A057145 -id:A057145 - cp's OEIS frontend

A139613 a(n) = 78*n + 13.

13, 91, 169, 247, 325, 403, 481, 559, 637, 715, 793, 871, 949, 1027, 1105, 1183, 1261, 1339, 1417, 1495, 1573, 1651, 1729, 1807, 1885, 1963, 2041, 2119, 2197, 2275, 2353, 2431, 2509, 2587, 2665, 2743, 2821, 2899, 2977, 3055, 3133

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 13th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 13th column in the square array A057145.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016921, A057145, A139600.

[13*(6*n + 1): n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

78*Range[0, 50] + 13 (* Paolo Xausa, Apr 04 2024 *)

a(n)=78*n+13 \\ Charles R Greathouse IV, Oct 05 2011

From Elmo R. Oliveira, Apr 03 2024: (Start)
G.f.: 13*(1+5*x)/(x-1)^2.
E.g.f.: 13*exp(x)*(1 + 6*x).
a(n) = 13*A016921(n).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A139617 a(n) = 136*n + 17.

Original entry on oeis.org

17, 153, 289, 425, 561, 697, 833, 969, 1105, 1241, 1377, 1513, 1649, 1785, 1921, 2057, 2193, 2329, 2465, 2601, 2737, 2873, 3009, 3145, 3281, 3417, 3553, 3689, 3825, 3961, 4097, 4233, 4369, 4505, 4641, 4777, 4913, 5049, 5185, 5321

Offset: 0

Omar E. Pol, May 21 2008

Numbers of the 17th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 17th column in the square array A057145.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017077, A057145, A139600.

[17*(8*n + 1): n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

136*Range[0,40]+17  (* or *) LinearRecurrence[{2,-1},{17,153},40] (* Harvey P. Dale, Jul 27 2015 *)

a(n)=136*n+17 \\ Charles R Greathouse IV, Oct 05 2011

From Elmo R. Oliveira, Apr 04 2024: (Start)
G.f.: 17*(1+7*x)/(x-1)^2.
E.g.f.: 17*exp(x)*(1 + 8*x).
a(n) = 17*A017077(n).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A344410 a(n) = (3n^2 - 1) (3n^2 - 2) (3n^3 - 3n + 1)/2.

Original entry on oeis.org

1, 1, 1045, 23725, 195661, 975061, 3578401, 10680265, 27453385, 63016921, 132361021, 258815701, 477132085, 837244045, 1408778281, 2286380881, 3595928401, 5501691505, 8214519205, 12001111741, 17194450141, 24205450501, 33535911025, 45792819865, 61704091801

Offset: 0

Seiichi Manyama, May 17 2021

a(n) is both (3*n+2)-gonal number and (3*n+2)-gonal pyramidal number.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Numberphile, Cannon Ball Numbers.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A016789, A027669, A057145, A344376, A344377.

Table[PolygonalNumber[3*n + 2, 3*n^3 - 3*n + 1], {n, 0, 24}] (* Amiram Eldar, May 17 2021 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,1,1045,23725,195661,975061,3578401,10680265},30] (* Harvey P. Dale, Aug 10 2021 *)

a(n) = (3*n^2-1)*(3*n^2-2)*(3*n^3-3*n+1)/2;

p(k, n) = n*((k-2)*n-k+4)/2;
a(n) = p(3*n+2, 3*n^3-3*n+1);

my(N=40, x='x+O('x^N)); Vec((1-7*x+1065*x^2+15337*x^3+35135*x^4+15567*x^5+943*x^6-x^7)/(1-x)^8)

Let p(k,m) = A057145(k,m) denote m-th k-gonal number. Then
  a(n) = p(3*n+2, 3*n^3-3*n+1);
  a(n) = Sum_{j=1..3*n^2-2} p(3*n+2, j) for n > 0.
G.f.: (1-7*x+1065*x^2+15337*x^3+35135*x^4+15567*x^5+943*x^6-x^7)/(1-x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Wesley Ivan Hurt, Sep 05 2022

A350207 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero n-gonal numbers in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

25, 65, 144, 305, 456, 622, 731, 1006, 1434, 1499, 1711, 1806, 2446, 2742, 3001, 3051, 3544, 3699, 3962, 4345, 5362, 5039, 5756, 5712, 6251, 6655, 7399, 7698, 7591, 8304, 8673, 9241, 9362, 9299, 10312, 10606, 11121, 10736, 12113, 12737, 12934

Offset: 3

Ilya Gutkovskiy, Dec 19 2021

nonn

Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A057145, A097563, A274046, A350196, A350197, A350198, A350199, A350209, A350210.

