A080851 -id:A080851 - cp's OEIS frontend

A051797 Partial sums of A007585.

1, 12, 50, 140, 315, 616, 1092, 1800, 2805, 4180, 6006, 8372, 11375, 15120, 19720, 25296, 31977, 39900, 49210, 60060, 72611, 87032, 103500, 122200, 143325, 167076, 193662, 223300, 256215, 292640, 332816, 376992, 425425, 478380, 536130

Offset: 0

Barry E. Williams, Dec 11 1999

a(n-1) is the n-th antidiagonal sum of the convolution array A213835. - Clark Kimberling, Jul 04 2012

Convolution of A000027 with A001107 (excluding 0). - Bruno Berselli, Dec 07 2012

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Index to sequences related to pyramidal numbers.

Cf. A000027, A001107, A007585, A080851.
Cf. A093565 ((8, 1) Pascal, column m=4).
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.

List([0..40], n-> (2*n+1)*Binomial(n+3,3)); # G. C. Greubel, Aug 30 2019

/* A000027 convolved with A001107 (excluding 0): */
A001107:=func; [&+[(n-i+1)*A001107(i): i in [1..n]]: n in [1..35]]; // Bruno Berselli, Dec 07 2012

[(2*n+1)*Binomial(n+3,3): n in [0..40]]; // G. C. Greubel, Aug 30 2019

seq((2*n+1)*binomial(n+3,3), n=0..40); # G. C. Greubel, Aug 30 2019

Table[(2*n+1)*Binomial[n+3,3], {n,0,40}] (*  Vladimir Joseph Stephan Orlovsky, Apr 19 2011, modified by G. C. Greubel, Aug 30 2019 *)

vector(40, n, (2*n-1)*binomial(n+2,3)) \\ G. C. Greubel, Aug 30 2019

[(2*n+1)*binomial(n+3,3) for n in (0..40)] # G. C. Greubel, Aug 30 2019

a(n) = binomial(n+3,3)*(2*n+1) = (n+1)*(n+2)*(n+3)*(2*n+1)/6.
G.f.: (1+7*x)/(1-x)^5.
a(n) = A080851(8,n). - R. J. Mathar, Jul 28 2016
E.g.f.: (6 + 66*x + 81*x^2 + 25*x^3 + 2*x^4)*exp(x)/6. - G. C. Greubel, Aug 30 2019
From Amiram Eldar, Feb 11 2022: (Start)
Sum_{n>=0} 1/a(n) = (32*log(2) - 11)/10.
Sum_{n>=0} (-1)^n/a(n) = (8*Pi - 56*log(2) + 23)/10. (End)

A374498 Square array read by antidiagonals: row n lists n-gonal pyramidal numbers that are products of smaller n-gonal pyramidal numbers.

Original entry on oeis.org

1, 36, 1, 45, 560, 1, 210, 19600, 4900, 1, 300, 43680, 513590, 56448, 1, 378, 45760, 333833500, 127008, 4750, 1, 630, 893200, 711410700, 259200, 1926049000, 58372180608, 1

Offset: 2

Pontus von Brömssen, Jul 09 2024

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 pyramidal number for every n and it equals the empty product.

			Array begins:
  n\k| 1     2          3            4
  ---+--------------------------------
  2  | 1    36         45          210
  3  | 1   560      19600        43680
  4  | 1  4900     513590    333833500
  5  | 1 56448     127008       259200
  6  | 1  4750 1926049000 655578709500

Cf. A080851, A374370, A374499 (second column).

Rows: A068143 (n=2 except the first term), A364151 (n=3), A374500 (n=4), A374501 (n=5), A374502 (n=6).

A261720 Array of pyramidal (3-dimensional figurate numbers) read by antidiagonals.

Original entry on oeis.org

1, 1, 4, 1, 5, 10, 1, 6, 14, 20, 1, 7, 18, 30, 35, 1, 8, 22, 40, 55, 56, 1, 9, 26, 50, 75, 91, 84, 1, 10, 30, 60, 95, 126, 140, 120, 1, 11, 34, 70, 115, 161, 196, 204, 165, 1, 12, 38, 80, 135, 196, 252, 288, 285, 220, 1, 13, 42, 90, 155, 231, 308, 372, 405, 385, 286

Offset: 1

Gary W. Adamson, Aug 29 2015

First few sequences in the array:
  1,  4, 10, 20,  35,  56,  84, 120, 165,  220, ... A000292
  1,  5, 14, 30,  55,  91, 140, 204, 285,  385, ... A000330
  1,  6, 18, 40,  75, 126, 196, 288, 405,  550, ... A002411
  1,  7, 22, 50,  95, 161, 252, 372, 525,  715, ... A002412
  1,  8, 26, 60, 115, 196, 308, 456, 645,  880, ... A002413
  1,  9, 30, 70, 135, 231, 364, 540, 765, 1045, ... A002414
  1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, ... A007584
  ...
The corresponding bases to rows are: Triangle, Square, Pentagon, Hexagon, Heptagon, Octagon, ...

