A113428 -id:A113428 - cp's OEIS frontend

A057569 Numbers of the form k(5k+1)/2 or k(5k-1)/2.

0, 2, 3, 9, 11, 21, 24, 38, 42, 60, 65, 87, 93, 119, 126, 156, 164, 198, 207, 245, 255, 297, 308, 354, 366, 416, 429, 483, 497, 555, 570, 632, 648, 714, 731, 801, 819, 893, 912, 990, 1010, 1092, 1113, 1199, 1221, 1311, 1334, 1428, 1452, 1550

Offset: 1

N. J. A. Sloane, Oct 04 2000

a(n) is the set of all m such that 40*m+1 is a perfect square. - Gary Detlefs, Feb 22 2010
Integers of the form (n^2 - n) / 10. Numbers of the form  n * (5*n - 1) / 2 where n is an integer. - Michael Somos, Jan 13 2012
Also integers of the form sum_{k=1..n} k/5. - Alonso del Arte, Jan 20 2012
These numbers appear in a theta function identity. See the Hardy-Wright reference, Theorem 356 on p. 284. See the G.f. of A113428. - Wolfdieter Lang, Oct 28 2016

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A074378, A001318, A057570, A154260. - Vladimir Joseph Stephan Orlovsky, Jan 06 2009

Cf. A001622, A113428.

[(10*(n^2-n)+12*(-1)^n*(n div 2))/16: n in [1..60]]; // Vincenzo Librandi, Oct 29 2016

Select[Table[Plus@@Range[n]/5, {n, 0, 199}], IntegerQ] (* Alonso del Arte, Jan 20 2012 *)
LinearRecurrence[{1,2,-2,-1,1},{0,2,3,9,11},50] (* Harvey P. Dale, Jul 05 2021 *)

{a(n) = (10 * (n^2 - n) + 12 * (-1)^n * (n\2)) / 16}; \\ Michael Somos, Jan 13 2012

Vec(x^2*(2*x^2+x+2) / ((1-x)^3*(1+x)^2) + O(x^60)) \\ Colin Barker, Jun 13 2017

A005475 UNION A005476. G.f.: x^2*(2x^2+x+2)/((1-x)^3*(1+x)^2). a(n) = A132356(n+1)/4. - R. J. Mathar, Apr 07 2008
a(n) = (A090771(n)^2 -1)/40. - Gary Detlefs, Feb 22 2010
|A113428(n)| is the characteristic function of the numbers a(n).
a(n) = a(1 - n) for all n in Z. - Michael Somos, Jan 13 2012
From Colin Barker, Jun 13 2017: (Start)
a(n) = n*(5*n - 2)/8 for n even.
a(n) = (5*n - 3)*(n - 1)/8 for n odd.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.
(End)
From Amiram Eldar, Mar 17 2022: (Start)
Sum_{n>=2} 1/a(n) = 10 - 2*sqrt(1+2/sqrt(5))*Pi.
Sum_{n>=2} (-1)^n/a(n) = 2*sqrt(5)*log(phi) - 5*(2-log(5)), where phi is the golden ratio (A001622). (End)

A113429 Expansion of f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 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, 1, 0, 0, 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, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Oct 31 2005

For the g.f. identity see the Hardy-Wright reference, Theorem 355 on p. 284. - Wolfdieter Lang, Oct 28 2016

			G.f. = 1 - x - x^4 + x^7 + x^13 - x^18 - x^27 + x^34 + x^46 - x^55 - x^70 + ...
G.f. = q^9 - q^49 - q^169 + q^289 + q^529 - q^729 - q^1089 + q^1369 + q^1849 + ...

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 93.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities

Cf. A085787, A113428.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^5] QPochhammer[ x^4, x^5] QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Jun 26 2017 *)
a[ n_] := Module[{m = 40 n + 9, k}, If[IntegerQ[k = Sqrt[m]], If[Mod[k, 10] == 7, k = -k]; (-1)^Quotient[k, 10], 0]]; (* Michael Somos, Jun 26 2017 *)

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 - x^k*[1, 1, 0, 0, 1][k%5 + 1], 1 + x * O(x^n)), n))};

{a(n) = my(m, k); if( n<0, 0, issquare(m = 40*n + 9, &k), if( k%10==7, k=-k); (-1)^(k\10), 0)}; /* Michael Somos, Oct 29 2016 */

