A006938 -id:A006938 - cp's OEIS frontend

A033687 Theta series of hexagonal lattice A_2 with respect to deep hole divided by 3.

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

Offset: 0

N. J. A. Sloane

nonn

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
a(n)=0 if and only if A000731(n)=0 (see the Han-Ono paper). - Emeric Deutsch, May 16 2008
Number of 3-core partitions of n (denoted c_3(n) in Granville and Ono, p. 340). - Brian Hopkins, May 13 2008
Denoted by g_1(q) in Cynk and Hulek in Remark 3.4 on page 12 (but not explicitly listed).
This is a member of an infinite family of integer weight modular forms. g_1 = A033687, g_2 = A030206, g_3 = A130539, g_4 = A000731. - Michael Somos, Aug 24 2012
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x + 2*x^2 + 2*x^4 + x^5 + 2*x^6 + x^8 + 2*x^9 + 2*x^10 + 2*x^12 + 2*x^14 + ...
G.f. = q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + q^25 + 2*q^28 + 2*q^31 + 2*q^37 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.35) and (32.351).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Johnathan M. Borwein and Peter B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012) see page 697.
Slawomir Cynk and Klaus Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
Andrew Granville and Ken Ono, Defect Zero p-blocks for Finite Simple Groups, Transactions of the American Mathematical Society, Vol. 348 (1996), pp. 331-347.
Guo-Niu Han and Ken Ono, Hook Lengths and 3-Cores, Ann. Comb. (2011) Vol. 15, 305-312. See also arXiv:0805.2461, [math.NT], 2008.
Jeremy Lovejoy and Olivier Mallet, n-color overpartitions, twisted divisor functions, and Rogers-Ramanujan identities, South East Asian J. Math. Math. Sci., 6 (2008), 23-36. [From _Jeremy Lovejoy_, Jun 12 2009]
Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000731, A002324, A005882, A006938, A033685.

Cf. A045831, A053723, A081622.

Basis( ModularForms( Gamma1(9), 1), 316) [2]; /* Michael Somos, May 06 2015 */

a[ n_] := If[ n < 0, 0, DivisorSum[ 3 n + 1, KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Sep 23 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Sep 01 2015 *)

{a(n) = if( n<0, 0, sumdiv( 3*n + 1, d, kronecker( -3, d)))}; /* Michael Somos, Nov 03 2005 */

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

{a(n) = my(A, p, e); if( n<0, 0, A = factor( 3*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, p%6==1, e+1, 1-e%2)))}; /* Michael Somos, May 06 2015 */

Euler transform of period 3 sequence [1, 1, -2, ...].
Expansion of q^(-1/3) * eta(q^3)^3 / eta(q) in powers of q.
a(4*n + 1) = a(n). - Michael Somos, Dec 06 2004
a(n) = b(3*n + 1) where b(n) is multiplicative and b(p^e) = 0^e if p = 3, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 2, 5 (mod 6). - Michael Somos, May 20 2005
Given g.f. A(x), B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2*w - 2*u*w^2 - v^3. - Michael Somos, Dec 06 2004
Given g.f. A(x), B(q)= q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u3^2 + u1*u6^2 - u1*u3*u6 - u2^2*u3. - Michael Somos, May 20 2005
Given g.f. A(x), B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u2*u3^2 + 2*u2*u3*u6 + 4*u2*u6^2 - u1^2*u6. - Michael Somos, May 20 2005
G.f.: Product_{k>0} (1 - x^(3*k))^3 / (1 - x^k).
G.f.: Sum_{k in Z} x^k / (1 - x^(3*k + 1)) = Sum_{k in Z} x^k / (1 - x^(6*k + 2)). - Michael Somos, Nov 03 2005
Expansion of q^(-1) * c(q^3) / 3 = q^(-1) * (a(q) - b(q)) / 9 in powers of q^3 where a(), b(), c() are cubic AGM theta functions. - Michael Somos, Dec 25 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A005928.
a(n) = Sum_{d|3n+1} LegendreSymbol{d,3} - Brian Hopkins, May 13 2008
q-series for a(n): Sum_{n >= 0} q^(n^2+n)(1-q)(1-q^2)...(1-q^n)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))). [From Jeremy Lovejoy, Jun 12 2009]
a(n) = A002324(3*n + 1). 3*a(n) = A005882(n) = A033685(3*n + 1). - Michael Somos, Apr 04 2003
G.f.: (2 * psi(x^2) * f(x^2, x^4) + phi(x) * f(x^1, x^5)) / 3 where phi(), psi() are Ramanujan theta functions and f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 07 2018
Sum_{k=1..n} a(k) ~ 2*Pi*n/3^(3/2). - Vaclav Kotesovec, Dec 17 2022

A008559 a(1)=2; thereafter, convert a(n-1) from base 10 to base 2 but regard the result as a base 10 number.

Original entry on oeis.org

2, 10, 1010, 1111110010, 1000010001110100011000101111010

Offset: 1

N. J. A. Sloane

The previous number is converted to binary digits and then the digits are regarded as decimal digits in the next number in the sequence. - Michael Somos, May 16 2014
The next term has 100 digits. - Harvey P. Dale, Jul 16 2011
The number of decimal digits of a(n) is A242347(n). - Robert G. Wilson v, Jul 10 2013
log(a(n))/log(a(n-1)) ~ log_2(10) = A020862. - Robert G. Wilson v, Jul 10 2013

Clifford Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 350.

Vincenzo Librandi, Table of n, a(n) for n = 1..7

Cf. A006937 (essentially the same sequence).

