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.

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A139620 a(n) = 190*n + 20.

Original entry on oeis.org

20, 210, 400, 590, 780, 970, 1160, 1350, 1540, 1730, 1920, 2110, 2300, 2490, 2680, 2870, 3060, 3250, 3440, 3630, 3820, 4010, 4200, 4390, 4580, 4770, 4960, 5150, 5340, 5530, 5720, 5910, 6100, 6290, 6480, 6670, 6860, 7050, 7240, 7430
Offset: 0

Views

Author

Omar E. Pol, May 21 2008

Keywords

Comments

Numbers of the 20th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 20th column in the square array A057145.

Links

Crossrefs

Cf. A008601, A057145, A139600.

Programs

Formula

From Elmo R. Oliveira, Apr 08 2024: (Start)
G.f.: 10*(2+17*x)/(x-1)^2.
E.g.f.: 10*exp(x)*(2 + 19*x).
a(n) = 10*(A008601(n) + 2).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A321156 Numbers that have exactly 5 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

561, 1485, 1701, 2016, 2556, 2601, 2850, 3025, 3060, 3256, 3321, 4186, 4761, 4851, 5226, 5320, 5565, 5841, 6175, 6216, 6336, 6525, 6670, 7425, 7821, 7840, 8001, 8100, 8625, 8646, 9730, 9856, 9945, 9976, 10116, 10296, 10450, 10585, 11025, 11305, 11340, 12025, 12090
Offset: 1

Views

Author

Hugh Erling, Oct 28 2018

Keywords

Comments

n | 2*m where m is a term in this sequence. - David A. Corneth, Oct 29 2018

Examples

			561 has representations P(3, 188)=P(6, 39)=P(11, 12)=P(17, 6)=P(33, 3).
1485 has representations P(3, 496)=P(5, 150)=P(9, 43)=P(15, 16)=P(54, 3).
1701 has representations P(3, 568)=P(6, 115)=P(9, 49)=P(18, 13)=P(21, 10).

Links

Crossrefs

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321157, A321158, A321159, A321160, A320943.

Programs

  • PARI
    isok(n) = sum(k=3, n-1, ispolygonal(n, k)) == 5; \\ Michel Marcus, Oct 29 2018
  • PARI
    is(n) = my(d=divisors(n<<1)); sum(i=2, #d, k=2*(d[i]^2 - 2 * d[i] + n) / (d[i] - 1) / d[i]; k == k\1 && min(d[i], k) >=3) == 5 \\ David A. Corneth, Oct 29 2018

A321157 Numbers that have exactly 7 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

11935, 12376, 21736, 24220, 41041, 45441, 51360, 52326, 53361, 54145, 54405, 58311, 58696, 73360, 82720, 89425, 90321, 96580, 101025, 102025, 108801, 113050, 117216, 118405, 122265, 122500, 122760, 123201, 123256, 127281, 128961, 135201, 144585, 152076, 165376, 166635, 169456, 174097
Offset: 1

Views

Author

Hugh Erling, Oct 29 2018

Keywords

Examples

			11935 has representations P(n,k) = P(5, 1195) = P(7, 570) = P(10, 267) = P(14, 133) = P(35, 22) = P(55, 10) = P(154, 3).
12376 has representations P(n,k) = P(4, 2064) = P(7, 591) = P(16, 105) = P(26, 40) = P(34, 24) = P(56, 10) = P(91, 5).
21736 has representations P(n,k) = P(4, 3624) = P(8, 778) = P(11, 397) = P(16, 183) = P(19, 129) = P(22, 96) = P(208, 3).

Links

Crossrefs

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321156, A321158, A321159, A321160, A320943.

A321158 Numbers that have exactly 8 representations as a k-gonal number, P(m,k) = m*((k-2)*m - (k-4))/2, k and m >= 3.

Original entry on oeis.org

11781, 61776, 75141, 133056, 152361, 156520, 176176, 179740, 188650, 210925, 241605, 266085, 292825, 298936, 338625, 342585, 354025, 358281, 360801, 365365, 371925, 391392, 395200, 400960, 417340, 419805, 424270, 438516
Offset: 1

Views

Author

Hugh Erling, Oct 29 2018

Keywords

Examples

			a(1) 11781 has representations P(m,k) = P(3, 3928)=P(6, 787)=P(9,329)=P(11, 216)=P(21, 58)=P(63, 8)=P(77, 6)=P(153, 3).
a(2) 61776 has representations P(m,k) = P(3, 20593)=P(6, 4120)=P(8,2208)=P(11, 1125)=P(26, 192)=P(36, 100)=P(176, 6)=P(351, 3).
a(3) 75141 has representations P(m,k) = P(3, 25048)=P(6, 5011)=P(9,2089)=P(11, 1368)=P(18, 493)=P(27, 216)=P(66, 37)=P(69, 34).

Links

Crossrefs

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321156, A321157, A321159, A321160, A320943.

