A002411 -id:A002411 - cp's OEIS frontend

A177708 Pentagonal triangle.

1, 6, 12, 18, 57, 51, 40, 156, 209, 145, 75, 330, 531, 534, 330, 126, 600, 1074, 1278, 1122, 651, 196, 987, 1895, 2488, 2559, 2081, 1162, 288, 1512, 3051, 4275, 4824, 4563, 3537, 1926, 405, 2196, 4599, 6750, 8100, 8370, 7506, 5634, 3015

Offset: 1

Jonathan Vos Post, Dec 11 2010

This is to A093445 as pentagonal numbers A000326 are to triangular numbers A000217. The n-th row of the triangular table begins by considering A000217(n) pentagonal numbers (starting with 1) in order. Now segregate them into n chunks beginning with n members in the first chunk, n-1 members in the second chunk, and so forth. Now sum each chunk. Thus the first term is the sum of first n numbers = n*(3n-1)/2, the second term is the sum of the next n-1 terms (from n+1 to 2n-1), the third term is the sum of the next n-2 terms (2n to 3n-3)... This triangle can be called the pentagonal triangle. The sequence contains the triangle by rows. The first column is A002411 (Pentagonal pyramidal numbers: n^2*(n+1)/2).

			The row for n = 4 is (1+5+12+22), (35+51+70), (92+117), 145 => 40, 156, 209, 145.
    1;
    6,   12;
   18,   57,   51;
   40,  156,  209,   145;
   75,  330,  531,   534,   330;
  126,  600, 1074,  1278,  1122,   651;
  196,  987, 1895,  2488,  2559,  2081,  1162;
  288, 1512, 3051,  4275,  4824,  4563,  3537,  1926;
  405, 2196, 4599,  6750,  8100,  8370,  7506,  5634, 3015;
  550, 3060, 6596, 10024, 12570, 13775, 13450, 11631, 8534, 4510;

Cf. A000217, A000326, A002411, A093445, A236770 (right border).

A000326 :=proc(n) n*(3*n-1)/2 ; end proc:
A177708 := proc(n,k) kc := 1 ; nsk := n ; ns := 1 ; while kc < k do ns := ns+nsk ; kc := kc+1 ; nsk := nsk-1 ; end do: add(A000326(i),i=ns..ns+nsk-1) ; end proc: # R. J. Mathar, Dec 14 2010

Table[Total/@TakeList[PolygonalNumber[5,Range[60]],Range[n,1,-1]],{n,10}]//Flatten (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Feb 17 2018 *)

T(n,1) = A002411(n).
T(n,2) = n*(n-1)*(7*n-2)/2.
T(n,3) = (n-2)*(19*n^2-26*n+9)/2 = Sum_{i=2n..3(n-1)} A000326(i).

A272039 a(n) = 10n^2 + 4n + 1.

Original entry on oeis.org

1, 15, 49, 103, 177, 271, 385, 519, 673, 847, 1041, 1255, 1489, 1743, 2017, 2311, 2625, 2959, 3313, 3687, 4081, 4495, 4929, 5383, 5857, 6351, 6865, 7399, 7953, 8527, 9121, 9735, 10369, 11023, 11697, 12391, 13105, 13839, 14593, 15367, 16161, 16975, 17809, 18663, 19537

Offset: 0

Vincenzo Librandi, Apr 20 2016

Polynomials from the table "Coefficients and roots of Ehrhart polynomials" in Beck et al. paper (see Links section):
. Cube:                   A000578;
. Cube minus corner:      A004068;
. Prism:                  A002411;
. Octahedron:             A005900;
. Square pyramid:         A000330;
. Bypyramid:              A006003;
. Unimodular tetrahedron: A000292;
. Fat tetrahedron:        A167875;
. Cyclic(2,5), which has the same polynomial form of this sequence.
a(n) for n = 0, -1, 1, -2, 2, -3, 3, ... gives all x such that (5*x - 3)/2 is a square.
Squares in sequence: 1, 49, 1385329, 101263969, 2880599856289, ...
Is this 1 followed by A228219?

M. Beck, J. A. De Loera, M. Develin, J. Pfeifle and R. P. Stanley, Coefficients and roots of Ehrhart Polynomials, page 19.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000292, A000330, A000578, A002411, A004068, A005900, A006003, A167875, A168668, A228219, A272124, A272130.

Magma
```
[10*n^2+4*n+1: n in [0..50]];
```

Table[10 n^2 + 4 n + 1, {n, 0, 50}]
LinearRecurrence[{3,-3,1},{1,15,49},50] (* Harvey P. Dale, Dec 26 2021 *)

a(n)=10*n^2+4*n+1 \\ Charles R Greathouse IV, Jun 17 2017

O.g.f.: (1 + 12*x + 7*x^2)/(1 - x)^3.
E.g.f.: (1 + 14*x + 10*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A168668(n) + 1.

Edited by Bruno Berselli, Apr 22 2016

A329754 Doubly pentagonal pyramidal numbers.

Original entry on oeis.org

0, 1, 126, 3078, 32800, 213750, 1008126, 3783976, 11985408, 33297075, 83338750, 191592126, 410450976, 828497488, 1589341950, 2917620000, 5154021376, 8801526501, 14585352318, 23529456550, 37052820000, 57089119626, 86233820926, 127923156648, 186649920000, 268221484375, 380065968126

Offset: 0

Ilya Gutkovskiy, Nov 20 2019

Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A000326, A002411, A140236, A232713, A329753, A329755, A329756, A329757.

A002411[n_] := n^2 (n + 1)/2; a[n_] := A002411[A002411[n]]; Table[a[n], {n, 0, 26}]
Table[Sum[k (3 k - 1)/2, {k, 0, n^2 (n + 1)/2}], {n, 0, 26}]
nmax = 26; CoefficientList[Series[x (1 + 116 x + 1863 x^2 + 7570 x^3 + 9350 x^4 + 3474 x^5 + 304 x^6 + 2 x^7)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 126, 3078, 32800, 213750, 1008126, 3783976, 11985408, 33297075}, 27]

G.f.: x*(1 + 116*x + 1863*x^2 + 7570*x^3 + 9350*x^4 + 3474*x^5 + 304*x^6 + 2*x^7)/(1 - x)^10.
a(n) = A002411(A002411(n)).
a(n) = Sum_{k=0..A002411(n)} A000326(k).
a(n) = n^4 *(n^3+n^2+2) *(n+1)^2 /16. - R. J. Mathar, Nov 28 2019

A368607 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y| + |y-z| = k, where x,y,z are in {1,2,...,n} and x != y and y < z.

Original entry on oeis.org

1, 3, 2, 1, 5, 6, 4, 2, 1, 7, 10, 10, 6, 4, 2, 1, 9, 14, 16, 14, 9, 6, 4, 2, 1, 11, 18, 22, 22, 19, 12, 9, 6, 4, 2, 1, 13, 22, 28, 30, 29, 24, 16, 12, 9, 6, 4, 2, 1, 15, 26, 34, 38, 39, 36, 30, 20, 16, 12, 9, 6, 4, 2, 1, 17, 30, 40, 46, 49, 48, 44, 36, 25

Offset: 1

Clark Kimberling, Jan 25 2024

Row n consists of 2n-1 positive integers.

			First six rows:
 1
 3    2   1
 5    6   4   2  1
 7   10  10   6  4   2   1
 9   14  16  14  9   6   4  2  1
 11  18  22  22  19  12  9  6  4  2  1
For n=3, there are 6 triples (x,y,z) having x != y and y < z:
  123:  |x-y| + |y-z| = 2
  212:  |x-y| + |y-z| = 2
  213:  |x-y| + |y-z| = 3
  312:  |x-y| + |y-z| = 3
  313:  |x-y| + |y-z| = 4
  323:  |x-y| + |y-z| = 2
so row 2 of the array is (3,2,1), representing three 2s, two 3s, and one 4.

Cf. A005408 (column 1), A002411 (row sums), A002620 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368521, A368522, A368604, A368605, A368606, A368609.

t1[n_] := t1[n] = Tuples[Range[n], 3];
t[n_] := t[n] = Select[t1[n], #[[1]] != #[[2]] < #[[3]] &];
a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
u = Table[Length[a[n, k]], {n, 2, 15}, {k, 2, 2 n - 2}];
v = Flatten[u] (* sequence *)
Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 2, 2 n - 2}]] (* array *)

A372583 a(n) = (3n^5 + 5n^3)/8.

