A051030 -id:A051030 - cp's OEIS frontend

A261004 Expansion of (-3-164x-x^2)/(1-99x+99*x^2-x^3).

-3, -461, -45343, -4443321, -435400283, -42664784581, -4180713488823, -409667257120241, -40143210484294963, -3933624960203786301, -385455102889486762703, -37770666458209498958761, -3701139857801641411196043, -362673935398102648798253621, -35538344529156257940817658983

Offset: 0

N. J. A. Sloane, Aug 12 2015

Mc Laughlin (2010) gives an identity relating ten sequences, denoted a_k, b_k, ..., f_k, p_k, q_k, r_k, s_k. This is the sequence a_k.

Seiichi Manyama, Table of n, a(n) for n = 0..500
Kwang-Wu Chen, Extensions of an amazing identity of Ramanujan, Fib. Q., 50 (2012), 227-230.
J. Mc Laughlin, An identity motivated by an amazing identity of Ramanujan, Fib. Q., 48 (No. 1, 2010), 34-38.
Index entries for linear recurrences with constant coefficients, signature (99, -99, 1).

Cf. A051028, A051029, A051030.

Cf. A010701, A269548, A269549, A269550, A269551, A269552, A269553, A269554, A269555, A269556.

LinearRecurrence[{99,-99,1},{-3,-461,-45343},30] (* Harvey P. Dale, Dec 02 2017 *)

Vec((-3-164*x-x^2)/(1-99*x+99*x^2-x^3) + O(x^20)) \\ Michel Marcus, Feb 29 2016

A051028 Ramanujan's a-series: expansion of (1+53x+9x^2)/(1-82x-82x^2+x^3).

Original entry on oeis.org

1, 135, 11161, 926271, 76869289, 6379224759, 529398785665, 43933719985479, 3645969360009049, 302571523160765631, 25109790452983538281, 2083810036074472911735, 172931123203728268135681, 14351199415873371782349831, 1190976620394286129666900249

Offset: 0

Eric W. Weisstein

The "amazing" identity of Ramanujan is a(n)^3 + b(n)^3 = c(n)^3 + (-1)^n, where a(n)=A051028(n), b(n)=A051029(n) and c(n)=A051030(n). - Emeric Deutsch, Oct 14 2006

Robert Israel, Table of n, a(n) for n = 0..468
Kwang-Wu Chen, Extensions of an amazing identity of Ramanujan, Fib. Q., 50 (2012), 227-230.
Jung Hun Han and Michael D. Hirschhorn, Another Look at an Amazing Identity of Ramanujan, Mathematics Magazine, Vol. 79 (2006), pp. 302-304.
Michael D. Hirschhorn, An amazing identity of Ramanujan, Math. Mag. 68 (1995), no. 3, 199--201. MR1335148
Michael D. Hirschhorn, A Proof in the Spirit of Zeilberger of an Amazing Identity of Ramanujan, Math. Mag., 69.4 (1996), 267-269.
Michael D. Hirschhorn, Ramanujan and Fermat's Last Theorem, The Australian Mathematical Society, Gazette, Volume 31 Number 4, September 2004.
J. Mc Laughlin, An identity motivated by an amazing identity of Ramanujan, Fib. Q., 48 (No. 1, 2010), 34-38.
Eric Rowland and Jesus Sistos Barron, Complexity of powers of a constant-recursive sequence, arXiv:2501.14643 [math.NT], 2025. See p. 2.
Eric Weisstein's World of Mathematics, Ramanujan's Sum Identity.
Index entries for linear recurrences with constant coefficients, signature (82,82,-1).

Cf. A051029, A051030.

g:=(1+53*x+9*x^2)/(1-82*x-82*x^2+x^3): gser:=series(g,x=0,20): seq(coeff(gser,x,n),n=0..12); # Emeric Deutsch, Oct 14 2006

CoefficientList[Series[(1 + 53 x + 9 x^2)/(1 - 82 x - 82 x^2 + x^3), {x, 0, 33}], x] (* Vincenzo Librandi, Jul 22 2015 *)

Vec((1+53*x+9*x^2)/(1-82*x-82*x^2+x^3) + O(x^30)) \\ Michel Marcus, Feb 29 2016

G.f.: (1+53*x+9*x^2)/((1+x)*(1-83*x+x^2)).

X(n+1) = A*X(n), where X(n) = transpose(A051028(n), A051029(n), A051030(n)) and A = matrix(3,3,[63,104,-68; 64,104,-67; 80,131,-85]). - Emeric Deutsch, Oct 14 2006

Minor edits (g.f. and name) by M. F. Hasler, May 08 2016

A051029 Ramanujan's b-series: expansion of (2-26x-12x^2)/(1-82x-82x^2+x^3).