			Row 2: (1, 5, 14, 30, 55, ...) = (1, 4, 10, 20, 35, ...) + (0, 1, 4, 10, 20, 35, ...).
(1, 7, 22, 50, ...) is the binomial transform of (1, 6, 9, 4, 0, 0, 0, ...) 3rd row in Pascal's triangle (1,4) followed by zeros. (1, 7, 22, 50, ...) is the third partial sum of (1, 4, 4, 4, ...).

Albert H. Beiler, "Recreations in the Theory of Numbers"; Dover, 1966, p. 194.

Cf. A000292, A000330, A002411, A002412, A002413, A002414, A007584, A220084.

Similar to A080851 but without row n=0.

T[n_,k_]:=k(k+1)((k-1)n+3)/6; Flatten[Table[T[n-k+1,k],{n,11},{k,n}]] (* Stefano Spezia, Aug 15 2024 *)

T(n,k) = A080851(n,k).
Given: first sequence in the array is A000292: (1, 4, 10, 20, 35, ...) Subsequent rows are generated by adding (0, 1, 4, 10, 20, 35, ...) to the current row.
n-th row is the binomial transform of row 3 in Pascal's triangle (1,n) followed by zeros.  Alternatively, begin with (1, 4, 10, 20, ...) being the binomial transform of (1, 3, 3, 1, 0, 0, 0, ...). Add (0, 1, 2, 1, 0, 0, 0, ...) to the latter to obtain the inverse binomial transform of the next row: (1, 5, 14, 30, 55,..); then repeat the operation.
The row starting (1, N, ...) is the 3rd partial sum of (1, (N-3), (N-3), (N-3), ...).
From Stefano Spezia, Aug 15 2024: (Start)
T(n,k) = k*(k + 1)*((k - 1)*n + 3)/6.
G.f. as array: x*y*(1 + x*(y - 1))/((1 - x)^2*(1 - y)^4).
E.g.f. as array: exp(y)*y*(exp(x)*(6 + 3*(1 + x)*y + x*y^2) - 3*(2 + y))/6. (End)

A350397 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal pyramidal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

305, 1980, 1900, 3321, 5256, 8310, 12516, 17108, 24832, 34249, 42381, 61697, 78766, 106956, 132994, 170325, 203415, 266595, 322943, 393828, 475520, 569416, 695799, 823447, 958300, 1149125, 1313545, 1565055, 1802736, 2088119, 2376250, 2748270, 3135195, 3548876

Offset: 3

Ilya Gutkovskiy, Dec 29 2021

nonn

			For n = 3: 305 = 1 + 84 + 220 = 20 + 120 + 165 = 56 + 84 + 165.

Martin Ehrenstein, Table of n, a(n) for n = 3..52
Eric Weisstein's World of Mathematics, Pyramidal Number

Cf. A080851, A350210, A350405.

a(10)-a(22) from Michael S. Branicky, Dec 29 2021

a(23)-a(36) from Martin Ehrenstein, Jan 14 2022

A350210 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero n-gonal pyramidal numbers in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

140, 490, 1055, 1872, 2610, 4255, 5011, 8708, 7497, 10819, 12860, 15636, 18055, 24275, 27373, 28146, 30826, 38178, 41849, 44025, 36165, 47621, 57896, 64648, 60064, 67125, 71975, 81820, 77701, 91584, 91320, 99835, 98916, 108686, 112606, 123180, 120919, 142270

Offset: 3

Ilya Gutkovskiy, Dec 19 2021

nonn

			For n = 3: 140 = 1 + 20 + 35 + 84 = 56 + 84 = 20 + 120. - _Martin Ehrenstein_, Jan 09 2022

Martin Ehrenstein, Table of n, a(n) for n = 3..1002
Eric Weisstein's World of Mathematics, Pyramidal Number

Cf. A080851, A350205, A350206, A350207, A350209, A350397.

a(35)-a(40) from Martin Ehrenstein, Jan 09 2022

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

Kelvin Voskuijl, Jun 14 2024

nonn

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

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

Pontus von Brömssen, Table of n, a(n) for n = 1..46
Brady Haran and Matt Parker, 90,525,801,730 Cannon Balls, Numberphile video (2019).
Brady Haran and Matt Parker, Cannon Ball Numbers, Numberphile (2019).
Index to sequences related to polygonal numbers.

