A180447 -id:A180447 - 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

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

A180446 Number of non-pentagonal numbers <= n.

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 63, 64, 65, 66

Offset: 0

William A. Tedeschi, Sep 07 2010

			a(5) = 5 - floor((sqrt(24*5+1)+1)/6) = 3.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

f[n_] := n - Floor[(Sqrt[24 n + 1] + 1)/6]; Array[f, 74, 0] (* Robert G. Wilson v, Sep 10 2010 *)
Accumulate[Table[If[IntegerQ[(1+Sqrt[1+24n])/6],0,1],{n,0,80}]]-1 (* Harvey P. Dale, May 22 2023 *)

l = [n-floor((sqrt(24*n+1)+1)/6) for n in range(0,101)]

from math import isqrt
def A180446(n): return n-(m:=isqrt((k:=n<<1)//3))-(k>m*(3*m+5)) # Chai Wah Wu, Nov 04 2024

a(n) = n - floor((sqrt(24n+1)+1)/6) = n - A180447(n).

More terms from Robert G. Wilson v, Sep 10 2010

A145172 Number of pentagonal numbers needed to represent n with greedy algorithm.

Original entry on oeis.org

1, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 2, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 2, 3, 4, 5, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 2, 3, 4, 5, 6, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 2, 3, 4, 5, 6, 3, 4, 5, 6

Offset: 1

Christina Steffan (christina.steffan(AT)gmx.at), Oct 03 2008

nonn

Sequence is unbounded.

			a(21)=6 since 21 = 12+5+1+1+1+1.

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000326 (pentagonal numbers), A053610, A057945, A180447, A192988.

a(n)={my(s=0); forstep(k=(sqrtint(24*n+1)+1)\6, 1, -1, my(t=k*(3*k-1)/2); s+=n\t; n%=t); s} \\ Andrew Howroyd, Apr 21 2021

Terms a(41) and beyond from Andrew Howroyd, Apr 21 2021

A183294 Complement of A005449.

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Clark Kimberling, Jan 03 2011

Cf. A005449, A180447.

a=3/2; b=1/2;
F[n_]:=a*n^2+b*n;
R[n_]:=(n/a+((b-1)/(2a))^2)^(1/2);
G[n_]:=n-1+Ceiling[R[n]-(b-1)/(2a)];
Table[F[n], {n,60}]
Table[G[n], {n,100}]

a(n) = n + (sqrtint(24*n)+1)\6; \\ Kevin Ryde, Sep 01 2024

from math import isqrt
def A183294(n): return n+(m:=isqrt((k:=n<<1)//3))+(k>=m*(3*m+5)+4) # Chai Wah Wu, Nov 04 2024

(See the Mathematica code.)
a(n) = n + A180447(n-1). - Kevin Ryde, Sep 01 2024
a(n) = n+m+1 if 2n>=m(3m+5)+4 and a(n) = n+m otherwise where m = floor(sqrt(2n/3)). - Chai Wah Wu, Nov 04 2024

A368088 Index of smallest pentagonal number with n digits.

Original entry on oeis.org

1, 3, 9, 26, 82, 259, 817, 2583, 8166, 25821, 81650, 258200, 816497, 2581990, 8164966, 25819890, 81649659, 258198890, 816496582, 2581988898, 8164965810, 25819888975, 81649658093, 258198889748, 816496580928, 2581988897472, 8164965809278, 25819888974717, 81649658092773

Offset: 1

Kelvin Voskuijl, Dec 17 2023

The digits of the odd- and even-indexed terms converge to those in the decimal expansions of sqrt(2/3) and sqrt(20/3), respectively.

			a(4) = 26 as the 26th pentagonal number is 26*(3*26-1)/2 = 1001 which has 4 digits (while the 25th is 925 which is only 3).

Cf. A000326, A180447.
Cf. A068092 (for triangular numbers), A017936 (for squares).
Cf. A157697 (square root of 2/3), A020772 (square root of 20/3)

a[n_] := Ceiling[(Sqrt[24*10^(n-1) + 1] + 1)/6]; Array[a, 40] (* Amiram Eldar, Dec 30 2023 *)

a(n) = 1 + (sqrtint(24*10^(n-1)) + 1)\6 \\ Andrew Howroyd, Dec 30 2023

a(n) = ceiling((sqrt(24*10^(n-1) + 1) + 1)/6).

cp's OEIS Frontend

A080995 Characteristic function of generalized pentagonal numbers A001318.

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

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A180446 Number of non-pentagonal numbers <= n.

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A145172 Number of pentagonal numbers needed to represent n with greedy algorithm.

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A183294 Complement of A005449.

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A368088 Index of smallest pentagonal number with n digits.

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