A272854 -id:A272854 - cp's OEIS frontend

A272853 Ramanujan's alpha-series.

9, 791, 65601, 5444135, 451797561, 37493753471, 3111529740489, 258219474707159, 21429104870953665, 1778357484814447079, 147582242134728153849, 12247547739697622322431

Offset: 0

Robert Munafo, May 08 2016

nonn

Ramanujan's notes define this by the same G.f. as A051028 (the a-series) but using Laurent series expansion. These give identities of the form alpha(n)^3 + beta(n)^3 = gamma(n)^3 + (-1)^n, where alpha(n)=A272853(n), beta(n)=A272854(n) and gamma(n)=A272855(n). They are from page 82 of the "lost notebook" of Ramanujan. A051028,A051029,A051030 give his examples (135, 138, 172) and (11161, 11468, 14258) while A272853,A272854,A272855 give the examples (9, 10, 12), (791, 812, 1010), and (65601, 67402, 83802).

			a(3)=5444135 because 5444135^3 + 5593538^3 = 6954572^3 - 1.

S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341. New Delhi (Narosa publ. house).

Seiichi Manyama, Table of n, a(n) for n = 0..520
Robert Munafo, Sequences Related to the Work of Srinivasa Ramanujan
Index entries for linear recurrences with constant coefficients, signature (82, 82, -1).

Cf. A051028, A051029, A051030.

Cf. A272854, A272855.

Rest@ CoefficientList[Normal@ Series[(1 + 53*a + 9*a^2)/(1 - 82*a - 82*a^2 + a^3), {a, Infinity, 20}], 1/a] (* Giovanni Resta, May 08 2016 *)

G.f.: (9+53*x+x^2)/(1-82*x-82*x^2+x^3).
a(-3)=-11161; a(-2)=-135; a(-1)=-1; a(n) = 82*a(n-1)+82*a(n-2)-a(n-3).
A272853(n)^3 + A272854(n)^3 = A272855(n)^3 + (-1)^n.

A272855 Ramanujan's gamma-series.

Original entry on oeis.org

12, 1010, 83802, 6954572, 577145658, 47896135058, 3974802064140, 329860675188578, 27374461238587818, 2271750422127600332, 188527910575352239722, 15645544827332108296610

Offset: 0

Robert Munafo, May 08 2016

nonn

Ramanujan's notes define this by the same G.f. as A051029 (the b-series) but using Laurent series expansion. It is mislabeled as "beta" in Ramanujan's notes. These give identities of the form alpha(n)^3 + beta(n)^3 = gamma(n)^3 + (-1)^n, where alpha(n)=A272853(n), beta(n)=A272854(n) and gamma(n)=A272855(n). They are from page 82 of the "lost notebook" of Ramanujan. A051028,A051029,A051030 give his examples (135, 138, 172) and (11161, 11468, 14258) while A272853,A272854,A272855 give the examples (9, 10, 12), (791, 812, 1010), and (65601, 67402, 83802).

			a(3)=6954572 because 5444135^3 + 5593538^3 = 6954572^3 - 1.

S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341. New Delhi (Narosa publ. house).

Seiichi Manyama, Table of n, a(n) for n = 0..520
Robert Munafo, Sequences Related to the Work of Srinivasa Ramanujan
Index entries for linear recurrences with constant coefficients, signature (82, 82, -1).

Cf. A051028, A051029, A051030.

Cf. A272853, A272854.

Rest@ CoefficientList[ Normal@ Series[-1*(2 - 26 a - 12 a^2)/(1 - 82*a - 82*a^2 + a^3), {a, Infinity, 10}], 1/a] (* Giovanni Resta, May 08 2016 *)

G.f.: x*(12+26*x-2*x^2)/(1-82*x-82*x^2+x^3).
a(-3)=11468; a(-2)=138; a(-1)=2; a(n) = 82*a(n-1)+82*a(n-2)-a(n-3).
A272853(n)^3 + A272854(n)^3 = A272855(n)^3 + (-1)^n.

cp's OEIS Frontend

A272853 Ramanujan's alpha-series.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula