A057145 -id:A057145 - cp's OEIS frontend

A256857 a(n) = n(n^2 + 3n - 2)/2.

0, 1, 8, 24, 52, 95, 156, 238, 344, 477, 640, 836, 1068, 1339, 1652, 2010, 2416, 2873, 3384, 3952, 4580, 5271, 6028, 6854, 7752, 8725, 9776, 10908, 12124, 13427, 14820, 16306, 17888, 19569, 21352, 23240, 25236, 27343, 29564, 31902, 34360, 36941, 39648, 42484, 45452, 48555

Offset: 0

Luciano Ancora, Apr 11 2015

The sequence is the 5th upper diagonal of the array in A139600.

Eric Weisstein's World of Mathematics, Polygonal Number
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A057145, A139600.

[n*(n^2+3*n-2)/2: n in [0..50]]; // Vincenzo Librandi, Apr 14 2015

Mathematica
```
Table[n (n^2 + 3 n - 2)/2, {n, 0, 40}]
```

vector(50, n, n--; n*(n^2+3*n-2)/2) \\ Bruno Berselli, Apr 15 2015

G.f.: x*(1 + 4*x -2*x^2)/(1 - x)^4.
a(n) = A057145(n+6,n). - R. J. Mathar, Jul 28 2016
a(n) = Sum_{i=1..n} (n-i-1) mod (n+1). - Wesley Ivan Hurt, Sep 15 2017
E.g.f.: exp(x)*x*(2 + 6*x + x^2)/2. - Stefano Spezia, Jan 20 2024

a(41)-a(45) from Stefano Spezia, Jan 20 2024

A275490 Square array of 5D pyramidal numbers, read by antidiagonals.

Original entry on oeis.org

1, 1, 5, 1, 6, 15, 1, 7, 21, 35, 1, 8, 27, 56, 70, 1, 9, 33, 77, 126, 126, 1, 10, 39, 98, 182, 252, 210, 1, 11, 45, 119, 238, 378, 462, 330, 1, 12, 51, 140, 294, 504, 714, 792, 495, 1, 13, 57, 161, 350, 630, 966, 1254, 1287, 715, 1, 14, 63, 182, 406, 756, 1218, 1716, 2079, 2002, 1001

Offset: 2

R. J. Mathar, Jul 30 2016

Let F(r,n,d) = binomial(r+d-2,d-1)* ((r-1)*(n-2)+d) /d be the d-dimensional pyramidal numbers. Then A(n,k) = F(k,n,5).

Sum of the n-th antidiagonal is binomial(n+4,7) + binomial(n+4,5) = A055797(n-1). - Mathew Englander, Oct 27 2020

			The array starts in rows n>=2 and columns k>=1 as
   1    5   15   35   70  126  210  330  495   715  1001  1365  1820
   1    6   21   56  126  252  462  792 1287  2002  3003  4368  6188
   1    7   27   77  182  378  714 1254 2079  3289  5005  7371 10556
   1    8   33   98  238  504  966 1716 2871  4576  7007 10374 14924
   1    9   39  119  294  630 1218 2178 3663  5863  9009 13377 19292
   1   10   45  140  350  756 1470 2640 4455  7150 11011 16380 23660
   1   11   51  161  406  882 1722 3102 5247  8437 13013 19383 28028
   1   12   57  182  462 1008 1974 3564 6039  9724 15015 22386 32396
   1   13   63  203  518 1134 2226 4026 6831 11011 17017 25389 36764

Michael De Vlieger, Table of n, a(n) for n = 2..11176 (rows 2 <= n <= 150, flattened)
Index to sequences related to polygonal numbers

Cf. Row sums of A080852 (4D), A080851 (3D), A057145 (2D), A077028 (1D).

Cf. A055797.

Table[Binomial[k + 3, 4] + (# - 2)*Binomial[k + 3, 5] &[m - k + 1], {m, 2, 12}, {k, m - 1}] // Flatten (* Michael De Vlieger, Nov 05 2020 *)

A(n+2,k) = Sum_{j=0..k-1} A080852(n,j).

A(n,k) = binomial(k+3,4) + (n-2)*binomial(k+3,5). - Mathew Englander, Oct 27 2020

A275709 a(n) = 2n^3 + 3n^2.

