A085141 -id:A085141 - cp's OEIS frontend

A080995 Characteristic function of generalized pentagonal numbers A001318.

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

Offset: 0

Michael Somos, Feb 27 2003

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Repeatedly [1,[0,]^2k,1,[0,]^k], k>=0; characteristic function of generalized pentagonal numbers: a(A001318(n))=1, a(A090864(n))=0. - Reinhard Zumkeller, Apr 22 2006
Starting with offset 1 with 1's signed (++--++,...), i.e., (1, 1, 0, 0, -1, 0, -1, 0, ...); is the INVERTi transform of A000041 starting (1, 2, 3, 5, 7, 11, ...). - Gary W. Adamson, May 17 2013
Number 9 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

			G.f. = 1 + x + x^2 + x^5 + x^7 + x^12 + x^15 + x^22 + x^26 + x^35 + x^40 + x^51 + ...
G.f. = q + q^25 + q^49 + q^121 + q^169 + q^289 + q^361 + q^529 + q^625 + ...

Percy A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p. 81, Article 331.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1001 from T. D. Noe)
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See P(q).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Don Zagier, Elliptic modular forms and their applications, in "The 1-2-3 of modular forms", Springer-Verlag, 2008.
Index entries for characteristic functions.

Cf. A001318 (support), A010815 (absolute values), A033683, A089806.
Cf. A000326, A180447, A085141, A133985.
Cf. A000009, A000041, A000122, A000700, A010054, A090864, A121373.

a080995 = a033683 . (+ 1) . (* 24)  -- Reinhard Zumkeller, Nov 14 2015

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (Series[ EllipticTheta[ 3, Log[y] / (2 I), x^(3/2)], {x, 0, n + Floor@Sqrt[n]}] // Normal // TrigToExp) /. {y -> x^(1/2)}, {x, 0, n}]]; (* Michael Somos, Nov 18 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] / QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Jun 08 2013 *)
a[ n_] := If[ n < 0, 0, Boole[ IntegerQ[ Sqrt[ 24 n + 1]]]]; (* Michael Somos, Jun 08 2013 *)

{a(n) = if( n<0, 0, abs( polcoeff( eta(x + x * O(x^n)), n)))};

{a(n) = issquare( 24*n + 1)}; /* Michael Somos, Apr 13 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A) * eta(x^6 + A)), n))};

Expansion of phi(-x^3) / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions. - Michael Somos, Sep 14 2007
Expansion of psi(x) - x * psi(x^9) in powers of x^3 where psi() is a Ramanujan theta function. - Michael Somos, Sep 14 2007
Expansion of f(x, x^2) in powers of x where f() is Ramanujan's two-variable theta function.
Expansion of q^(-1/24) * eta(q^2) * eta(q^3)^2 / (eta(q) * eta(q^6)) in powers of q.
a(n) = b(24*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p>3. - Michael Somos, Jun 06 2005
Euler transform of period 6 sequence [ 1, 0, -1, 0, 1, -1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 2^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089810.
G.f.: Product_{k>0} (1 - x^(3*k)) / (1 - x^k + x^(2*k)). - Michael Somos, Jan 26 2008
G.f.: Sum x^(n*(3n+1)/2), n=-inf..inf [the exponents are the pentagonal numbers, A000326].
a(n) = |A010815(n)| = A089806(2*n) = A033683(24*n + 1).
For n > 0, a(n) = b(n) - b(n-1) + c(n) - c(n-1), where b(n) = floor(sqrt(2n/3+1/36)+1/6) (= A180447(n)) and c(n) = floor(sqrt(2n/3+1/36)-1/6) (= A085141(n)). - Mikael Aaltonen, Mar 08 2015
a(n) = (-1)^n * A133985(n). - Michael Somos, Jul 12 2015
a(n) = A000009(n) (mod 2). - John M. Campbell, Jun 29 2016
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 2*sqrt(2/3) = 1.632993... . - Amiram Eldar, Jan 13 2024

Minor edits by N. J. A. Sloane, Feb 03 2012

A090864 Complement of generalized pentagonal numbers (A001318).

Original entry on oeis.org

3, 4, 6, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84

Offset: 1

Jon Perry, Feb 12 2004

Also n for which A006906(n) is even, or equivalently n for which A000009(n) is even (since A006906 and A000009 have the same parity).