Original entry on oeis.org

1, 17, 108, 424, 1250, 3051, 6517, 12608, 22599, 38125, 61226, 94392, 140608, 203399, 286875, 395776, 535517, 712233, 932824, 1205000, 1537326, 1939267, 2421233, 2994624, 3671875, 4466501, 5393142, 6467608, 7706924, 9129375, 10754551, 12603392, 14698233

Offset: 1

Kelvin Voskuijl, May 05 2024

Sum of pentagonal numbers in increasing groups 1, 5+12, 22+35+51, 70+92+117+145 etc.

			The first ten pentagonal numbers are 1, 5, 12, 22, 35, 51, 70, 92, 117, and 145.  Taking them in groups, respectively, of 1, 2, 3, and 4, i.e., (1), (5, 12), (22, 35, 51), and (70, 92, 117, 145), and summing each group separately gives 1, 17, 108, 424.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A260513 (for triangular numbers), A072474 (for squares).

Cf. A000326 (pentagonal numbers), A002411 (their partial sums).

A372583[n_] := (3*n^5 + 5*n^3)/8; Array[A372583, 50] (* Paolo Xausa, May 25 2024 *)

From Stefano Spezia, May 06 2024: (Start)
G.f.: x*(1 + 11*x + 21*x^2 + 11*x^3 + x^4)/(1 - x)^6.
E.g.f.: exp(x)*x*(8 + 60*x + 80*x^2 + 30*x^3 + 3*x^4)/8. (End)

A374501 Pentagonal pyramidal numbers that are products of smaller pentagonal pyramidal numbers.

Original entry on oeis.org

1, 56448, 127008, 259200, 5644800, 31840200, 42688800, 60766200, 116493300, 130662720, 193179168, 442828800, 499000500, 897544800, 917632800, 3624409800, 6914880000, 13831171200, 15410656800, 31246000128, 53936416800, 64024732800, 72945774720, 88957620000

Offset: 1

Pontus von Brömssen, Jul 09 2024

nonn

			1 is a term because it is a pentagonal pyramidal number and equals the empty product.
56448 is a term because it is a pentagonal pyramidal number and equals the product of the pentagonal pyramidal numbers 196 and 288.
127008 is a term because it is a pentagonal pyramidal number and equals the product of the pentagonal pyramidal numbers 6, 6, 18, and 196.

Row n=5 of A374498.

Cf. A002411, A374500, A374502.

A376180 Triangle read by rows (blocks). Each row consists of a permutation of the numbers of its constituents. The length of row number n is the n-th pentagonal number n(3n-1)/2 = A000326(n); see Comments.

Original entry on oeis.org

1, 4, 5, 3, 6, 2, 13, 12, 14, 11, 15, 10, 16, 9, 17, 8, 18, 7, 30, 29, 31, 28, 32, 27, 33, 26, 34, 25, 35, 24, 36, 23, 37, 22, 38, 21, 39, 20, 40, 19, 58, 59, 57, 60, 56, 61, 55, 62, 54, 63, 53, 64, 52, 65, 51, 66, 50, 67, 49, 68, 48, 69, 47, 70, 46, 71, 45, 72, 44, 73, 43, 74, 42, 75, 41, 101, 102, 100, 103, 99, 104, 98, 105, 97, 106, 96, 107

Offset: 1

Boris Putievskiy, Sep 14 2024

A209278 presents an algorithm for generating permutations.

The sequence is an intra-block permutation of integer positive numbers.

			Triangle begins:
     k =  1   2   3   4   5   6   7   8   9  10  11  12
  n=1:    1;
  n=2:    4,  5,  3,  6,  2;
  n=3:   13, 12, 14, 11, 15, 10, 16,  9, 17,  8, 18,  7;
Subtracting (n-1)^2*n/2 from each term in row n is a permutation of 1 .. n(3n-1)/2:
  1;
  3,4,2,5,1;
  7,6,8,5,9,4,10,3,11,2,12,1;

Boris Putievskiy, Table of n, a(n) for n = 1..9126
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000326, A002411, A209278, A375602.

a[n_]:=Module[{L,R,P,Result},L=Ceiling[Max[x/.NSolve[x^2 (x+1)-2 n==0,x,Reals]]];
R=n-((L-1)^2)*L/2; P=Which[OddQ[R]&&OddQ[L*(3*L-1)/2],(L*(3*L-1)/2-R+2)/2,OddQ[R]&&EvenQ[L*(3*L-1)/2],(R+L*(3*L-1)/2+1)/2,EvenQ[R]&&OddQ[L*(3*L-1)/2],Ceiling[(L*(3*L-1)/2+1)/2]+R/2, EvenQ[R]&&EvenQ[L*(3*L-1)/2],Ceiling[(L*(3*L-1)/2+1)/2]-R/2 ];
Result=P+(L-1)^2*L/2; Result]
Nmax=18; Table[a[n],{n,1,Nmax}]

Linear sequence:
a(n) = P(n) + (L(n)-1)^2*L(n)/2. a(n) = P(n) + A002411(L(n)-1), where P = (L(n)(3L(n) - 1)/2 - R(n) + 2)/2 if R(n) is odd and L(n)(3L(n) - 1)/2 is odd, P = (R(n) + L(n)(3L(n) - 1)/2 + 1)/2 if R(n) is odd and L(n)(3L(n) - 1)/2 is even, P = ceiling((L(n)(3L(n) - 1)/2 + 1)/2) + R(n)/2 if R(n) is even and L(n)(3L(n) - 1)/2 is odd, P = ceiling((L(n)(3L(n) - 1)/2 + 1)/2) - R(n)/2 if R(n) is even and L(n)(3L(n) - 1)/2 is even. L(n) = ceiling(x(n)), x(n) is largest real root of the equation x^2*(x+1)-2*n = 0.
Triangular array T(n,k) for 1 <= k <= n(3n-1)/2 (see Example):
T(n,k) = P(n,k) + (n-1)^2*n/2, T(n,k) = P(n,k) + A002411(n-1), where P(n,k) = (n(3n - 1)/2 - k + 2)/2 if k is odd and n(3n - 1)/2 is odd,
P(n,k) = (k + n(3n - 1)/2 + 1)/2 if k is odd and n(3n - 1)/2 is even, P(n,k) = ceiling((n(3n - 1)/2 + 1)/2) + k/2 if k is even and n(3n - 1)/2 is odd, P(n,k) = ceiling((n(3n - 1)/2 + 1)/2) - k/2 if k is even and n(3n - 1)/2 is even.

A094930 Triangle T(n,m) read by rows, defined by squaring a matrix with row entries 2+3*(m-1).

Original entry on oeis.org

4, 14, 25, 30, 65, 64, 52, 120, 152, 121, 80, 190, 264, 275, 196, 114, 275, 400, 462, 434, 289, 154, 375, 560, 682, 714, 629, 400, 200, 490, 744, 935, 1036, 1020, 860, 529, 252, 620, 952, 1221, 1400, 1462, 1380, 1127, 676, 310, 765, 1184, 1540, 1806, 1955, 1960

Offset: 1

Gary W. Adamson, Jun 17 2004

Matrix square of the matrix B(n,m) = 2+3*(m-1), B containing the first terms of A016789

in its row n, n>0, 1<=m<=n.

			The matrix B starts as
  2 ;
  2,5 ;
  2,5,8 ;
  2,5,8,11 ;
  2,5,8,11,14 ;
and interpreting this as a lower triangular matrix, its square T = B^2 starts
  4;
  14,25;
  30,65,64;
  52,120,152,121;

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)

Cf. A024212, A096037, A011379, A002411, A096038, A095794.

A094930 := proc(n,m) (3*m-1)*(3*m+3*n-2)*(n+1-m)/2 ; end: seq(seq(A094930(n,m),m=1..n),n=1..20) ; # R. J. Mathar, Oct 09 2009

T(n,m) = sum_{k=m..n} B(n,k)*B(k,m) = (3*m-1)*(3*m+3*n-2)*(n+1-m)/2.
Row sums: sum_{m=1..n} T(n,m) = A024212(n).
G.f. as triangle: x*y*(4+2*x+13*x*y-16*x^2*y+x^2*y^2-4*x^3*y^2)/((1-x)*(1-x*y))^3. - Robert Israel, May 06 2019

Edited and extended by R. J. Mathar, Oct 09 2009

A096037 Triangle T(n,m) = (3n+3m-2)*(n+1-m)/2 read by rows.

