A057145 -id:A057145 - cp's OEIS frontend

A139614 a(n) = 91*n + 14.

14, 105, 196, 287, 378, 469, 560, 651, 742, 833, 924, 1015, 1106, 1197, 1288, 1379, 1470, 1561, 1652, 1743, 1834, 1925, 2016, 2107, 2198, 2289, 2380, 2471, 2562, 2653, 2744, 2835, 2926, 3017, 3108, 3199, 3290, 3381, 3472, 3563, 3654

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 14th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 14th 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. A057145, A139600, A153080.

[7*(13*n + 2): n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

91*Range[0,40]+14 (* or *) LinearRecurrence[{2,-1},{14,105},50] (* Harvey P. Dale, Feb 25 2015 *)

a(n)=91*n+14 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = 2*a(n-1) - a(n-2); a(0)=14, a(1)=105. - Harvey P. Dale, Feb 25 2015
From Elmo R. Oliveira, Apr 04 2024: (Start)
G.f.: 7*(2+11*x)/(x-1)^2.
E.g.f.: 7*exp(x)*(2 + 13*x).
a(n) = 7*A153080(n). (End)

A139615 a(n) = 105*n + 15.

Original entry on oeis.org

15, 120, 225, 330, 435, 540, 645, 750, 855, 960, 1065, 1170, 1275, 1380, 1485, 1590, 1695, 1800, 1905, 2010, 2115, 2220, 2325, 2430, 2535, 2640, 2745, 2850, 2955, 3060, 3165, 3270, 3375, 3480, 3585, 3690, 3795, 3900, 4005, 4110, 4215

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 15th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 15th 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. A016993, A057145, A139600.

[15*(7*n + 1): n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

105*Range[0,40]+15 (* or *) LinearRecurrence[{2,-1},{15,120},50] (* Harvey P. Dale, May 07 2016 *)

a(n)=105*n+15 \\ Charles R Greathouse IV, Oct 05 2011

From Elmo R. Oliveira, Apr 04 2024: (Start)
G.f.: 15*(1+6*x)/(x-1)^2.
E.g.f.: 15*exp(x)*(1 + 7*x).
a(n) = 15*A016993(n).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A139616 a(n) = 120*n + 16.

Original entry on oeis.org

16, 136, 256, 376, 496, 616, 736, 856, 976, 1096, 1216, 1336, 1456, 1576, 1696, 1816, 1936, 2056, 2176, 2296, 2416, 2536, 2656, 2776, 2896, 3016, 3136, 3256, 3376, 3496, 3616, 3736, 3856, 3976, 4096, 4216, 4336, 4456, 4576, 4696, 4816

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 16th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 16th 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. A008597, A057145, A139600.

[8*(15*n + 2): n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

a(n)=120*n+16 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = 2*a(n-1)-a(n-2). - Wesley Ivan Hurt, Jun 17 2021
From Elmo R. Oliveira, Apr 04 2024: (Start)
G.f.: 8*(2+13*x)/(x-1)^2.
E.g.f.: 8*exp(x)*(2 + 15*x).
a(n) = 8*(A008597(n) + 2). (End)

A139619 a(n) = 171*n + 19.

Original entry on oeis.org

19, 190, 361, 532, 703, 874, 1045, 1216, 1387, 1558, 1729, 1900, 2071, 2242, 2413, 2584, 2755, 2926, 3097, 3268, 3439, 3610, 3781, 3952, 4123, 4294, 4465, 4636, 4807, 4978, 5149, 5320, 5491, 5662, 5833, 6004, 6175, 6346, 6517, 6688

Offset: 0

Omar E. Pol, May 21 2008

Numbers of the 19th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 19th 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. A016957, A017329, A057145, A139600, A139606-A139618, A139620.

Cf. A017173, A051682.

[19*(9*n + 1): n in [0..50]]; // Vincenzo Librandi, Jul 23 2011

Range[0, 100]*171 + 19 (* Paolo Xausa, Apr 17 2024 *)

a(n)=171*n+19 \\ Charles R Greathouse IV, Oct 05 2011

From Chai Wah Wu, Apr 14 2017: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 1.
G.f.: (152*x + 19)/(x - 1)^2. (End)
From Elmo R. Oliveira, Apr 11 2024: (Start)
E.g.f.: 19*exp(x)*(1 + 9*x).
a(n) = 19*A017173(n) = 19*(A051682(n+1) - A051682(n)). (End)

A180266 a(0) = 0; a(n) = C(2n-2,n-1)(n^2-2*n+2)/n for n >= 1.