Programs

  • Mathematica
    r[n_] := Module[{k}, Sum[Boole[d >= 3 && (k = 2(d^2 - 2d + n)/(d^2 - d); IntegerQ[k] && k >= 3)], {d, Divisors[2n]}]];
Select[Range[500000], r[#] == 8&] (* Jean-François Alcover, Sep 23 2019, after Andrew Howroyd *)
  • PARI
    r(n)={sumdiv(2*n, d, if(d>=3, my(k=2*(d^2 - 2*d + n)/(d^2 - d)); !frac(k) && k>=3))}
for(n=1, 5*10^5, if(r(n)==8, print1(n, ", "))) \\ Andrew Howroyd, Nov 26 2018
  • Python
    # See link.

A321159 Numbers that have exactly 9 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

27405, 126225, 194481, 201825, 273105, 478401, 538461, 615681, 718641, 859600, 862785, 1056160, 1187145, 1257201, 1328481, 1413126, 1439361, 1532601, 1540540, 1619541, 1625625, 1708785, 1842400, 1849926, 1890945
Offset: 1

Views

Author

Hugh Erling, Oct 29 2018

Keywords

Examples

			a(1) 27405 has representations P(n,k) = P(3, 9136)=P(5, 2742)=P(9, 763)=P(14, 303)=P(18, 181)=P(27, 80)=P(35, 48)=P(63, 16)=P(105, 7).
a(2) 126225 has representations P(n,k) = P(3, 42076)=P(5, 12624)=P(9, 3508)=P(15, 1204)=P(17, 930)=P(33, 241)=P(50, 105)=P(99, 28)=P(225, 7).
a(3) 194481 has representations P(n,k) = P(3, 64828)=P(6, 12967)=P(9, 5404)=P(14, 2139)=P(18, 1273)=P(21, 928)=P(27, 556)=P(81, 62)=P(441, 4).

Links

Crossrefs

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321156, A321157, A321158, A321160, A320943.

Programs

  • PARI
    isok(n) = sum(k=3, n-1, ispolygonal(n, k)) == 9; \\ Michel Marcus, Nov 02 2018
  • Python
    # See Erling link.

A321160 Numbers that have exactly 10 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.

Original entry on oeis.org

220780, 519156, 1079001, 1154440, 1324576, 1447551, 2429505, 2454705, 2491776, 2603601, 2665125, 2700621, 2772225, 2953665, 3000025, 3086721, 3316600, 3665376, 4488561, 4741660, 5142501, 5388201, 5785101, 6076225
Offset: 1

Views

Author

Hugh Erling, Oct 29 2018

Keywords

Examples

			a(1) 220780 has representations P(n,k) = P(4, 36798) = P(7, 10515) = P(10, 4908) = P(14, 2428) = P(19, 1293) = P(28, 586) = P(35, 373) = P(38, 316) = P(40, 285) = P(664, 3).
a(2) 519156 has representations P(n,k) = P(3, 173053) = P(6, 34612) = P(8, 18543) = P(11, 9441) = P(27, 1481) = P(36, 826) = P(66, 244) = P(92, 126) = P(99, 109) = P(456, 7).
a(3) 1079001 has representations P(n,k) = P(3, 359668) = P(6, 71935) = P(9, 29974) = P(11, 19620) = P(14, 11859) = P(21, 5140) = P(27, 3076) = P(66, 505) = P(81, 335) = P(126, 139).

Links

Crossrefs

Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321156, A321157, A321158, A321159, A320943.

Programs

  • PARI
    isok(n) = sum(k=3, n-1, ispolygonal(n, k)) == 10; \\ Michel Marcus, Nov 02 2018
  • Python
    # See links.

A373711 Numbers that are simultaneously k-gonal and k-gonal pyramidal for some k >= 3.

Original entry on oeis.org

0, 1, 10, 120, 175, 441, 946, 1045, 1540, 4900, 5985, 7140, 23001, 23725, 48280, 195661, 245905, 314755, 801801, 975061, 1169686, 3578401, 10680265, 27453385, 55202400, 63016921, 101337426, 132361021, 197427385, 258815701, 432684460, 477132085, 837244045
Offset: 1

Views

Author

Kelvin Voskuijl, Jun 14 2024

Keywords

Comments

Matt Parker calls these numbers cannonball numbers, after the cannonball problem involving finding a number both square and square pyramidal.
If m==2 (mod 3), the m-gonal number A057145(m,(m^3-6*m^2+3*m+19)/9) = (m^2-4*m-2)*(m^2-4*m+1)*(m^3-6*m^2+3*m+19)/162 = A344410((m-2)/3) is a term. See comment in A027696. - Pontus von Brömssen, Dec 09 2024

Examples

			4900 is a term because it is both the 70th square and the 24th square pyramidal number.

Links

Crossrefs

Subsequences: A027568, A344376, A344410.
Cf. A027669, A027696, A057145, A080851, A379973, A379974, A379975.

Formula

a(n) = A057145(A379973(n),A379974(n)) = A080851(A379973(n)-2,A379975(n)-1). - Pontus von Brömssen, Jan 09 2025

Extensions

a(13)-a(33) from Pontus von Brömssen, Dec 08 2024

A374370 Square array read by antidiagonals: the n-th row lists n-gonal numbers that are products of smaller n-gonal numbers.

