A195313 -id:A195313 - cp's OEIS frontend

A281815 Expansion of f(x, x^10) in powers of x where f(, ) is Ramanujan's general theta function.

1, 1, 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, 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, 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, 0, 0

Offset: 0

Michael Somos, Jan 30 2017

nonn

			G.f. = 1 + x + x^10 + x^13 + x^31 + x^36 + x^63 + x^70 + x^106 + x^115 + ...
G.f. = q^81 + q^169 + q^961 + q^1225 + q^2809 + q^3249 + q^5625 + q^6241 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. f(x, x^k) for k=2..11: A080995, A010054, A133100, A089801, A274179, A214263, A281814, A205988, A281815, A186742.

Cf. A195313.

a[ n_] := SquaresR[ 1, 88 n + 81] / 2;
a[ n_] := If[ n < 0, 0, Boole @ IntegerQ @ Sqrt @ (88 n + 81)];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^11] QPochhammer[ -x^10, x^11] QPochhammer[ x^11], {x, 0, n}];

PARI
```
{a(n) = issquare(88*n + 81)};
```

f(x,x^m) = 1 + Sum_{k>=1} x^((m+1)*k*(k-1)/2) (x^k + x^(m*k)). - N. J. A. Sloane, Jan 30 2017
Euler transform of period 22 sequence [1, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, ...].
Characteristic function of generalized 13-gonal numbers A195313.
G.f.: Sum_{k in Z} x^(k*(11*k + 9)/2).
G.f.: Product_{k>0} (1 + x^(11*k-10)) * (1 + x^(11*k-1)) * (1 - x^(11*k)).
Sum_{k=1..n} a(k) ~ (2*sqrt(2/11)) * sqrt(n). - Amiram Eldar, Jan 13 2024

A154617 Eleven times hexagonal numbers: a(n) = 11n(2*n-1).

Original entry on oeis.org

0, 11, 66, 165, 308, 495, 726, 1001, 1320, 1683, 2090, 2541, 3036, 3575, 4158, 4785, 5456, 6171, 6930, 7733, 8580, 9471, 10406, 11385, 12408, 13475, 14586, 15741, 16940, 18183, 19470, 20801, 22176, 23595, 25058, 26565, 28116, 29711, 31350, 33033, 34760, 36531, 38346

Offset: 0

Omar E. Pol, Jan 13 2009

Sequence found by reading the line from 0, in the direction 0, 11, ..., in the square spiral whose vertices are the generalized tridecagonal numbers A195313. - Omar E. Pol, Sep 18 2011

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000384, A002939, A094159, A152745, A153781, A195313.

a(n)=11*n*(2*n-1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 22*n^2 - 11*n = 11*A000384(n).
a(n) = a(n-1) + 44*n - 33 (with a(0)=0). - Vincenzo Librandi, Dec 15 2010
From Elmo R. Oliveira, Dec 15 2024: (Start)
G.f.: 11*x*(1 + 3*x)/(1 - x)^3.
E.g.f.: 11*x*(1 + 2*x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

A194801 Square array read by antidiagonals: T(n,k) = k((n+1)k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 3, 0, 1, 2, 4, 1, 0, 1, 3, 5, 4, 6, 0, 1, 4, 6, 7, 9, 3, 0, 1, 5, 7, 10, 12, 9, 10, 0, 1, 6, 8, 13, 15, 15, 16, 6, 0, 1, 7, 9, 16, 18, 21, 22, 16, 15, 0, 1, 8, 10, 19, 21, 27, 28, 26, 25, 10, 0, 1, 9, 11, 22, 24, 33, 34, 36, 35

Offset: 0

Omar E. Pol, Feb 05 2012

Note that a single formula gives several types of numbers. Row 0 lists 0 together the Molien series for 3-dimensional group [2,k]+ = 22k. Row 1 lists, except first zero, the squares repeated. If n >= 2, row n lists the generalized (n+3)-gonal numbers, for example: row 2 lists the generalized pentagonal numbers A001318. See some other examples in the cross-references section.

			Array begins:
(A008795): 0, 1,  0,  3,  1,  6,  3, 10,   6,  15,  10...
(A008794): 0, 1,  1,  4,  4,  9,  9, 16,  16,  25,  25...
A001318:   0, 1,  2,  5,  7, 12, 15, 22,  26,  35,  40...
A000217:   0, 1,  3,  6, 10, 15, 21, 28,  36,  45,  55...
A085787:   0, 1,  4,  7, 13, 18, 27, 34,  46,  55,  70...
A001082:   0, 1,  5,  8, 16, 21, 33, 40,  56,  65,  85...
A118277:   0, 1,  6,  9, 19, 24, 39, 46,  66,  75, 100...
A074377:   0, 1,  7, 10, 22, 27, 45, 52,  76,  85, 115...
A195160:   0, 1,  8, 11, 25, 30, 51, 58,  86,  95, 130...
A195162:   0, 1,  9, 12, 28, 33, 57, 64,  96, 105, 145...
A195313:   0, 1, 10, 13, 31, 36, 63, 70, 106, 115, 160...
A195818:   0, 1, 11, 14, 34, 39, 69, 76, 116, 125, 175...

