A002411 -id:A002411 - cp's OEIS frontend

A133459 Sums of two nonzero pentagonal pyramidal numbers.

2, 7, 12, 19, 24, 36, 41, 46, 58, 76, 80, 81, 93, 115, 127, 132, 144, 150, 166, 197, 201, 202, 214, 236, 252, 271, 289, 294, 306, 322, 328, 363, 392, 406, 411, 414, 423, 445, 480, 484, 531, 551, 556, 568, 576, 590, 601, 625, 676, 693, 727, 732, 744, 746, 766

Offset: 1

Jonathan Vos Post, Dec 23 2007

nonn

Does this sequence ever include a pentagonal pyramidal number? That is, is it ever the case that A002411(i)=A002411(j)+A002411(k) as is often true for triangular pyramidal numbers (tetrahedral numbers) or square pyramidal numbers?

The answer to the above question is yes: A002411(30) + A002411(36) = 13950 + 23976 = 37926 = A002411(42) (see A172425). - Chai Wah Wu, Apr 16 2016

Cf. A002311, A002411, A053721, A172425.

nn = 12; Take[Union@ Map[Total, Tuples[#^2 (# + 1)/2 &@ Range@ nn, 2]], # (# - 1)/2 &[nn - 1]] (* Michael De Vlieger, Apr 16 2016 *)

{A002411(i) + A002411(j) for i, j > 0} = {i^2*(i+1)/2 + j^2*(j+1)/2 for i, j > 0}.

A213761 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3n-5+3h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 6, 4, 18, 15, 7, 40, 36, 24, 10, 75, 70, 54, 33, 13, 126, 120, 100, 72, 42, 16, 196, 189, 165, 130, 90, 51, 19, 288, 280, 252, 210, 160, 108, 60, 22, 405, 396, 364, 315, 255, 190, 126, 69, 25, 550, 540, 504, 448, 378, 300

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A172073.
Antidiagonal sums: A002419.
Row 1, (1,2,3,4,5,...)**(1,4,7,10,13,...): A002411.
Row 2, (1,2,3,4,5,...)**(4,7,10,13,16,...): A077414.
Row 3, (1,2,3,4,5,...)**(7,10,13,16,...): (k^3 + 7*k^2 + 6*k)/2.
Row 4, (1,2,3,4,5,...)**(10,13,16,...): (k^3 + 10*k^2 + 9*k)/2.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
1....6....18...40....75....126
4....15...36...70....120...189
7....24...54...100...165...252
10...33...72...130...210...315
13...42...90...160...255...378

Clark Kimberling, Antidiagonals n = 1..45, flattened

Cf. A212500.

b[n_]:=n;c[n_]:=3n-2;
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213761 *)
Table[t[n,n],{n,1,40}] (* A172073 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A002419 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = x*(3*n - 2 - (3*n - 5)*x) and g(x) = (1 - x)^4.

A213771 Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 6, 2, 18, 11, 3, 40, 30, 16, 4, 75, 62, 42, 21, 5, 126, 110, 84, 54, 26, 6, 196, 177, 145, 106, 66, 31, 7, 288, 266, 228, 180, 128, 78, 36, 8, 405, 380, 336, 279, 215, 150, 90, 41, 9, 550, 522, 472, 406, 330, 250, 172, 102, 46

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A213772
Antidiagonal sums: A132117
Row 1, (1,4,7,10,...)**(1,2,3,4,...): A002411
Row 2, (1,4,7,10,...)**(2,3,4,5,...): A162260
Row 3, (1,2,3,4,5,...)**(7,10,13,16,...): (k^3 + 7*k^2 - 2*k)/2
Row 4, (1,2,3,4,5,...)**(10,13,16,...): (k^3 + 10*k^2 - 3*k)/2
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
1....6....18...40....75....126
2....11...30...62....110...177
3....16...42...84....145...228
4....21...54...106...180...279
5....26...66...128...215...330

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A212500

b[n_]:=3n-2;c[n_]:=n;
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213771 *)
Table[t[n,n],{n,1,40}] (* A213772 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A132117 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = x*(n + (n+1)*x - (n+2)*x^2) and g(x) = (1 - x)^4.

A338494 Least number of pentagonal pyramidal numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 4, 5, 6, 1, 2, 3, 4, 5, 2, 2, 3, 4, 5, 6, 3, 3, 4, 5, 5, 6, 4, 2, 3, 4, 5, 6, 3, 3, 4, 5

Offset: 1

Ilya Gutkovskiy, Oct 30 2020

nonn

Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A002411, A100878, A104246, A338491, A338493, A338495.

A366014 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 2 * A(x)).

Original entry on oeis.org

0, 1, 6, 54, 580, 6873, 86688, 1141500, 15512220, 215928900, 3063184410, 44124882750, 643692232404, 9490176205006, 141184118174640, 2116751269990968, 31951313566227228, 485159929343783532, 7405637373574690968, 113572576254948487800, 1749075343256441443320

Offset: 0

Ilya Gutkovskiy, Sep 26 2023

nonn

Reversion of g.f. for pentagonal pyramidal numbers (with signs).

Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number
Eric Weisstein's World of Mathematics, Series Reversion

Cf. A002293, A002411, A064063, A365754, A366015, A366016, A366017.

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^4/(1 - 2 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 2 x)/(1 + x)^4, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[4 n, n - k - 1] 2^k, {k, 0, n - 1}], {n, 1, 20}]]

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(4*n,n-k-1) * 2^k for n > 0.

A062707 Table by antidiagonals of nk(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

Henry Bottomley, Jul 11 2001

			  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

G. C. Greubel, Antidiagonals n = 0..100, flattened

Main diagonal is A002411. Sum of antidiagonals is A000332.

Flat(List([0..12], n-> List([0..n], k-> k*Binomial(n-k+1,2)))); # G. C. Greubel, Sep 02 2019

[k*Binomial(n-k+1,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 02 2019

seq(seq(k*binomial(n-k+1,2), k=0..n), n=0..12); # G. C. Greubel, Sep 02 2019

Table[k*Binomial[n-k+1, 2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 02 2019 *)

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

[[k*binomial(n-k+1,2) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Sep 02 2019

T(n, k) = T(n, 1)*T(1, k) = A001477(n)*A000217(k).

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

A095833 Triangle read by rows: T(n,k) = (n-k+1)*n, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 4, 2, 9, 6, 3, 16, 12, 8, 4, 25, 20, 15, 10, 5, 36, 30, 24, 18, 12, 6, 49, 42, 35, 28, 21, 14, 7, 64, 56, 48, 40, 32, 24, 16, 8, 81, 72, 63, 54, 45, 36, 27, 18, 9, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 121, 110, 99, 88, 77, 66, 55, 44, 33, 22, 11, 144, 132, 120, 108, 96, 84

Offset: 1

Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 10 2004

			   1
   4   2
   9   6   3
  16  12   8   4
  25  20  15  10   5
  36  30  24  18  12   6
  49  42  35  28  21  14   7
  64  56  48  40  32  24  16   8

Cf. A002411 (row sums).

A096038 Triangle T(n,m) = (3n^2-3m^2+5*m-4+n)/2 read by rows.

Original entry on oeis.org

1, 6, 4, 14, 12, 7, 25, 23, 18, 10, 39, 37, 32, 24, 13, 56, 54, 49, 41, 30, 16, 76, 74, 69, 61, 50, 36, 19, 99, 97, 92, 84, 73, 59, 42, 22, 125, 123, 118, 110, 99, 85, 68, 48, 25, 154, 152, 147, 139, 128, 114, 97, 77, 54, 28, 186, 184, 179, 171, 160, 146, 129, 109, 86, 60, 31

Offset: 1

Gary W. Adamson, Jun 17 2004

The triangle is obtained by subtracting the triangle A094930 from
its square root (also described in A094930) and then dividing each element of column m through 3*m-1.
For the first three rows n=1 to 3 this yields for example:
4;.................2;............2......................1;
14,25;......minus..2,5;.......=..12,20;......->.divide..6,4;
30,65,64;..........2,5,8;........28,60,56;..............14;12,7;

Cf. A095794, A011379, A002411, A096037, A096036, A024212.

def A096038(n,m):
    return (3*n**2-3*m**2+5*m-4+n)//2
print( [A096038(n,m) for n in range(20) for m in range(1,n+1)] )
# R. J. Mathar, Oct 11 2009

T(n,1) = A095794(n).
T(n,n) = 3*n-2.
T(n,m) = A094930(n,m)/(3*m-1)-1.

Edited, T(3,2) corrected, and extended by R. J. Mathar, Oct 11 2009

A127739 Triangle read by rows, in which row n contains the triangular number T(n) = A000217(n) repeated n times; smallest triangular number greater than or equal to n.

Original entry on oeis.org

1, 3, 3, 6, 6, 6, 10, 10, 10, 10, 15, 15, 15, 15, 15, 21, 21, 21, 21, 21, 21, 28, 28, 28, 28, 28, 28, 28, 36, 36, 36, 36, 36, 36, 36, 36, 45, 45, 45, 45, 45, 45, 45, 45, 45, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66

Offset: 1

Gary W. Adamson, Jan 27 2007

Seen as a sequence, these are the triangular numbers applied to the Kruskal-Macaulay function A123578. - Peter Luschny, Oct 29 2022

			First few rows of the triangle are:
   1;
   3,  3;
   6,  6,  6;
  10, 10, 10, 10;
  15, 15, 15, 15, 15;
  ...