Original entry on oeis.org

2, 138, 11468, 951690, 78978818, 6554290188, 543927106802, 45139395574362, 3746025905565260, 310875010766342202, 25798879867700837522, 2140996154008403172108, 177676881902829762447458, 14745040201780861879966890, 1223660659865908706274804428

Offset: 0

Eric W. Weisstein

The "amazing" identity of Ramanujan is a(n)^3 + b(n)^3 = c(n)^3 + (-1)^n, where a(n) = A051028(n), b(n) = A051029(n) and c(n) = A051030(n). - Emeric Deutsch, Oct 14 2006

For additional references and links see A051028.

Robert Israel, Table of n, a(n) for n = 0..468
Kwang-Wu Chen, Extensions of an amazing identity of Ramanujan, Fib. Q., 50 (2012), 227-230.
Jung Hun Han and Michael D. Hirschhorn, Another Look at an Amazing Identity of Ramanujan, Mathematics Magazine, Vol. 79 (2006), pp. 302-304.
Michael D. Hirschhorn, An amazing identity of Ramanujan, Math. Mag. 68 (1995), no. 3, 199--201. MR1335148
Michael D. Hirschhorn, A Proof in the Spirit of Zeilberger of an Amazing Identity of Ramanujan, Math. Mag., 69.4 (1996), 267-269.
J. Mc Laughlin, An identity motivated by an amazing identity of Ramanujan, Fib. Q., 48 (No. 1, 2010), 34-38.
Eric Rowland and Jesus Sistos Barron, Complexity of powers of a constant-recursive sequence, arXiv:2501.14643 [math.NT], 2025. See p. 2.
Eric Weisstein's World of Mathematics, Ramanujan's Sum Identity.
Index entries for linear recurrences with constant coefficients, signature (82,82,-1).

Cf. A051028, A051030.

g:=(2-26*x-12*x^2)/(1-82*x-82*x^2+x^3): gser:=series(g,x=0,20): seq(coeff(gser,x,n),n=0..12); # Emeric Deutsch, Oct 14 2006

CoefficientList[Series[(2 - 26 x - 12 x^2)/(1 - 82 x - 82 x^2 + x^3), {x, 0, 33}], x] (* Vincenzo Librandi, Jul 22 2015 *)

Vec((2-26*x-12*x^2)/(1-82*x-82*x^2+x^3) + O(x^30)) \\ Michel Marcus, Feb 29 2016

G.f.: (2-26*x-12*x^2)/((1+x)*(1-83*x+x^2)).

X(n+1) = A*X(n), where X(n) = transpose(A051028(n), A051029(n), A051030(n)) and A = matrix(3,3,[63,104,-68; 64,104,-67; 80,131,-85]). - Emeric Deutsch, Oct 14 2006

Minor edits (g.f. and name) by M. F. Hasler, May 08 2016

A272853 Ramanujan's alpha-series.

Original entry on oeis.org

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.

A272854 Ramanujan's beta-series.

Original entry on oeis.org

10, 812, 67402, 5593538, 464196268, 38522696690, 3196919629018, 265305806511788, 22017185020849402, 1827161050923988562, 151632350041670201260, 12583657892407702716002

Offset: 0

Robert Munafo, May 08 2016

Ramanujan's notes define this by the same G.f. as A051030 (the c-series) but using Laurent series expansion. It is mislabeled as "gamma" 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)=5593538 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, A272855.

Rest@ CoefficientList[ Normal@Series[-(2 + 8*x - 10*x^2)/(1 - 82*x - 82*x^2 + x^3), {x, Infinity, 20}], 1/x] (* Giovanni Resta, May 08 2016 *)

G.f.: (10-8*x-2*x^2)/(1-82*x-82*x^2+x^3).
a(-3)=14258; a(-2)=172; 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.

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

A261004 Expansion of (-3-164*x-x^2)/(1-99*x+99*x^2-x^3).

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A051028 Ramanujan's a-series: expansion of (1+53x+9x^2)/(1-82x-82x^2+x^3).

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A051029 Ramanujan's b-series: expansion of (2-26x-12x^2)/(1-82x-82x^2+x^3).

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A272853 Ramanujan's alpha-series.

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A272854 Ramanujan's beta-series.

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A272855 Ramanujan's gamma-series.

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A261004 Expansion of (-3-164x-x^2)/(1-99x+99*x^2-x^3).