Rows (0-11): 0 together with A008795, (truncated A008794), A001318, A000217, A085787, A001082, A118277, A074377, A195160, A195162, A195313, A195818
Columns (0-9): A000004, A000012, A001477, (truncated A000027), A016777, (truncated A008585), A016945, (truncated A016957), A017341, (truncated A017329).
Cf. A139600.

A317300 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.

Original entry on oeis.org

0, 1, -3, 0, -8, -3, -15, -8, -24, -15, -35, -24, -48, -35, -63, -48, -80, -63, -99, -80, -120, -99, -143, -120, -168, -143, -195, -168, -224, -195, -255, -224, -288, -255, -323, -288, -360, -323, -399, -360, -440, -399, -483, -440, -528, -483, -575, -528, -624, -575, -675, -624, -728, -675, -783

Offset: 0

Omar E. Pol, Jul 29 2018

Taking the same formula with k = 1 we have A317301.
Taking the same formula with k = 2 we have A001057 (canonical enumeration of integers).
Taking the same formula with k = 3 we have 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
Taking the same formula with k = 4 we have A008794 (squares repeated) except the initial zero.
Taking the same formula with k >= 5 we have the generalized k-gonal numbers (see Crossrefs section).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Row 0 of A303301.
Cf. A001057, A008795, A008794, A174474, A317301.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

concat(0, Vec(x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50))) \\ Colin Barker, Aug 01 2018

a(n) = -A174474(n+1).
From Colin Barker, Aug 01 2018: (Start)
G.f.: x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = -n*(n + 4) / 4 for n even.
a(n) = -(n - 3)*(n + 1) / 4 for n odd.
(End)

A317301 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 1.

Original entry on oeis.org

0, 1, -2, 1, -5, 0, -9, -2, -14, -5, -20, -9, -27, -14, -35, -20, -44, -27, -54, -35, -65, -44, -77, -54, -90, -65, -104, -77, -119, -90, -135, -104, -152, -119, -170, -135, -189, -152, -209, -170, -230, -189, -252, -209, -275, -230, -299, -252, -324, -275, -350, -299, -377, -324, -405, -350, -434

Offset: 0

Omar E. Pol, Jul 29 2018

Taking the same formula with k = 0 we have A317300.
Taking the same formula with k = 2 we have A001057 (canonical enumeration of integers).
Taking the same formula with k = 3 we have 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
Taking the same formula with k = 4 we have A008794 (squares repeated) except the initial zero.
Taking the same formula with k >= 5 we have the generalized k-gonal numbers (see Crossrefs section).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Row 1 of A303301.
Cf. A000096, A001057, A008794, A008795, A317300.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

/* By definition: */ k:=1; [0] cat [m*i*((k-2)*m*i-k+4)/2: i in [1,-1], m in [1..30]]; // Bruno Berselli, Jul 30 2018

Table[(-2 n (n + 1) - 5 (2 n + 1) (-1)^n + 5)/16, {n, 0, 60}] (* Bruno Berselli, Jul 30 2018 *)

concat(0, Vec(x*(1 - 3*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^50))) \\ Colin Barker, Aug 01 2018

From Bruno Berselli, Jul 30 2018: (Start)
O.g.f.: x*(1 - 3*x + x^2)/((1 + x)^2*(1 - x)^3).
E.g.f.: (-5*(1 + 2*x) + (5 - 2*x^2)*exp(2*x))*exp(-x)/16.
a(n) = a(-n+1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (-2*n*(n + 1) - 5*(2*n + 1)*(-1)^n + 5)/16. Therefore:
a(n) = -n*(n + 6)/8 for even n;
a(n) = -(n - 5)*(n + 1)/8 for odd n. Also:
a(n) = a(n-5) for odd n > 3.
2*(2*n - 1)*a(n) + 2*(2*n + 1)*a(n-1) + n*(n^2 - 3) = 0. (End)

cp's OEIS Frontend

A281815 Expansion of f(x, x^10) in powers of x where f(, ) is Ramanujan's general theta function.

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A154617 Eleven times hexagonal numbers: a(n) = 11*n*(2*n-1).

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A194801 Square array read by antidiagonals: T(n,k) = k*((n+1)*k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

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A317300 Sequence obtained by taking the general formula for generalized k-gonal numbers: m*((k - 2)*m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.

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Formula

A317301 Sequence obtained by taking the general formula for generalized k-gonal numbers: m*((k - 2)*m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 1.

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Magma

Mathematica

PARI

Formula

A154617 Eleven times hexagonal numbers: a(n) = 11n(2*n-1).

A194801 Square array read by antidiagonals: T(n,k) = k((n+1)k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.

A317300 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.

A317301 Sequence obtained by taking the general formula for generalized k-gonal numbers: m((k - 2)m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 1.