s-3)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018
a(n) = Sum_{p|n, p prime} p^3. - Wesley Ivan Hurt, Feb 04 2022
a(n) = Sum_{d|n} d^3 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

More terms from Antti Karttunen, Jul 10 2017

A079979 Characteristic function of multiples of six.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1

Offset: 0

Vladimir Baltic, Feb 17 2003

Period 6: repeat [1, 0, 0, 0, 0, 0].
a(n)=1 if n=6k, a(n)=0 otherwise.
Decimal expansion of 1/999999.
Number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=3, r=3, I={-2,-1,0,1,2}.
Also, number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=1, r=5, I={0,1,2,3,4}.
a(n) is also the number of partitions of n such that each part is six (a(0)=1 because the empty partition has no parts to test equality with six). Hence a(n) is also the number of 2-regular graphs on n vertices with each part having girth exactly six. - Jason Kimberley, Oct 10 2011
This sequence is the Euler transformation of A185016. - Jason Kimberley, Oct 10 2011

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

Antti Karttunen, Table of n, a(n) for n = 0..65538
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).
Index entries for characteristic functions

Cf. A002524-A002529, A010875, A072827, A072850-A072856, A079955-A080014, A097325, A122841.

Characteristic function of multiples of g: A000007 (g=0), A000012 (g=1), A059841 (g=2), A079978 (g=3), A121262 (g=4), A079998 (g=5), this sequence (g=6), A082784 (g=7).

Magma
```
&cat[[1,0^^5]^^30];
```

A079979 := func; [A079979:n in [0..59]];  // Jason Kimberley, Oct 10 2011

PadRight[{},120,{1,0,0,0,0,0}] (* Harvey P. Dale, Feb 19 2013 *)

a(n)=!(n%6) \\ Charles R Greathouse IV, Oct 10 2011

(define (A079979 n) (if (zero? (modulo n 6)) 1 0)) ;; Antti Karttunen, Dec 22 2017

a(n) = a(n-6).
G.f.: 1/(1-x^6).
a(n) = floor((1/2)*cos(n*Pi/3) + 1/2). - Gary Detlefs, May 16 2011
a(n) = floor(n/6) - floor((n-1)/6). - Tani Akinari, Oct 23 2012
a(n) = (((((v^n - w^n)^2)*(2 - (-1)^n)*(w^(2*n) + w^n - 3))^2 - 144)^2)/20736, where w = (-1+i*sqrt(3))/2, v = (1+i*sqrt(3))/2. - Bogart B. Strauss, Sep 20 2013
E.g.f.: (2*cos(sqrt(3)*x/2)*cosh(x/2) + cosh(x))/3. - Vaclav Kotesovec, Feb 15 2015

More terms from Antti Karttunen, Dec 22 2017

A082784 Characteristic function of multiples of 7.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 0

Reinhard Zumkeller, May 22 2003

This sequence is the Euler transformation of A185017. - Jason Kimberley, Oct 14 2011

			a(14) = a(2*7) = 1; a(41) = a(5*7+6) = 0.

Index entries for characteristic functions.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,1).

Cf. A008589, A076309.
Characteristic function of multiples of g: A000007 (g=0), A000012 (g=1), A059841 (g=2), A079978 (g=3), A121262 (g=4), A079998 (g=5), A079979 (g=6), this sequence (g=7). - Jason Kimberley, Oct 14 2011
Cf. A000007, A047304, A109720, A185017.

a082784 = a000007 . (`mod` 7)
a082784_list = cycle [1,0,0,0,0,0,0]
-- Reinhard Zumkeller, Oct 27 2012

[Binomial(n-1,6) mod 7 : n in [0..100]]; // Wesley Ivan Hurt, Oct 07 2014

A082784:=n->0^(n mod 7): seq(A082784(n), n=0..100); # Wesley Ivan Hurt, Oct 07 2014

Table[Mod[Binomial[n - 1, 6], 7], {n, 0, 100}] (* Wesley Ivan Hurt, Oct 07 2014 *)
Table[Boole[Divisible[n, 7]], {n, 0, 100}] (* Amiram Eldar, Oct 31 2023 *)

a(n)=!(n%7) \\ Charles R Greathouse IV, Dec 03 2012

a(n) = 0^(n mod 7).
a(0)=1, a(n)=0 for 1<=n<7, a(n+7)=a(n).
a(n) = 1 - (n^6 mod 7). - Paolo P. Lava, Oct 02 2006
a(n) = 1 - A109720(n); a(A008589(n)) = 1; a(A047304(n)) = 0. - Reinhard Zumkeller, Nov 30 2009
a(n) = floor(n/7)-floor((n-1)/7). - Tani Akinari, Oct 26 2012
a(n) = C(n-1,6) mod 7. - Wesley Ivan Hurt, Oct 07 2014
From Wesley Ivan Hurt, Jul 11 2016: (Start)
G.f.: 1/(1-x^7).
a(n) = a(n-7) for n>6.
a(n) = (gcd(n,7) - 1)/6. (End)

Wrong formula and keyword mult removed by Amiram Eldar, Oct 31 2023

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A121262 The characteristic function of the multiples of four.

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A077267 Number of zeros in base-3 expansion of n.

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A103221 Number of partitions of n into parts 2 and 3.

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A005063 Sum of squares of primes dividing n.

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A000086 Number of solutions to x^2 - x + 1 == 0 (mod n).

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