Original entry on oeis.org

2, 7, 5, 15, 13, 8, 26, 24, 19, 11, 40, 38, 33, 25, 14, 57, 55, 50, 42, 31, 17, 77, 75, 70, 62, 51, 37, 20, 100, 98, 93, 85, 74, 60, 43, 23, 126, 124, 119, 111, 100, 86, 69, 49, 26, 155, 153, 148, 140, 129, 115, 98, 78, 55, 29, 187, 185, 180, 172, 161, 147, 130, 110, 87, 61, 32

Offset: 1

Gary W. Adamson, Jun 17 2004

			The triangle starts in row n=1 as
2;
7,5;
15,13,8;
26,24,19,11;

Cf. A094930, A024212, A011379, A002411, A096038, A095794.

def A096037(n,m):
    return (3*n+3*m-2)*(n+1-m)//2
print( [A096037(n,m) for n in range(20) for m in range(1,n+1)] )
# R. J. Mathar, Oct 11 2009

T(n,m) = (3*n+3*m-2)*(n+1-m)/2 .
T(n,m) = A094930(n,m)/(3*m-1).
T(n,1) = A005449(n).
T(n,n) = A016768(n-1).
Row sums: sum_{m=1..n} T(n,m) = n^2*(n+1) = A011379(n).

Edited and extended, A-numbers corrected by R. J. Mathar, Oct 11 2009

A136359 Perfect squares in A133459; or perfect squares that are the sums of two nonzero pentagonal pyramidal numbers.

Original entry on oeis.org

36, 81, 144, 289, 484, 576, 625, 676, 3600, 7396, 9801, 14400, 35344, 40000, 40804, 44100, 45796, 56644, 59049, 71824, 112896, 121104, 172225, 226576, 231361, 254016, 274576, 290521, 319225, 362404, 480249, 495616, 518400, 527076, 535824

Offset: 1

Alexander Adamchuk, Dec 25 2007

nonn

Corresponding numbers m such that m^2 = a(n) are listed in A136360.
Note that some numbers in A136360 are also perfect squares. The corresponding numbers k such that m = k^2 are listed in A136361.
Includes all nonzero members of A099764: this occurs when the two pentagonal pyramidal numbers are both equal to i^2*(i+1)/2 where i+1 is a square. - Robert Israel, Feb 04 2020

			A133459 begins {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, ...}.
Thus a(1) = 36, a(2) = 81, a(3) = 144, a(4) = 289 that are the perfect squares in A133459.

Robert Israel, Table of n, a(n) for n = 1..985

Cf. A136360, A136361, A133459, A002311, A002411, A053721, A099764.

N:= 200: # for terms up to N^2*(N+1)/2.
PP:= [seq(i^2*(i+1)/2, i=1..N)]:
PP2:= sort(convert(select(`<=`,{seq(seq(PP[i]+PP[j],j=i..N),i=1..N)},PP[-1]),list)):
select(issqr,PP2); # Robert Israel, Feb 04 2020

Select[ Intersection[ Flatten[ Table[ i^2*(i+1)/2 + j^2*(j+1)/2, {i,1,300}, {j,1,i} ] ] ], IntegerQ[ Sqrt[ # ] ] & ]

a(n) = A136360(n)^2.

cp's OEIS Frontend

A177708 Pentagonal triangle.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A272039 a(n) = 10*n^2 + 4*n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A329754 Doubly pentagonal pyramidal numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A368607 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y| + |y-z| = k, where x,y,z are in {1,2,...,n} and x != y and y < z.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A372583 a(n) = (3*n^5 + 5*n^3)/8.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A374501 Pentagonal pyramidal numbers that are products of smaller pentagonal pyramidal numbers.

Views

Author

Keywords

Examples

Crossrefs

A376180 Triangle read by rows (blocks). Each row consists of a permutation of the numbers of its constituents. The length of row number n is the n-th pentagonal number n(3n-1)/2 = A000326(n); see Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A094930 Triangle T(n,m) read by rows, defined by squaring a matrix with row entries 2+3*(m-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A272039 a(n) = 10n^2 + 4n + 1.

A372583 a(n) = (3n^5 + 5n^3)/8.

A096037 Triangle T(n,m) = (3n+3m-2)*(n+1-m)/2 read by rows.