Original entry on oeis.org

1, 4, 1, 6, 36, 1, 8, 45, 16, 1, 9, 210, 36, 10045, 1, 10, 300, 64, 11310, 2850, 1, 12, 378, 81, 20475, 61776, 6426, 1, 14, 630, 100, 52360, 79800, 9828, 1408, 1, 15, 780, 144, 197472, 103740, 35224, 61920, 265926, 1, 16, 990, 196, 230300, 145530, 60606, 67200, 391950, 69300, 1
Offset: 2

Views

Author

Pontus von Brömssen, Jul 07 2024

Keywords

Comments

If there are only finitely many solutions for a certain value of n, the rest of that row is filled with 0's.
The first term in each row is 1, because 1 is an n-gonal number for every n and it equals the empty product.

Examples

			Array begins:
   n=2: 1,      4,       6,       8,       9,       10,       12,       14
   n=3: 1,     36,      45,     210,     300,      378,      630,      780
   n=4: 1,     16,      36,      64,      81,      100,      144,      196
   n=5: 1,  10045,   11310,   20475,   52360,   197472,   230300,   341055
   n=6: 1,   2850,   61776,   79800,  103740,   145530,   437580,   719400
   n=7: 1,   6426,    9828,   35224,   60606,  1349460,  2077992,  3333330
   n=8: 1,   1408,   61920,   67200,  276640,   297045,   870485,  1022000
   n=9: 1, 265926,  391950, 1096200, 1767546,  1787500,  9909504, 28123200
  n=10: 1,  69300, 1297890, 4257000, 5756400,  9140040,  9729720, 10648800
  n=11: 1,  79135,  792330, 2382380, 5570565, 15361500, 22230000, 49888395
  n=12: 1,   9504,   45696,  604128, 1981980,  2208465,  4798080, 13837824

Crossrefs

Cf. A057145, A374371 (second column), A374498.
Rows: A018252 (n=2), A068143 (n=3 except first term), A062312 (n=4), A374372 (n=5), A374373 (n=6).

A062707 Table by antidiagonals of n*k*(k+1)/2.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 3, 2, 0, 0, 6, 6, 3, 0, 0, 10, 12, 9, 4, 0, 0, 15, 20, 18, 12, 5, 0, 0, 21, 30, 30, 24, 15, 6, 0, 0, 28, 42, 45, 40, 30, 18, 7, 0, 0, 36, 56, 63, 60, 50, 36, 21, 8, 0, 0, 45, 72, 84, 84, 75, 60, 42, 24, 9, 0, 0, 55, 90, 108, 112, 105, 90, 70, 48, 27, 10, 0
Offset: 0

Views

Author

Henry Bottomley, Jul 11 2001

Keywords

Examples

			  0   0   0   0   0   0   0   0   0
  0   1   3   6  10  15  21  28  36
  0   2   6  12  20  30  42  56  72
  0   3   9  18  30  45  63  84 108
  0   4  12  24  40  60  84 112 144
  0   5  15  30  50  75 105 140 180
  0   6  18  36  60  90 126 168 216
  0   7  21  42  70 105 147 196 252
  0   8  24  48  80 120 168 224 288

Links

Crossrefs

Main diagonal is A002411. Sum of antidiagonals is A000332.

Programs

  • GAP
    Flat(List([0..12], n-> List([0..n], k-> k*Binomial(n-k+1,2)))); # G. C. Greubel, Sep 02 2019
  • Magma
    [k*Binomial(n-k+1,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 02 2019
  • Maple
    seq(seq(k*binomial(n-k+1,2), k=0..n), n=0..12); # G. C. Greubel, Sep 02 2019
  • Mathematica
    Table[k*Binomial[n-k+1, 2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 02 2019 *)
  • PARI
    T(n,k) = k*binomial(n-k+1,2);
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Sep 02 2019
  • Sage
    [[k*binomial(n-k+1,2) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Sep 02 2019

Formula

T(n, k) = T(n, 1)*T(1, k) = A001477(n)*A000217(k).
T(n, k) = A057145(n+2, k+1)-(k+1).

A139609 a(n) = 36*n + 9.

Original entry on oeis.org

9, 45, 81, 117, 153, 189, 225, 261, 297, 333, 369, 405, 441, 477, 513, 549, 585, 621, 657, 693, 729, 765, 801, 837, 873, 909, 945, 981, 1017, 1053, 1089, 1125, 1161, 1197, 1233, 1269, 1305, 1341, 1377, 1413, 1449, 1485, 1521, 1557, 1593, 1629, 1665, 1701
Offset: 0

Views

Author

Omar E. Pol, Apr 27 2008

Keywords

Comments

Numbers of the 9th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

Links

Crossrefs

Cf. A016813, A044102, A057145, A139600, A152994.

Programs

Formula

a(n) = A057145(n+2,9).
G.f.: 9*(1+3*x)/(x-1)^2. - R. J. Mathar, Jul 28 2016
From Elmo R. Oliveira, Apr 16 2024: (Start)
E.g.f.: 9*exp(x)*(1 + 4*x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 9*A016813(n) = A044102(n) + 9 = A152994(n+1) - A152994(n). (End)
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