A374144 a(n) is the smallest number which can be represented as the sum of two distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

81, 1105, 205427, 483031, 9402323, 6232341, 79324200, 768459127, 2265692766, 2413112833, 6737406626, 150437989675, 45319359337, 15140186701

Offset: 3

Ilya Gutkovskiy, Jun 28 2024

			a(3) = 81 = 3 + 78 = 15 + 66 = 36 + 45.

Eric Weisstein's World of Mathematics, Polygonal Number
Michael S. Branicky, Python program for A374144

Cf. A057145, A093195, A332989, A342326, A350405, A374141, A374142, A374143.

Python
```
# see linked program
```

a(9)-a(16) from Michael S. Branicky, Jun 30 2024

A139607 a(n) = 21*n + 7.

Original entry on oeis.org

7, 28, 49, 70, 91, 112, 133, 154, 175, 196, 217, 238, 259, 280, 301, 322, 343, 364, 385, 406, 427, 448, 469, 490, 511, 532, 553, 574, 595, 616, 637, 658, 679, 700, 721, 742, 763, 784, 805, 826, 847, 868, 889, 910, 931, 952, 973, 994

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 7th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

7th transversal numbers (or 7-transversal numbers): (A000217(7)-7)*n + 7.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. - From N. J. A. Sloane, Dec 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000217, A008603, A016777, A057145, A139600, A152744.

[21*n+7: n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

Range[7, 1500, 21] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)
21*Range[0,50]+7 (* or *) LinearRecurrence[{2,-1},{7,28},50] (* Harvey P. Dale, Feb 23 2020 *)

a(n)=21*n+7 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = A057145(n+2,7).
G.f.: 7*(1+2*x)/(x-1)^2. - R. J. Mathar, Jul 28 2016
From Elmo R. Oliveira, Apr 12 2024: (Start)
E.g.f.: 7*exp(x)*(1 + 3*x).
a(n) = 7*A016777(n) = A008603(n) + 7 = A152744(n+1) - A152744(n).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A139608 a(n) = 28*n + 8.

Original entry on oeis.org

8, 36, 64, 92, 120, 148, 176, 204, 232, 260, 288, 316, 344, 372, 400, 428, 456, 484, 512, 540, 568, 596, 624, 652, 680, 708, 736, 764, 792, 820, 848, 876, 904, 932, 960, 988, 1016, 1044, 1072, 1100, 1128, 1156, 1184, 1212, 1240, 1268, 1296, 1324, 1352, 1380

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 8th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. - From N. J. A. Sloane, Dec 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017005, A057145, A135628, A139600, A316466.

[4*(7*n + 2): n in [0..50]]; // Vincenzo Librandi, Jul 13 2011

Range[8, 7000, 28] (* Vladimir Joseph Stephan Orlovsky, Jul 13 2011 *)
LinearRecurrence[{2,-1},{8,36},50] (* or *) NestList[28+#&,8,50] (* Harvey P. Dale, Dec 14 2012 *)

a(n)=28*n+8 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = A057145(n+2,8).
a(n) = 2*a(n-1) - a(n-2); a(0)=8, a(1)=36. - Harvey P. Dale, Dec 14 2012
G.f.: 4*(2+5*x)/(x-1)^2. - R. J. Mathar, Jul 28 2016
From Elmo R. Oliveira, Apr 16 2024: (Start)
E.g.f.: 4*exp(x)*(2 + 7*x).
a(n) = 4*A017005(n) = A135628(n) + 8 = A316466(n+1) - A316466(n). (End)

A139610 a(n) = 45*n + 10.