Original entry on oeis.org

0, 5, 28, 81, 176, 325, 540, 833, 1216, 1701, 2300, 3025, 3888, 4901, 6076, 7425, 8960, 10693, 12636, 14801, 17200, 19845, 22748, 25921, 29376, 33125, 37180, 41553, 46256, 51301, 56700, 62465, 68608, 75141, 82076, 89425, 97200, 105413, 114076, 123201, 132800, 142885

Offset: 0

Joshua Giambalvo, Aug 06 2016

Apart from the initial zero this sequence gives the 2nd pentagonal number, the 4th hexagonal number, the 6th heptagonal number, the 8th octagonal number, the 10th nonagonal number, etc. as well as the 5th nonnegative number, the 7th triangular number, the 9th square, the 11th pentagonal number, the 13th hexagonal number, etc. This is a reliable pattern that does not seem to appear on any other pairs of polygonal numbers (see link).

a(n) is the maximal determinant of a 3 X 3 matrix with integer elements from {1, ..., n+1}, so (for example) the maximum determinant of a 3 X 3 matrix with integer elements from {1, ..., 5} = det(1, 5, 5; 5, 1, 5; 5, 5, 1) = a(4) = 176. - Matthew Scroggs, Dec 31 2022

Carauleanu Marc and Colin Barker, Table of n, a(n) for n = 0..3030 (first 1000 terms from Colin Barker)
Joshua Giambalvo, Illustration of initial terms in a square array, Imgur, (2016).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000290, A005408, A005563, A033428, A033431, A057145, A139600.

[n^2*(2*n + 3): n in [0..30]]; // G. C. Greubel, Oct 19 2018

seq(2*n^3+3*n^2, n=0..30); # Robert Israel, Aug 09 2016

Table[2 n^3 + 3 n^2, {n, 0, 41}] (* or *)
CoefficientList[Series[x (5 + 8 x - x^2)/(1 - x)^4, {x, 0, 41}], x] (* Michael De Vlieger, Aug 11 2016 *)

concat(0, Vec(x*(5+8*x-x^2)/(1-x)^4 + O(x^50))) \\ Colin Barker, Aug 28 2016

a(n)=n^2*(2*n+3) \\ Charles R Greathouse IV, Aug 28 2016

for n in range(0,50): print(n**2*(2*n+3), end=' ') # Stefano Spezia, Oct 19 2018

From Colin Barker, Aug 06 2016: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.
G.f.: x*(5+8*x-x^2) / (1-x)^4. (End)
a(n) = A033431(n) + A033428(n). - Omar E. Pol, Aug 09 2016
a(n) = A000290(n) * A005408(n+1). - Robert Israel, Aug 09 2016
a(n) = A320047(1, n, 0). - Kolosov Petro, Oct 04 2018
E.g.f.: x*(5 + 9*x + 2*x^2)*exp(x). - G. C. Greubel, Oct 19 2018
From Amiram Eldar, Jan 21 2023: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/18 + 4*log(2)/9 - 16/27.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/36 + Pi/9 -2*log(2)/9 - 8/27. (End)

A294958 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*((k-2)^2+k)/2).

Original entry on oeis.org

1, 1, 3, 9, 28, 75, 198, 494, 1243, 3061, 7500, 18055, 43057, 101292, 236178, 545218, 1248480, 2835059, 6390360, 14298631, 31778782, 70168935, 153993321, 335977369, 728962258, 1573189113, 3377881482, 7217395643, 15348900996, 32494548816, 68494383520, 143773075158, 300568066729

Offset: 0

Ilya Gutkovskiy, Nov 12 2017

nonn

Euler transform of A060354.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index to sequences related to polygonal numbers

Cf. A057145, A060354, A294959.

N:=100:
S:= series(mul(1/(1 - x^k)^(k*((k-2)^2+k)/2),k=1..N),x,N+1):
seq(coeff(S,x,k),k=0..N); # Robert Israel, Nov 12 2017

nmax = 32; CoefficientList[Series[Product[1/(1 - x^k)^(k ((k - 2)^2 + k)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 ((d - 2)^2 + d)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A060354(k).

a(n) ~ exp(2*Zeta'(-1) + 3*Zeta(3) / (8*Pi^2) - Pi^16 / (1036800000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (36000 * Zeta(5)^2) - 2*Zeta(3)^2 / (15*Zeta(5)) + Zeta'(-3)/2 + (-Pi^12 / (3600000 * 2^(2/5) * 3^(1/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (150 * 2^(2/5) * 3^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (12000 * 2^(4/5) * 3^(2/5) * Zeta(5)^(7/5)) + 2^(1/5) * Zeta(3) / (3*Zeta(5))^(2/5)) * n^(2/5) - (Pi^4 / (60 * 2^(1/5) * (3*Zeta(5))^(3/5))) * n^(3/5) + (5*(3*Zeta(5))^(1/5) / 2^(8/5)) * n^(4/5)) * (3*Zeta(5))^(53/400) / (2^(47/200) * sqrt(5*Pi) * n^(253/400)). - Vaclav Kotesovec, Nov 12 2017

A294959 Expansion of Product_{k>=1} (1 + x^k)^(k*((k-2)^2+k)/2).