Euler transform of period 5 sequence [-1, 0, 0, -1, -1, ...].
|a(n)| is the characteristic function of A085787.
G.f.: Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k-1)) * (1 - x^(5*k-4)) = Sum_{k in Z} (-1)^k * x^((5*k^2+3*k)/2).
f(a, b) = Sum_{k in Z} a^((k^2+k)/2) * b^((k^2-k)/2) is Ramanujan's general theta function.
G.f.: Sum_{n>=0} (x^(n*(n+1)) * Product_{k>=n+1} (1-x^k)). - Joerg Arndt, Apr 07 2011
From Wolfdieter Lang, Oct 30 2016: (Start)
a(n) = (-1)^k if n = b(2*k) for k >= 0, a(n) = (-1)^k if n = b(2*k-1), for k >= 1, and a(n) = 0 otherwise, where b(n) = A085787(n). See the second formula.
G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n+3)/2)*(1-x^(2*n+1)). See the Hardy reference, p. 93, G_1(x,x) from eq. (6.11.1) with C_n(x,x) = 1.
(End)
G.f.: Sum_{n>=0} (-1)^n*x^(n*(5*n-3)/2)*(1-x^(4*(2*n+1))). Reordered G_1(x,x) from the preceding formula. This is G_4(x,x) from Hardy, p. 93, eq. (6.11.1) with C_n(x,x) = 1. Note that Hardy uses only G_0, G_1 and G_2. - Wolfdieter Lang, Nov 01 2016
a(n) = -(1/n)*Sum_{k=1..n} A284361(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

A036822 Number of partitions satisfying cn(1,5) = cn(4,5) = 0.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 4, 4, 7, 6, 10, 11, 13, 18, 19, 25, 30, 33, 45, 47, 61, 70, 81, 100, 111, 135, 157, 177, 218, 238, 288, 328, 374, 443, 495, 579, 663, 747, 878, 973, 1134, 1281, 1448, 1670, 1863, 2135, 2414, 2705, 3103

Offset: 1

Olivier Gérard

nonn

For a given partition cn(i,n) means the number of its parts equal to i modulo n.
Short: (1=4 := 0).
a(n) is the number of partitions with parts congruent to 0, 2 or 3 mod 5. - George Beck, Aug 08 2020

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A036820.

c := proc(L,i,n)
    local a,p;
    a := 0 ;
    for p in L do
        if modp(p,n) = i then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
A036822 := proc(n)
    local a ,p;
    a := 0 ;
    for p in combinat[partition](n) do
        if c(p,1,5) = 0 then
            if c(p,4,5) = 0 then
                a := a+1 ;
            end if;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 19 2014

nmax = 50; Rest[CoefficientList[Series[Product[1/((1 - x^(5*k)) * (1 - x^(5*k-2)) * (1 - x^(5*k-3))), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2016 *)

Convolution inverse of A113428. - George Beck, May 21 2016
G.f.: Product_{k>=1} 1/((1 - x^(5*k)) * (1 - x^(5*k - 2)) * (1 - x^(5*k - 3))). - Vaclav Kotesovec, Jul 05 2016
a(n) ~ exp(Pi*sqrt(2*n/5)) / (2*sqrt(2*(5+sqrt(5)))*n). - Vaclav Kotesovec, Jul 05 2016

A133101 Expansion of f(x^2, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 11 2007

nonn

			G.f. = 1 + x^2 + x^3 + x^9 + x^11 + x^21 + x^24 + x^38 + x^42 + x^60 + x^65 + ...
G.f. = q + q^81 + q^121 + q^361 + q^441 + q^841 + q^961 + q^1521 + q^1681 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for characteristic functions

Cf. A057569, A113428.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^5] QPochhammer[ -x^3, x^5] QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Oct 31 2015 *)

{a(n) = if( n<0, 0, polcoeff( prod(k=1,n, 1 + x^k*[ -1, 0, 1, 1, 0][k%5 + 1], 1 + x * O(x^n)), n))};

The characteristic function of A057569.
Euler transform of period 10 sequence [ 0, 1, 1, -1, -1, -1, 1, 1, 0, -1, ...].
G.f.: Prod_{k>0} (1 - x^(5*k)) * (1 + x^(5*k - 2)) * (1 + x^(5*k - 3)) = Sum_{k in Z} x^((5*k^2 + k) / 2).
a(n) = abs(A113428(n)).