Subsequences: A027568, A344376, A344410.

Cf. A027669, A027696, A057145, A080851, A379973, A379974, A379975.

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

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

A257199 a(n) = n(n+1)(n+2)(n^2+2n+17)/120.

Original entry on oeis.org

1, 5, 16, 41, 91, 182, 336, 582, 957, 1507, 2288, 3367, 4823, 6748, 9248, 12444, 16473, 21489, 27664, 35189, 44275, 55154, 68080, 83330, 101205, 122031, 146160, 173971, 205871, 242296, 283712, 330616, 383537, 443037, 509712, 584193, 667147, 759278, 861328, 974078

Offset: 1

Luciano Ancora, Apr 18 2015

Antidiagonal sums of the array of pyramidal numbers shown in Table 2 of Sardelis and Valahas paper (see A261720).
This is the case j = 3 of (n^2 + (j-1)*n + (j+1)^2 + 1)*binomial(n+j-1, j)/((j+1)*(j+2)), where j is the space dimension: a(n) = (n^2+2*n+17)*binomial(n+2,3)/20.
The sequence is the binomial transform of (1, 4, 7, 7, 4, 1, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2015

D. A. Sardelis and T. M. Valahas, On Multidimensional Pythagorean Numbers, arXiv:0805.4070 [math.GM], 2008.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A257200, A257201.

For another version of the array, see A080851.

[n*(n+1)*(n+2)*(n^2+2*n+17)/120: n in [1..40]]; // Vincenzo Librandi, Apr 18 2015

Table[n (n + 1) (n + 2) (n^2 + 2n + 17)/120, {n, 40}]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,5,16,41,91,182},40] (* Harvey P. Dale, Mar 18 2018 *)

G.f.: x*(1 - x + x^2)/(1 - x)^6.

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)

A257055 a(n) = n(n + 1)(n^2 - n + 3)/6.

Original entry on oeis.org

0, 1, 5, 18, 50, 115, 231, 420, 708, 1125, 1705, 2486, 3510, 4823, 6475, 8520, 11016, 14025, 17613, 21850, 26810, 32571, 39215, 46828, 55500, 65325, 76401, 88830, 102718, 118175, 135315, 154256, 175120, 198033, 223125, 250530, 280386, 312835, 348023, 386100

Offset: 0

Bruno Berselli, Apr 15 2015

Partial sums of A037235.
After 0, this sequence is the 2nd diagonal of the square array in A080851.
For n > 2, a(n)-n is the 4th column of the triangular array in A208657.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000332, A037235, A080851, A208657.

Cf. similar sequences listed in A256859.

Magma
```
[n*(n+1)*(n^2-n+3)/6: n in [0..40]];
```

Table[n (n + 1) (n^2 - n + 3)/6, {n, 40}]

vector(40, n, n--; n*(n+1)*(n^2-n+3)/6)

Sage
```
[n*(n+1)*(n^2-n+3)/6 for n in (0..40)]
```

G.f.: x*(1 + 3*x^2)/(1 - x)^5.
a(n) = 3*A000332(n+1) + A000332(n+3).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Wesley Ivan Hurt, May 27 2021

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).

cp's OEIS Frontend

A051797 Partial sums of A007585.

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A374498 Square array read by antidiagonals: row n lists n-gonal pyramidal numbers that are products of smaller n-gonal pyramidal numbers.

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A261720 Array of pyramidal (3-dimensional figurate numbers) read by antidiagonals.

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A350397 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal pyramidal numbers in exactly n ways, or -1 if no such number exists.

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A350210 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero n-gonal pyramidal numbers in exactly n ways, or 0 if no such integer exists.

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

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A257199 a(n) = n*(n+1)*(n+2)*(n^2+2*n+17)/120.

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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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A257199 a(n) = n(n+1)(n+2)(n^2+2n+17)/120.

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

A257055 a(n) = n(n + 1)(n^2 - n + 3)/6.