Original entry on oeis.org

1, 1, 2, 8, 23, 64, 160, 397, 968, 2372, 5714, 13617, 32007, 74396, 171222, 390629, 883922, 1984631, 4423528, 9790146, 21524829, 47027558, 102135967, 220565018, 473743833, 1012274948, 2152271718, 4554344649, 9593260912, 20118418061, 42012556671, 87375161720, 181001416773

Offset: 0

Ilya Gutkovskiy, Nov 12 2017

nonn

Weigh transform of A060354.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index to sequences related to polygonal numbers

Cf. A057145, A060354, A294958.

g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(((i-2)^2+i)*i/2, j)*g(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> g(n$2):
seq(a(n), n=0..35);  # Alois P. Heinz, Nov 12 2017

nmax = 32; CoefficientList[Series[Product[(1 + x^k)^(k ((k - 2)^2 + k)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 ((d - 2)^2 + d)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

G.f.: Product_{k>=1} (1 + x^k)^A060354(k).

a(n) ~ exp(-2401 * Pi^16 / (3499200000000 * Zeta(5)^3) + 49 * Pi^8 * Zeta(3) / (2700000 * Zeta(5)^2) - 2*Zeta(3)^2 / (25*Zeta(5)) + (-343*Pi^12 / (810000000 * 2^(3/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (750 * 2^(3/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (360000 * 2^(1/5) * 3^(4/5) * 5^(2/5) * Zeta(5)^(7/5)) + (3/2)^(1/5) * Zeta(3) / (5*Zeta(5))^(2/5)) * n^(2/5) - (7*Pi^4 / (180 * 2^(4/5) * 3^(1/5) * (5*Zeta(5))^(3/5))) * n^(3/5) + (3^(2/5) * 5^(6/5) * Zeta(5)^(1/5) / 2^(12/5)) * n^(4/5)) * 3^(1/5) * Zeta(5)^(1/10) / (2^(69/80) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - Vaclav Kotesovec, Nov 12 2017

A350423 a(n) is the smallest n-gonal number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly n ways, or -1 if none exists.

Original entry on oeis.org

190, 289, 330, 561

Offset: 3

Ilya Gutkovskiy, Dec 31 2021

			For n = 3: 190 = 1 + 36 + 153 = 15 + 55 + 120 = 21 + 78 + 91.

Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A057145, A350405.

A360762 a(n) is the least n-gonal number that is the sum of two or more consecutive nonzero n-gonal numbers in more than one way, or -1 if no such number exists.

Original entry on oeis.org

9, 12880, 20449, 10764222, 794629045, 33205080888, 5985, 13925100

Offset: 2

Ilya Gutkovskiy, Feb 19 2023

			For n = 2: 9 = 2 + 3 + 4 = 4 + 5.
For n = 3: 12880 = 91 + ... + 903 = 300 + ... + 990.
For n = 4: 20449 = 7^2 + ... + 39^2 = 38^2 + ... + 48^2.
For n = 5: 10764222 = 1617 + ... + 115787 = 31032 + ... + 126005.
From _Michael S. Branicky_, Feb 19 2023: (Start)
n-th term and indices of n-gonal numbers summing to it:
a(2) = 9: 2..4, 4..5,
a(3) = 12880: 13..42, 24..44,
a(4) = 20449: 7..39, 38..48,
a(5) = 10764222: 33..278, 144..290,
a(6) = 794629045: 1305..1505, 5321..5334,
a(7) = 33205080888: 616..3422, 3235..4192,
a(8) = 5985: 1..18, 11..19,
a(9) = 13925100: 103..235, 220..282. (End)

Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A057145, A062681, A307608, A307614, A343777, A360777.

a(6)-a(9) from Michael S. Branicky, Feb 19 2023

A374274 a(n) is the smallest number which can be represented as the sum of four distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

47, 142, 344, 652, 834, 1542, 2263, 3090, 4792, 4570, 5844, 8480, 9571, 10542, 15892, 18202, 19587, 23166, 26732, 32040, 36371, 39730, 44709, 52940, 55141, 60362, 67705, 79624, 86906, 89266, 103591, 116246, 126610, 131462, 135324, 148190, 158152, 162422, 186126, 200254

Offset: 3

Ilya Gutkovskiy, Jul 02 2024

nonn

			a(3) = 47 = 1 + 3 + 15 + 28 = 1 + 10 + 15 + 21 = 3 + 6 + 10 + 28.
a(4) = 142 = 1^2 + 2^2 + 4^2 + 11^2 = 1^2 + 4^2 + 5^2 + 10^2 = 2^2 + 5^2 + 7^2 + 8^2 = 3^2 + 4^2 + 6^2 + 9^2.