The number of partitions of a(n) into distinct parts with an even number of parts equals the number of such partitions with an odd number of parts: A067661(a(n)) = A067659(a(n)). See, e.g., the Freitag-Busam reference, p. 410 given in A036499. - Wolfdieter Lang, Jan 19 2016

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000009, A001318, A067659, A067661, A080995, A006906, A010815, A118303.

Cf. A085141, A111651.

Complement[Range[200], Select[Accumulate[Range[0,200]]/3, IntegerQ]] (* G. C. Greubel, Jun 06 2017 *)

a(n) = my(q,r); [q,r]=divrem(sqrtint(24*n),3); n + q + (r >= bitnegimply(1,q)); \\ Kevin Ryde, Sep 15 2024

from math import isqrt
def A090864(n):
    def f(x): return n+(m:=isqrt(24*x+1)+1)//6+(m-2)//6
    kmin, kmax = 0,1
    while f(kmax) > kmax:
        kmax <<= 1
    while kmax-kmin > 1:
        kmid = kmax+kmin>>1
        if f(kmid) <= kmid:
            kmax = kmid
        else:
            kmin = kmid
    return kmax # Chai Wah Wu, Aug 29 2024

A080995(a(n)) = 0; A000009(a(n)) = A118303(n). - Reinhard Zumkeller, Apr 22 2006
A010815(a(n)) = A067661(a(n)) - A067659(a(n)) = 0, n >= 1. See a comment above. - Wolfdieter Lang, Jan 19 2016
a(n) = n+1 + A085141(n-1) + A111651(n). - Kevin Ryde, Sep 15 2024

More terms from Reinhard Zumkeller, Apr 22 2006
Edited by Ray Chandler, Dec 14 2011
Edited by Jon E. Schoenfield, Nov 25 2016

A235963 n appears (n+1)/(1 + (n mod 2)) times.

Original entry on oeis.org

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

Offset: 0

Mircea Merca, Jan 17 2014

n appears A001318(n+1) - A001318(n) = A026741(n+1) times.
Sum_{k=0...a(n)} (-1)^ceiling(k/2)*p(n-G(k)) = 0 for n>0, where p(n)=A000041(n) is the partition function, and G(k)=A001318(k) denotes the generalized pentagonal numbers.
Row lengths of A238442, n >= 1. - Omar E. Pol, Dec 22 2016

			As a triangle:
  0;
  1;
  2, 2, 2;
  3, 3;
  4, 4, 4, 4, 4;
  5, 5, 5;
  6, 6, 6, 6, 6, 6, 6;
  7, 7, 7, 7;
  8, 8, 8, 8, 8, 8, 8, 8, 8;
  ...

G. C. Greubel, Table of n, a(n) for n = 0..5000

First differences are A080995.

Cf. A000041, A001318, A026741, A085141, A180447, A238442.

T:= n-> n$(n+1)/(n mod 2+1):
seq(T(n), n=0..13);  # Alois P. Heinz, Nov 23 2024

Table[Table[n, {(n + 1)/(1 + Mod[n, 2])}], {n, 0, 14}]//Flatten (* T. D. Noe, Jan 29 2014 *)

from math import isqrt
def A235963(n): return (m:=isqrt((n+1<<3)//3))-(n+1<=(m*(3*m+4)+1 if m&1 else m*(3*m+2))>>3) # Chai Wah Wu, Nov 23 2024

A235963=lambda n: ((s:=isqrt(24*n+1))+1)//6+(s-1)//6 # Natalia L. Skirrow, May 13 2025

Let t = (sqrt(n*8/3 + 1) - 1)/2 + 1/3 and let k = floor(t); then a(n) = 2k if t - k < 2/3, 2k+1 otherwise. - Jon E. Schoenfield, Jun 13 2017
a(n) = m if n+1>A001318(m) and a(n) = m-1 otherwise where m = floor(sqrt(8(n+1)/3)). - Chai Wah Wu, Nov 23 2024
From Natalia L. Skirrow, May 13 2025: (Start)
a(n) = A180447(n) + A085141(n).
a(n) = floor((s+1)/6) + floor((s-1)/6) where s=floor(sqrt(24*n+1)).
G.f.: (f(x,x^2)-1)/(1-x), where f is Ramanujan's bivariate theta function. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/8 + (2-sqrt(2))*log(2)/8 + log(2+sqrt(2))/(2*sqrt(2)). - Amiram Eldar, May 28 2025

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A080995 Characteristic function of generalized pentagonal numbers A001318.

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A090864 Complement of generalized pentagonal numbers (A001318).

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A235963 n appears (n+1)/(1 + (n mod 2)) times.

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