Original entry on oeis.org

0, 1, 2, 10, 50, 238, 1092, 4884, 21450, 92950, 398684, 1696396, 7171892, 30161740, 126293000, 526864680, 2191034970, 9086921190, 37596989100, 155232577500, 639749274780, 2632212288420, 10814090022840, 44369043365400

Offset: 0

Robert G. Wilson v, Aug 22 2010

nonn

We may define Figurate Numbers F(r,n,d) with rank r, index n in dimension d as F(r,n,d) = binomial(r+d-2,d-1) *((r-1)*(n-2)+d) /d. These are polygonal numbers A057145 or A086271 in d=2, pyramidal numbers A080851 in d=3, and 4D pyramidal numbers A080852 in d=4, for example.
This sequence here is a(n) = F(n,n,n), the n-th n-gonal figurate number in n dimensions.
Limit_{n -> infinity} a(n+1)/a(n) = 4. - Robert G. Wilson v, Oct 30 2013

Albert H. Beiler, Recreations in the Theory of Numbers, The Queen of Mathematics Entertains, Second Edition, Dover, New York, 1966, Chptr. XVIII Ball Games, p. 196.

Cf. A000984, A024483, A057145, A080851, A080852, A086271.

Figurate[ngon_, rank_, dim_] := Binomial[rank + dim - 2, dim - 1] ((rank - 1)*(ngon - 2) + dim)/dim; Table[ Figurate[n, n, n], {n, 50}]
Join[{0},Table[Binomial[2n-2,n-1] (n^2-2n+2)/n,{n,30}]] (* Harvey P. Dale, Sep 22 2019 *)

a(n) = A000984(n-1) + (n-1)*A024483(n). [R. J. Mathar, Nov 18 2010]
From Ilya Gutkovskiy, Mar 29 2018: (Start)
O.g.f.: 1 - (1 - 7*x + 10*x^2)/(1 - 4*x)^(3/2).
E.g.f.: 1 - exp(2*x)*((1 - 3*x)*BesselI(0,2*x) + 2*x*BesselI(1,2*x)).
a(n) = [x^n] x*(1 - 3*x + n*x)/(1 - x)^(n+1). (End)

A379973 Least k >= 3 such that A373711(n) is both k-gonal and k-gonal pyramidal.

Original entry on oeis.org

3, 3, 3, 3, 10, 14, 6, 8, 3, 4, 8, 3, 30, 11, 88, 14, 43, 50, 276, 17, 322, 20, 23, 26, 41, 29, 145, 32, 823, 35, 2378, 38, 41, 44, 47, 50, 53, 56, 59, 374, 62, 65, 2386, 68, 71, 74

Offset: 1

Pontus von Brömssen, Jan 08 2025

For n <= 46, there is a unique k >= 3 such that A373711(n) is both k-gonal and k-gonal pyramidal. If this were true for all n, A027669 would be the sorted distinct terms of this sequence.

Cf. A027669, A057145, A080851, A373711, A379974, A379975.

A057145(a(n),A379974(n)) = A080851(a(n)-2,A379975(n)-1) = A373711(n).

A379974 A373711(n) is equal to the a(n)-th A379973(n)-gonal number.

Original entry on oeis.org

0, 1, 4, 15, 7, 9, 22, 19, 55, 70, 45, 119, 41, 73, 34, 181, 110, 115, 77, 361, 86, 631, 1009, 1513, 1683, 2161, 1191, 2971, 694, 3961, 604, 5149, 6553, 8191, 10081, 12241, 14689, 17443, 20521, 9000, 23941, 27721, 4970, 31879, 36433, 41401

Offset: 1

Pontus von Brömssen, Jan 08 2025

Indices to polygonal numbers are chosen so that the first k-gonal number is 1 (and the zeroth is 0).

Cf. A057145, A373711, A379973, A379975.

A057145(A379973(n),a(n)) = A373711(n).

A027696 Numbers k >= 2 such that for some m >= 2, the sum of the first m k-gonal numbers is again a k-gonal number, excluding the parametric solution m = (k^2-4*k-2)/3 when k==2 (mod 3).

Original entry on oeis.org

3, 4, 6, 8, 10, 11, 14, 17, 30, 41, 43, 50, 60, 88, 145, 276, 322, 374, 823, 1152

Offset: 1

Masanobu Kaneko (mkaneko(AT)math.kyushu-u.ac.jp)

The parametric solution: if k==2 (mod 3) and k >= 5, the sum of the first (k^2-4*k-2)/3 k-gonal numbers is the ((k^3-6*k^2+3*k+19)/9)-th k-gonal number A057145(k,(k^3-6*k^2+3*k+19)/9) = A344410((k-2)/3).
2378, 2386, and 31265 are also terms. See link "Cannon Ball Numbers". - Pontus von Brömssen, Jan 08 2025
Number k is a term iff the elliptic curve (3*k-6)*y^2 - (3*k-12)*y = (k-2)*x^3 + 3*x^2 - (k-5)*x has an integral point with x >= 2 different from (k^2-4*k-2)/3. The listed values may be incomplete. For example, I was not able to verify that k = 273 is not a term. - Max Alekseyev, Feb 27 2025

Brady Haran and Matt Parker, Cannon Ball Numbers, Numberphile (2019).

Cf. A027669, A057145, A344410, A373711.