A259357 Expansion of f(-x^5)^2 / f(-x, -x^4) in powers of x where f(,) is the Ramanujan general theta function.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 3, 3, 5, 5, 5, 6, 7, 7, 9, 9, 11, 11, 13, 13, 16, 17, 19, 20, 23, 24, 27, 29, 32, 34, 38, 40, 46, 48, 52, 56, 62, 65, 72, 76, 84, 89, 97, 102, 113, 119, 129, 137, 149, 157, 171, 181, 196, 208, 224, 236, 256, 270, 290, 308, 331

Offset: 0

Michael Somos, Jun 24 2015

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + ...
G.f. = q^23 + q^143 + q^263 + q^383 + 2*q^503 + q^623 + 2*q^743 + 2*q^863 + ...

G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part III, Springer, 2012, see p. 12, Entry 2.1.3, Equation (2.1.21).
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, equation 3.

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A035959, A113428, A259358.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^5] / (QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}];

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, -1, 0, 0, -1][k%5+1]), n))};

Expansion of f(-x^5) * f(-x^2, -x^3) / f(-x) in powers of x where f(,) is the Ramanujan general theta function.
Expansion of f(-x^5) * G(x) in powers of x where f() is a Ramanujan theta function and G() is a Rogers-Ramanujan function. - Michael Somos, Jul 09 2015
Euler transform of period 5 sequence [ 1, 0, 0, 1, -1, ...].
G.f.: Product_{k>0} (1 - x^(5*k)) / ((1 - x^(5*k-4)) * (1 - x^(5*k-1))).
Convolution of A035959 and A113428.

A205183 Euler transform of period 5 sequence [ 2, 1, 1, 2, 1, ...].

Original entry on oeis.org

1, 2, 4, 7, 13, 21, 35, 54, 84, 126, 188, 273, 396, 562, 794, 1106, 1533, 2099, 2862, 3864, 5194, 6927, 9198, 12131, 15938, 20817, 27087, 35070, 45246, 58114, 74400, 94870, 120602, 152768, 192957, 242928, 305025, 381857, 476839, 593830, 737776

Offset: 0

Michael Somos, Jan 23 2012

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			1 + 2*x + 4*x^2 + 7*x^3 + 13*x^4 + 21*x^5 + 35*x^6 + 54*x^7 + 84*x^8 + ...
q^-7 + 2*q^113 + 4*q^233 + 7*q^353 + 13*q^473 + 21*q^593 + 35*q^713 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000041, A003114, A113428.

f[x_,y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; CoefficientList[Series[f[-q^2,-q^3]/QPochhammer[q]^2, {q,0,100}], q] (* G. C. Greubel, Apr 14 2018 *)

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 / (1 - x^k)^[ 1, 2, 1, 1, 2][k%5 + 1], 1 + x * O(x^n)), n))}

Expansion of G(x) / f(-x) = f(-x^2, -x^3) / f(-x)^2 in powers of x where f() is a Ramanujan theta function and G() is a Rogers-Ramanujan function.
G.f. is the limit as n goes to infinity of Sum_{k=0..n} x^k^2 / ((x;x)k * (x;x){n-k}) = Sum_{k=-n..n} (-1)^k * x^(k*(5*k - 1)/2) / ((x;x){n-k} * (x;x){n+k}).
G.f.: (Sum_{k>=0} x^k^2 / ((1 - x) ... (1 - x^k))) / Product_{k>0} (1 - x^k).
G.f.: (Sum_{k} (-1)^k * x^(k*(5*k - 1)/2)) / Product_{k>0} (1 - x^k)^2.
Convolution of A000041 and A003114.
a(n) ~ exp(Pi*sqrt(14*n/15)) * sqrt(7*phi) / (4*3^(1/2)*5^(3/4)*n), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 24 2018

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A057569 Numbers of the form k*(5*k+1)/2 or k*(5*k-1)/2.

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A113429 Expansion of f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function.

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A036822 Number of partitions satisfying cn(1,5) = cn(4,5) = 0.

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A133101 Expansion of f(x^2, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

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A259357 Expansion of f(-x^5)^2 / f(-x, -x^4) in powers of x where f(,) is the Ramanujan general theta function.

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A205183 Euler transform of period 5 sequence [ 2, 1, 1, 2, 1, ...].

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A057569 Numbers of the form k(5k+1)/2 or k(5k-1)/2.