For initial terms 2 through 12 see A008559, A006938, A260025, A260024, A260026, A260027, A260028, A260029, A008559 (again), A006938 (again), A260030 respectively.

f:=proc(n) local i,j,r; i:=convert(n,base,2); j:=add(i[r]*10^(r-1),r=1..nops(i)); end;
g:=proc(n,M) global f; local a,b,t1; a:=n; t1:=[a]; for i from 1 to M do b:=f(a); t1:=[op(t1),b]; a:=b; od; t1; end; g(2,5); # N. J. A. Sloane, Jul 14 2015

NestList[FromDigits[IntegerDigits[#,2]]&,2,5] (* Harvey P. Dale, Jul 16 2011 *)

lista(nn) = my(k=2); print1(k); for(n=2, nn, print1(", ", k=fromdigits(binary(k)))); \\ Jinyuan Wang, Jan 18 2025

A008559_list = [2]
for _ in range(5):
    A008559_list.append(int(bin(A008559_list[-1])[2:]))
# Chai Wah Wu, Dec 26 2014

Comment corrected by Chai Wah Wu, Dec 26 2014

A260024 Convert the last term from decimal to binary! a(1)=5.

Original entry on oeis.org

5, 101, 1100101, 100001100100101000101, 1010110101111001011010001101011011001100110110011001100101110100101

Offset: 1

N. J. A. Sloane, Jul 14 2015

C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 350.

For initial terms 2 through 12 see A008559, A006938, A260025, A260024, A260026, A260027, A260028, A260029, A008559 (again), A006938 (again), A260030 respectively.

NestList[FromDigits[IntegerDigits[#,2]]&,5,5] (* Harvey P. Dale, Jun 23 2025 *)

A260025 Convert the last term from decimal to binary! a(1)=4.

Original entry on oeis.org

4, 100, 1100100, 100001100100101000100, 1010110101111001011010001101011011001100110110011001100101110100100

Offset: 1

N. J. A. Sloane, Jul 14 2015

For initial terms 2 through 12 see A008559, A006938, A260025, A260024, A260026, A260027, A260028, A260029, A008559 (again), A006938 (again), A260030 respectively.

NestList[FromDigits[IntegerDigits[#,2]]&,4,5] (* Harvey P. Dale, Jun 11 2017 *)

a(n) = my(k=4); for(i=2, n, k=fromdigits(binary(k))); k; \\ Jinyuan Wang, Jan 18 2025

A260026 Convert the last term from decimal to binary! a(1)=6.

Original entry on oeis.org

6, 110, 1101110, 100001100110100110110, 1010110101111001011010001101011100010111010110010110001101100011110

Offset: 1

N. J. A. Sloane, Jul 14 2015

For initial terms 2 through 12 see A008559, A006938, A260025, A260024, A260026, A260027, A260028, A260029, A008559 (again), A006938 (again), A260030 respectively.

a(n) = my(k=6); for(i=2, n, k=fromdigits(binary(k))); k; \\ Jinyuan Wang, Jan 18 2025

A260027 Convert the last term from decimal to binary! a(1)=7.

Original entry on oeis.org

7, 111, 1101111, 100001100110100110111, 1010110101111001011010001101011100010111010110010110001101100011111

Offset: 1

N. J. A. Sloane, Jul 14 2015

For initial terms 2 through 12 see A008559, A006938, A260025, A260024, A260026, A260027, A260028, A260029, A008559 (again), A006938 (again), A260030 respectively.

A260028 Convert the last term from decimal to binary! a(1)=8.

Original entry on oeis.org

8, 1000, 1111101000, 1000010001110100000111001001000

Offset: 1

N. J. A. Sloane, Jul 14 2015

For initial terms 2 through 12 see A008559, A006938, A260025, A260024, A260026, A260027, A260028, A260029, A008559 (again), A006938 (again), A260030 respectively.

a(n) = my(k=8); for(i=2, n, k=fromdigits(binary(k))); k; \\ Jinyuan Wang, Jan 18 2025

A260029 Convert the last term from decimal to binary! a(1)=9.

Original entry on oeis.org

9, 1001, 1111101001, 1000010001110100000111001001001

Offset: 1

N. J. A. Sloane, Jul 14 2015

For initial terms 2 through 12 see A008559, A006938, A260025, A260024, A260026, A260027, A260028, A260029, A008559 (again), A006938 (again), A260030 respectively.

A260030 Convert the last term from decimal to binary! a(1)=12.

Original entry on oeis.org

12, 1100, 10001001100, 1001010100000110110010101010001100

Offset: 1

N. J. A. Sloane, Jul 14 2015

The next term has 110 digits. - Harvey P. Dale, May 15 2018

For initial terms 2 through 12 see A008559, A006938, A260025, A260024, A260026, A260027, A260028, A260029, A008559 (again), A006938 (again), A260030 respectively.

NestList[FromDigits[IntegerDigits[#,2]]&,12,4] (* Harvey P. Dale, May 15 2018 *)

cp's OEIS Frontend

A033687 Theta series of hexagonal lattice A_2 with respect to deep hole divided by 3.

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A008559 a(1)=2; thereafter, convert a(n-1) from base 10 to base 2 but regard the result as a base 10 number.

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A260024 Convert the last term from decimal to binary! a(1)=5.

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A260025 Convert the last term from decimal to binary! a(1)=4.

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A260026 Convert the last term from decimal to binary! a(1)=6.

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A260027 Convert the last term from decimal to binary! a(1)=7.

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A260028 Convert the last term from decimal to binary! a(1)=8.

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A260029 Convert the last term from decimal to binary! a(1)=9.

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A260030 Convert the last term from decimal to binary! a(1)=12.

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