Michael S. Branicky, Table of n, a(n) for n = 3..204
Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A025417, A057145, A350405, A374144, A374273.

a(36) and beyond from Michael S. Branicky, Jul 08 2024

A374371 Least n-gonal number that can be written as a product of two or more smaller n-gonal numbers, or 0 if no such number exists.

Original entry on oeis.org

4, 36, 16, 10045, 2850, 6426, 1408, 265926, 69300, 79135, 9504, 195615, 145236, 126630, 42120, 81356859, 9410205, 165550, 1379840, 11340, 3009069, 8321351148, 316200, 47555937, 218338146, 9042726, 822528, 12300400, 300186051, 46955700, 766737400, 206898615

Offset: 2

Pontus von Brömssen, Jul 07 2024

nonn

			For 2 <= n <= 33, the n-gonal number a(n) can be written as a product of smaller n-gonal numbers in the following ways:
   n |          a(n)
  ---+---------------------------
   2 |          4 = 2*2
   3 |         36 = 6*6
   4 |         16 = 4*4
   5 |      10045 = 35*287
   6 |       2850 = 15*190
   7 |       6426 = 34*189
   8 |       1408 = 8*176
   9 |     265926 = 46*5781
  10 |      69300 = 10*6930
  11 |      79135 = 95*833
  12 |       9504 = 33*288
  13 |     195615 = 115*1701
  14 |     145236 = 14*10374
  15 |     126630 = 15*42*201
  16 |      42120 = 45*936
  17 |   81356859 = 549*148191
  18 |    9410205 = 343*27435
  19 |     165550 = 175*946
  20 |    1379840 = 20*68992
  21 |      11340 = 21*540
  22 |    3009069 = 427*7047
  23 | 8321351148 = 23*66*5481786
  24 |     316200 = 136*2325
  25 |   47555937 = 351*135487
  26 |  218338146 = 26*8397621
  27 |    9042726 = 154*58719
  28 |     822528 = 28*29376
  29 |   12300400 = 764*16100
  30 |  300186051 = 13051*23001
  31 |   46955700 = 3060*15345
  32 |  766737400 = 5720*134045
  33 |  206898615 = 12615*16401

Second column of A374370.

Cf. A057145.

A117669 The n-th n-gonal number divisible by n.

Original entry on oeis.org

1, 4, 15, 64, 210, 630, 1288, 736, 4959, 9850, 14421, 6336, 34840, 57134, 20805, 28288, 141933, 208818, 253099, 89200, 107856, 583462, 680340, 56304, 1045675, 1367210, 1553553, 497056, 2242164, 687090, 3155986, 979456, 1092861, 5336674, 1580845, 1779408, 7797898, 9370838, 2622789, 756400

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 11 2006

nonn

			The 5th number in the sequence must be the 5th pentagonal (5-gonal) number which can be divided by 5, namely 210.

Cf. A057145.

A117669 := proc(n)
    fdiv := 0 ;
    for i from 1 do
        if modp(A057145(n,i),n) = 0 then
            fdiv := fdiv+1 ;
            if fdiv = n then
                return A057145(n,i) ;
            end if;
        end if;
    end do:
end proc:
seq(A117669(n),n=1..40) ; # R. J. Mathar, Nov 22 2022

Table[Select[PolygonalNumber[n,Range[2000]],Mod[#,n]==0&][[n]],{n,40}] (* Harvey P. Dale, Jun 08 2024 *)

Offset and 2nd term corrected by R. J. Mathar, Nov 22 2022

cp's OEIS Frontend

A256857 a(n) = n*(n^2 + 3*n - 2)/2.

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A275490 Square array of 5D pyramidal numbers, read by antidiagonals.

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A275709 a(n) = 2*n^3 + 3*n^2.

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A294958 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*((k-2)^2+k)/2).

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A294959 Expansion of Product_{k>=1} (1 + x^k)^(k*((k-2)^2+k)/2).

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A350423 a(n) is the smallest n-gonal number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly n ways, or -1 if none exists.

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Author

Keywords

Examples

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A360762 a(n) is the least n-gonal number that is the sum of two or more consecutive nonzero n-gonal numbers in more than one way, or -1 if no such number exists.

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Keywords

Examples

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A374274 a(n) is the smallest number which can be represented as the sum of four distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

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A256857 a(n) = n(n^2 + 3n - 2)/2.

A275709 a(n) = 2n^3 + 3n^2.