More terms from Masanobu Kaneko (mkaneko(AT)math.kyushu-u.ac.jp), Jan 05 1998

Name clarified by Max Alekseyev, Feb 27 2025

A167149 10000-gonal numbers: a(n) = n + 4999 * n * (n-1).

Original entry on oeis.org

0, 1, 10000, 29997, 59992, 99985, 149976, 209965, 279952, 359937, 449920, 549901, 659880, 779857, 909832, 1049805, 1199776, 1359745, 1529712, 1709677, 1899640, 2099601, 2309560, 2529517, 2759472, 2999425, 3249376, 3509325, 3779272, 4059217, 4349160, 4649101

Offset: 0

Michael G. Fenner (sidk.20c(AT)gmail.com), Oct 28 2009

There are infinitely many 10000-gonal numbers that are also squares. The first seven are at n = 0, 1, 2, 21, 9800, 173774514938177, 1042188013912456. - Muniru A Asiru, Apr 10 2016

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

Cf. A057145. - R. J. Mathar, Nov 02 2009

A167149:=List([1..10^2],n->n+499*n*(n-1)); # Muniru A Asiru, Sep 27 2017

P := proc(n,k) n*((k-2)*n-k+4)/2 ; end: A167149 := proc(n) P(n,10000) ; end: seq(A167149(n),n=0..50) ; # R. J. Mathar, Nov 02 2009

Table[n + 4999 n (n - 1), {n, 0, 31}] (* or *)
CoefficientList[Series[x (1 + 9997 x)/(1 - x)^3, {x, 0, 31}], x] (* Michael De Vlieger, Apr 10 2016 *)
LinearRecurrence[{3, -3, 1}, {0, 1, 10000}, 10] (* G. C. Greubel, Jun 04 2016 *)

x='x+O('x^99); concat(0, Vec(x*(1+9997*x)/(1-x)^3)) \\ Altug Alkan, Apr 10 2016

From R. J. Mathar, Nov 02 2009: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(1 + 9997*x)/(1-x)^3. (End)
E.g.f.: exp(x)*x*(1 + 4999*x). - Ilya Gutkovskiy, Apr 10 2016

Edited (but not checked) by N. J. A. Sloane, Nov 01 2009

Sequence extended by R. J. Mathar, Nov 02 2009

A177201 Multiples of nontrivially polygonal numbers A090466.

Original entry on oeis.org

6, 9, 10, 12, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 34, 35, 36, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 78, 80, 81, 82, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Jonathan Vos Post, May 04 2010

Multiples of numbers in the array of A057145 below the second row (which has every positive integer) and right of the 2nd column (which has every positive integer). That is, multiples of every triangular number >3, every square >4, every pentagonal number >5, every hexagonal number >6, every heptagonal number >7, every octagonal number >8, every 9-gonal (nonagonal) number >9, and so forth.

Nontrivially polygonal numbers {6, 9, 10, 12, 15, 16, 18, 21, 22, 24, 25, 27, 28, 30, 33, 34, 35, 36, 39, 40, 42, 45, 46, 48, 49,...} UNION 2*nontrivially polygonal = {12, 18, 20, 24, 30, 32, 36, 42, 44, 48, 50, 54, 56, 60, 66, 68, 70, 72, 78, 80, 84, 90, 92, 96, 98, ...} UNION 3*nontrivially polygonal = {18, 27, 30, 36, 45, 48, 54, 63, 66, 72, 75, 81, 84, 90, 99, ...} UNION 4*nontrivially polygonal = {24, 36, 40, 48, 60, 64, 72, 84, 88, 96, 100, ...} UNION 5*nontrivially polygonal = {30, 45, 50, 60, 75, 80, 90, ...} UNION 6*nontrivially polygonal = {36, 54, 60, 72, 90, 96, ...} UNION 7*nontrivially polygonal = {42, 63, 70, 84, ...} and so forth.

Cf. A057145, A090466, complement is A177202.

{A057145 * j for j = 1, 2, 3, 4, 5, ...} = {j * ((n-2)*k^2-(n-4)*k)/2, j>0, n > 2, k > 2}.

cp's OEIS Frontend

A139614 a(n) = 91*n + 14.

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A139615 a(n) = 105*n + 15.

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A139616 a(n) = 120*n + 16.

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A139619 a(n) = 171*n + 19.

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A180266 a(0) = 0; a(n) = C(2*n-2,n-1)*(n^2-2*n+2)/n for n >= 1.

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A379973 Least k >= 3 such that A373711(n) is both k-gonal and k-gonal pyramidal.

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A379974 A373711(n) is equal to the a(n)-th A379973(n)-gonal number.

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A027696 Numbers k >= 2 such that for some m >= 2, the sum of the first m k-gonal numbers is again a k-gonal number, excluding the parametric solution m = (k^2-4*k-2)/3 when k==2 (mod 3).

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A180266 a(0) = 0; a(n) = C(2n-2,n-1)(n^2-2*n+2)/n for n >= 1.