cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A195160 Generalized 11-gonal (or hendecagonal) numbers: m*(9*m - 7)/2 with m = 0, 1, -1, 2, -2, 3, -3, ...

Original entry on oeis.org

0, 1, 8, 11, 25, 30, 51, 58, 86, 95, 130, 141, 183, 196, 245, 260, 316, 333, 396, 415, 485, 506, 583, 606, 690, 715, 806, 833, 931, 960, 1065, 1096, 1208, 1241, 1360, 1395, 1521, 1558, 1691, 1730, 1870, 1911, 2058, 2101, 2255, 2300, 2461, 2508, 2676
Offset: 0

Views

Author

Omar E. Pol, Sep 10 2011

Keywords

Comments

Exponents of q in the expansion of Product_{n >= 1} (1 - q^(9*n))*(1 + q^(9*n-1))*(1 + q^(9*n-8)) = 1 + q + q^8 + q^11 + q^25 + q^30 + .... - Peter Bala, Nov 21 2024

Links

Crossrefs

Partial sums of A195159.
Column 7 of A195152.
Cf. A316672.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), this sequence (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).

Programs

  • Magma
    I:=[0, 1, 8, 11, 25]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Apr 09 2013
  • Mathematica
    CoefficientList[Series[x (1 + 7 x + x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 60}], x] (* Vincenzo Librandi, Apr 09 2013 *)
  • PARI
    a(n)=(18*n*(n+1)+5*(2*n+1)*(-1)^n-5)/16 \\ Charles R Greathouse IV, Sep 24 2015

Formula

From Bruno Berselli, Sep 14 2011: (Start)
G.f.: x*(1+7*x+x^2)/((1+x)^2*(1-x)^3).
a(n) = (18*n*(n+1)+5*(2*n+1)*(-1)^n-5)/16.
a(2n) = A062728(n), a(2n-1) = A051682(n). (End)
Sum_{n>=1} 1/a(n) = 18/49 + 2*Pi*cot(2*Pi/9)/7. - Vaclav Kotesovec, Oct 05 2016

A195313 Generalized 13-gonal numbers: m*(11*m-9)/2 with m = 0, 1, -1, 2, -2, 3, -3, ...

Original entry on oeis.org

0, 1, 10, 13, 31, 36, 63, 70, 106, 115, 160, 171, 225, 238, 301, 316, 388, 405, 486, 505, 595, 616, 715, 738, 846, 871, 988, 1015, 1141, 1170, 1305, 1336, 1480, 1513, 1666, 1701, 1863, 1900, 2071, 2110, 2290, 2331, 2520, 2563, 2761, 2806, 3013, 3060, 3276
Offset: 0

Views

Author

Omar E. Pol, Sep 14 2011

Keywords

Comments

Also generalized tridecagonal numbers or generalized triskaidecagonal numbers.
Also A211013 and positive terms of A051865 interleaved. - Omar E. Pol, Aug 04 2012
Numbers k for which 88*k + 81 is a square. - Bruno Berselli, Jul 10 2018

Links

Crossrefs

Partial sums of A195312.
Column 9 of A195152.
Cf. A316672.
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), this sequence (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).

Programs

  • Magma
    [(22*n*(n+1)+7*(2*n+1)*(-1)^n-7)/16: n in [0..50]]; // Vincenzo Librandi, Sep 16 2011
  • Magma
    A195313:=func; [0] cat [A195313(n*m): m in [1,-1], n in [1..25]]; // Bruno Berselli, Nov 13 2012
  • Maple
    a:= n-> (m-> m*(11*m-9)/2)(-ceil(n/2)*(-1)^n):
seq(a(n), n=0..60);  # Alois P. Heinz, Jul 10 2018
  • Mathematica
    lim = 50; Sort[Table[n*(11*n - 9)/2, {n, -lim, lim}]] (* T. D. Noe, Sep 15 2011 *)
Accumulate[With[{nn=30},Riffle[9Range[0,nn],Range[1,2nn+1,2]]]] (* Harvey P. Dale, Sep 24 2011 *)
  • PARI
    a(n)=(22*n*(n+1)+7*(2*n+1)*(-1)^n-7)/16 \\ Charles R Greathouse IV, Sep 24 2015

Formula

From Bruno Berselli, Sep 15 2011: (Start)
G.f.: x*(1 + 9*x + x^2)/((1 + x)^2*(1 - x)^3).
a(n) = (22*n*(n + 1) + 7*(2*n + 1)*(-1)^n - 7)/16.
a(n) - a(n-2) = A175885(n). (End)
Sum_{n>=1} 1/a(n) = 22/81 + 2*Pi*cot(2*Pi/11)/9. - Vaclav Kotesovec, Oct 05 2016

A277082 Generalized 15-gonal (or pentadecagonal) numbers: n*(13*n - 11)/2, n = 0,+1,-1,+2,-2,+3,-3, ...