Original entry on oeis.org

10, 55, 100, 145, 190, 235, 280, 325, 370, 415, 460, 505, 550, 595, 640, 685, 730, 775, 820, 865, 910, 955, 1000, 1045, 1090, 1135, 1180, 1225, 1270, 1315, 1360, 1405, 1450, 1495, 1540, 1585, 1630, 1675, 1720, 1765, 1810, 1855, 1900

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 10th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017185, A057145, A062708, A139600.

[5*(9*n + 2): n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

45*Range[0,50]+10 (* or *) LinearRecurrence[{2,-1},{10,55},50] (* Harvey P. Dale, Dec 27 2021 *)

a(n)=45*n+10 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = A057145(n+2,10).
G.f.: 5*(2+7*x)/(x-1)^2. - R. J. Mathar, Jul 28 2016
From Elmo R. Oliveira, Apr 16 2024: (Start)
E.g.f.: 5*exp(x)*(2 + 9*x).
a(n) = 5*A017185(n) = 5*(A062708(n+1) - A062708(n)).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A139611 a(n) = 55*n + 11.

Original entry on oeis.org

11, 66, 121, 176, 231, 286, 341, 396, 451, 506, 561, 616, 671, 726, 781, 836, 891, 946, 1001, 1056, 1111, 1166, 1221, 1276, 1331, 1386, 1441, 1496, 1551, 1606, 1661, 1716, 1771, 1826, 1881, 1936, 1991, 2046, 2101, 2156, 2211, 2266

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 11th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000566, A016861, A057145, A139600.

[11*(5*n + 1): n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

A139611:=n->55*n + 11; seq(A139611(n), n=0..50); # Wesley Ivan Hurt, Apr 16 2014

Table[55 n + 11, {n, 0, 50}] (* Wesley Ivan Hurt, Apr 16 2014 *)

a(n)=55*n+11 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = A057145(n+2,11).
a(n) = 11 * A016861(n). - Wesley Ivan Hurt, Apr 16 2014
G.f.: 11*(1+4*x)/(x-1)^2 . - R. J. Mathar, Jul 28 2016
From Elmo R. Oliveira, Apr 16 2024: (Start)
E.g.f.: 11*exp(x)*(1 + 5*x).
a(n) = 11*(A000566(n+1) - A000566(n)).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A139612 a(n) = 66*n + 12.

Original entry on oeis.org

12, 78, 144, 210, 276, 342, 408, 474, 540, 606, 672, 738, 804, 870, 936, 1002, 1068, 1134, 1200, 1266, 1332, 1398, 1464, 1530, 1596, 1662, 1728, 1794, 1860, 1926, 1992, 2058, 2124, 2190, 2256, 2322, 2388, 2454, 2520, 2586, 2652, 2718

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 12th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 12th column in the square array A057145.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017413, A057145, A139600.

[6*(11*n + 2): n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

66*Range[0,50]+12 (* or *) LinearRecurrence[{2,-1},{12,78},50] (* Harvey P. Dale, Jun 14 2017 *)

a(n)=66*n+12 \\ Charles R Greathouse IV, Oct 05 2011

From Elmo R. Oliveira, Apr 03 2024: (Start)
G.f.: 6*(2+9*x)/(x-1)^2.
E.g.f.: 6*exp(x)*(2 + 11*x).
a(n) = 6*A017413(n).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

cp's OEIS Frontend

A139613 a(n) = 78*n + 13.

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A139617 a(n) = 136*n + 17.

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A344410 a(n) = (3*n^2 - 1) * (3*n^2 - 2) * (3*n^3 - 3*n + 1)/2.

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A350207 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero n-gonal numbers in exactly n ways, or 0 if no such integer exists.

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A374144 a(n) is the smallest number which can be represented as the sum of two distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

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A139607 a(n) = 21*n + 7.

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A139608 a(n) = 28*n + 8.

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A344410 a(n) = (3n^2 - 1) (3n^2 - 2) (3n^3 - 3n + 1)/2.