Reinhard Zumkeller, Rows n=1..100 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A000217, A000384, A002024, A002411 (row sums), A003057, A057944, A123578.

a127739 n k = a127739_tabl !! (n-1) !! (k-1)
a127739_row n = a127739_tabl !! (n-1)
a127739_tabl = zipWith ($) (map replicate [1..]) $ tail a000217_list
-- Reinhard Zumkeller, Feb 03 2012, Mar 18 2011

A127739 := proc(n) local t, s; t := 1; s := 0;
while t <= n do s := s + 1; t := t + s od; s*(1 + s)/2 end:
seq(A127739(n), n = 1..66); # Peter Luschny, Oct 29 2022

Table[n(n+1)/2,{n,100},{n}]//Flatten (* Zak Seidov, Mar 19 2011 *)

A127739=n->binomial((sqrtint(8*n)+3)\2,2) \\ M. F. Hasler, Mar 09 2014

from math import isqrt
def A127739(n): return (r:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(r+1)>>1 # Chai Wah Wu, Nov 07 2024

Central terms: T(2*n-1,n) = A000384(n). - Reinhard Zumkeller, Mar 18 2011
a(n) = A003057(n)*A002024(n)/2; a(n) = (t+2)*(t+1)/2, where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Feb 08 2013
Sum_{n>=1} 1/a(n)^2 = 8 - 2*Pi^2/3. - Amiram Eldar, Aug 15 2022
a(n) = k(n)*(1 + k(n))/2 = A000217(A123578(n)), where k = A123578. - Peter Luschny, Oct 29 2022

Name edited by Michel Marcus, Apr 30 2020

A128629 A triangular array generated by moving Pascal sequences to prime positions and embedding new sequences at the nonprime locations. (cf. A007318 and A000040).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 9, 10, 5, 1, 1, 6, 10, 16, 15, 6, 1, 1, 5, 18, 20, 25, 21, 7, 1, 1, 8, 15, 40, 35, 36, 28, 8, 1, 1, 9, 27, 35, 75, 56, 49, 36, 9, 1

Offset: 1

Alford Arnold, Mar 29 2007

The array can be constructed by beginning with A007318 (Pascal's triangle) placing each diagonal on a prime row. The other rows are filled in by mapping the prime factorization of the row number to the known sequences on the prime rows and multiplying term by term.

			Row six begins 1 6 18 40 75 126 ... because rows two and three are
1 2 3 4 5 6 ...
1 3 6 10 15 21 ...
The array begins
1 1 1 1 1 1 1 1 1 A000012
1 2 3 4 5 6 7 8 9 A000027
1 3 6 10 15 21 28 36 45 A000217
1 4 9 16 25 36 49 64 81 A000290
1 4 10 20 35 56 84 120 165 A000292
1 6 18 40 75 126 196 288 405 A002411
1 5 15 35 70 126 210 330 495 A000332
1 8 27 64 125 216 343 512 729 A000578
1 9 36 100 225 441 784 1296 2025 A000537
1 8 30 80 175 336 588 960 1485 A002417
1 6 21 56 126 252 462 792 1287 A000389
1 12 54 160 375 756 1372 2304 3645 A019582
1 7 28 84 210 462 924 1716 3003 A000579
1 10 45 140 350 756 1470 2640 4455 A027800
1 12 60 200 525 1176 2352 4320 7425 A004302
1 16 81 256 625 1296 2401 4096 6561 A000583
1 8 36 120 330 792 1716 3432 6435 A000580
1 18 108 400 1125 2646 5488 10368 18225 A019584
1 9 45 165 495 1287 3003 6435 12870 A000581
1 16 90 320 875 2016 4116 7680 13365 A119771
1 15 90 350 1050 2646 5880 11880 22275 A001297
1 12 63 224 630 1512 3234 6336 11583 A027810
1 10 55 220 715 2002 5005 11440 24310 A000582
1 24 162 640 1875 4536 9604 18432 32805 A019583
1 16 100 400 1225 3136 7056 14400 27225 A001249
1 14 84 336 1050 2772 6468 13728 27027 A027818
1 27 216 1000 3375 9261 21952 46656 91125 A059827
1 20 135 560 1750 4536 10290 21120 40095 A085284

Cf. A000040 A007318.

Cf. A064553 (second diagonal), A080688 (second diagonal resorted).

A128629 := proc(n,m) if n = 1 then 1; elif isprime(n) then p := numtheory[pi](n) ; binomial(p+m-1,p) ; else a := 1 ; for p in ifactors(n)[2] do a := a* procname(op(1,p),m)^ op(2,p) ; od: fi; end: # R. J. Mathar, Sep 09 2009

A-number added to each row of the examples by R. J. Mathar, Sep 09 2009

cp's OEIS Frontend

A133459 Sums of two nonzero pentagonal pyramidal numbers.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A213761 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A213771 Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A338494 Least number of pentagonal pyramidal numbers needed to represent n.

Views

Author

Keywords

Links

Crossrefs

A366014 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 2 * A(x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A095833 Triangle read by rows: T(n,k) = (n-k+1)*n, n>=1, 1<=k<=n.

Views

Author

Keywords

Examples

Crossrefs

A096038 Triangle T(n,m) = (3*n^2-3*m^2+5*m-4+n)/2 read by rows.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

Formula

Extensions

A127739 Triangle read by rows, in which row n contains the triangular number T(n) = A000217(n) repeated n times; smallest triangular number greater than or equal to n.

A213761 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3n-5+3h, n>=1, h>=1, and ** = convolution.

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

A096038 Triangle T(n,m) = (3n^2-3m^2+5*m-4+n)/2 read by rows.