Original entry on oeis.org

0, 1, 12, 15, 37, 42, 75, 82, 126, 135, 190, 201, 267, 280, 357, 372, 460, 477, 576, 595, 705, 726, 847, 870, 1002, 1027, 1170, 1197, 1351, 1380, 1545, 1576, 1752, 1785, 1972, 2007, 2205, 2242, 2451, 2490, 2710, 2751, 2982, 3025, 3267, 3312, 3565, 3612, 3876, 3925, 4200, 4251, 4537, 4590, 4887, 4942
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 29 2016

Keywords

Comments

More generally, the ordinary generating function for the generalized k-gonal numbers is x*(1 + (k - 4)*x + x^2)/((1 - x)^3*(1 + x)^2). A general formula for the generalized k-gonal numbers is given by (k*(2*n^2 + 2*((-1)^n + 1)*n + (-1)^n - 1) - 2*(2*n^2 + 2*(3*(-1)^n + 1)*n + 3*((-1)^n - 1)))/16.
For k>4, Sum_{n>=1} 1/a(k,n) = 2*(k-2)/(k-4)^2 + 2*Pi*cot(2*Pi/(k-2))/(k-4). - Vaclav Kotesovec, Oct 05 2016
Numbers k for which 104*k + 121 is a square. - Bruno Berselli, Jul 10 2018
Partial sums of A317311. - Omar E. Pol, Jul 28 2018

Links

Crossrefs

Cf. A051867 (15-gonal numbers), A316672, A317311.
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), this sequence (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).

Programs

  • GAP
    a:=[0,1,12,15,37];;  for n in [6..60] do a[n]:=a[n-1]+2*a[n-2]-2*a[n-3]-a[n-4]+a[n-5]; od; a; # Muniru A Asiru, Jul 10 2018
  • Mathematica
    LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 12, 15, 37}, 56]
Table[(26 n^2 + 26 n + 9 (-1)^n (2 n + 1) - 9)/16, {n, 0, 55}]
  • PARI
    concat(0, Vec(x*(1+11*x+x^2)/((1-x)^3*(1+x)^2) + O(x^99))) \\ Altug Alkan, Oct 01 2016

Formula

G.f.: x*(1 + 11*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).
a(n) = (26*n^2 + 26*n + 9*(-1)^n*(2*n+1) - 9)/16.
Sum_{n>=1} 1/a(n) = 26/121 + 2*Pi*cot(2*Pi/13)/11 = 1.3032041594895857... . - Vaclav Kotesovec, Oct 05 2016

A303813 Generalized 19-gonal (or enneadecagonal) numbers: m*(17*m - 15)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

Original entry on oeis.org

0, 1, 16, 19, 49, 54, 99, 106, 166, 175, 250, 261, 351, 364, 469, 484, 604, 621, 756, 775, 925, 946, 1111, 1134, 1314, 1339, 1534, 1561, 1771, 1800, 2025, 2056, 2296, 2329, 2584, 2619, 2889, 2926, 3211, 3250, 3550, 3591, 3906, 3949, 4279, 4324, 4669, 4716, 5076, 5125, 5500, 5551, 5941, 5994, 6399
Offset: 0

Views

Author

Omar E. Pol, Jun 06 2018

Keywords

Comments

Numbers k for which 136*k + 225 is a square. - Bruno Berselli, Jul 10 2018
Partial sums of A317315. - Omar E. Pol, Jul 28 2018

Links

Crossrefs

Cf. A051871, A316672, A317315.
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), this sequence (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).

Programs

  • GAP
    a:=[0,1,16,19,49];;  for n in [6..60] do a[n]:=a[n-1]+2*a[n-2]-2*a[n-3]-a[n-4]+a[n-5]; od; a; # Muniru A Asiru, Jul 10 2018
  • Mathematica
    With[{nn = 54}, {0}~Join~Riffle[Array[PolygonalNumber[19, #] &, Ceiling[nn/2]], Array[PolygonalNumber[19, -#] &, Ceiling[nn/2]]]] (* Michael De Vlieger, Jun 06 2018 *)
CoefficientList[ Series[-x (x^2 + 15x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 16, 19, 49}, 51] (* Robert G. Wilson v, Jul 28 2018 *)
  • PARI
    concat(0, Vec(x*(1 + 15*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Jun 08 2018

Formula

From Colin Barker, Jun 08 2018: (Start)
G.f.: x*(1 + 15*x + x^2) / ((1 - x)^3*(1 + x)^2).
a(n) = (34*n^2 + 60*n)/16 for n even.
a(n) = (34*n^2 + 8*n - 26)/16 for n odd.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
(End)

A303815 Generalized 29-gonal (or icosienneagonal) numbers: m*(27*m - 25)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

Original entry on oeis.org

0, 1, 26, 29, 79, 84, 159, 166, 266, 275, 400, 411, 561, 574, 749, 764, 964, 981, 1206, 1225, 1475, 1496, 1771, 1794, 2094, 2119, 2444, 2471, 2821, 2850, 3225, 3256, 3656, 3689, 4114, 4149, 4599, 4636, 5111, 5150, 5650, 5691, 6216, 6259, 6809, 6854, 7429, 7476, 8076
Offset: 0

Views

Author

Omar E. Pol, Jun 06 2018

Keywords

Comments

Numbers k such that 216*k + 625 is a square. - Bruno Berselli, Jun 08 2018
Partial sums of A317325.

Links

Crossrefs

Cf. A255187, A277990 (see the third comment), A316672, A317325.
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), this sequence (k=29), A316729 (k=30).

Programs

  • Mathematica
    Table[(54 n (n + 1) + 23 (2 n + 1) (-1)^n - 23)/16, {n, 0, 50}] (* Bruno Berselli, Jun 07 2018 *)
CoefficientList[ Series[-x (x^2 + 25x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 26, 29, 79, 84}, 50] (* Robert G. Wilson v, Jul 28 2018 *)
With[{nn=25},Riffle[Table[1-(29x)/2+(27x^2)/2,{x,nn}],PolygonalNumber[ 29,Range[ nn]]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 26 2020 *)
  • PARI
    concat(0, Vec(x*(1 + 25*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^40))) \\ Colin Barker, Jun 12 2018

Formula

From Bruno Berselli, Jun 07 2018: (Start)
G.f.: x*(1 + 25*x + x^2)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n-1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = (54*n*(n + 1) + 23*(2*n + 1)*(-1)^n - 23)/16. Therefore:
a(n) = n*(27*n + 50)/8, if n is even, or (n + 1)*(27*n - 23)/8 otherwise.
2*(2*n - 1)*a(n) + 2*(2*n + 1)*a(n-1) - n*(27*n^2 - 25) = 0. (End)
Sum_{n>=1} 1/a(n) = 2*(27 + 25*Pi*cot(2*Pi/27))/625. - Amiram Eldar, Mar 01 2022

A303298 Generalized 21-gonal (or icosihenagonal) numbers: m*(19*m - 17)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

Original entry on oeis.org

0, 1, 18, 21, 55, 60, 111, 118, 186, 195, 280, 291, 393, 406, 525, 540, 676, 693, 846, 865, 1035, 1056, 1243, 1266, 1470, 1495, 1716, 1743, 1981, 2010, 2265, 2296, 2568, 2601, 2890, 2925, 3231, 3268, 3591, 3630, 3970, 4011, 4368, 4411, 4785, 4830, 5221, 5268, 5676, 5725, 6150, 6201, 6643, 6696, 7155, 7210
Offset: 0

Views

Author

Omar E. Pol, Jun 23 2018

Keywords

Comments

Numbers k for which 152*k + 289 is a square. - Bruno Berselli, Jul 10 2018
Partial sums of A317317. - Omar E. Pol, Jul 28 2018

Links

Crossrefs

Cf. A051873, A316672, A317317.
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), this sequence (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).

Programs

  • GAP
    a:=[0,1,18,21,55];;  for n in [6..60] do a[n]:=a[n-1]+2*a[n-2]-2*a[n-3]-a[n-4]+a[n-5]; od; a; # Muniru A Asiru, Jul 10 2018
  • Maple
    a:= n-> (m-> m*(19*m-17)/2)(-ceil(n/2)*(-1)^n):
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 23 2018
  • Mathematica
    CoefficientList[Series[-(x^2 + 17 x + 1) x/((x + 1)^2*(x - 1)^3), {x, 0, 55}], x] (* or *)
Array[PolygonalNumber[21, (1 - 2 Boole[EvenQ@ #]) Ceiling[#/2]] &, 56, 0] (* Michael De Vlieger, Jul 10 2018 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 18, 21, 55}, 51] (* Robert G. Wilson v, Jul 28 2018 *)
  • PARI
    concat(0, Vec(x*(1 + 17*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jun 24 2018

Formula

G.f.: -(x^2+17*x+1)*x/((x+1)^2*(x-1)^3). - Alois P. Heinz, Jun 23 2018
From Colin Barker, Jun 24 2018: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = (19*n^2 + 34*n) / 8 for n even.
a(n) = (19*n^2 + 4*n - 15) / 8 for n odd.
(End)
Sum_{n>=1} 1/a(n) = 38/289 + 2*Pi*cot(2*Pi/19)/17. - Amiram Eldar, Feb 28 2022
Showing 1-6 of 6 results.

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