A100165 -id:A100165 - cp's OEIS frontend

A260260 a(n) = n(16n^2 - 21*n + 7)/2.

0, 1, 29, 132, 358, 755, 1371, 2254, 3452, 5013, 6985, 9416, 12354, 15847, 19943, 24690, 30136, 36329, 43317, 51148, 59870, 69531, 80179, 91862, 104628, 118525, 133601, 149904, 167482, 186383, 206655, 228346, 251504, 276177, 302413, 330260, 359766, 390979

Offset: 0

Bruno Berselli, Jul 21 2015

Similar sequences, where P(s, m) = ((s-2)*m^2-(s-4)*m)/2 is the m-th s-gonal number:
A000578: P(3, m)*P( 3, m) - P(3, m-1)*P( 3, m-1);
A213772: P(3, m)*P( 4, m) - P(3, m-1)*P( 4, m-1) for m>0;
A005915: P(3, m)*P( 5, m) - P(3, m-1)*P( 5, m-1)   "    ;
A130748: P(3, m)*P( 6, m) - P(3, m-1)*P( 6, m-1) for m>1;
A027849: P(3, m)*P( 7, m) - P(3, m-1)*P( 7, m-1) for m>0;
A214092: P(3, m)*P( 8, m) - P(3, m-1)*P( 8, m-1)   "    ;
A100162: P(3, m)*P( 9, m) - P(3, m-1)*P( 9, m-1)   "    ;
A260260: P(3, m)*P(10, m) - P(3, m-1)*P(10, m-1), this sequence;
A100165: P(3, m)*P(11, m) - P(3, m-1)*P(11, m-1) for m>0.

Bruno Berselli, Table of n, a(n) for n = 0..1000
OEIS Wiki, Figurate numbers.
Wikipedia, Polygonal numbers: Table of values.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000217, A000292, A001107.
Subsequence of A047275.
Sequences of the same type (see comment): A000578, A005915, A027849, A100162, A100165, A130748, A213772, A214092.

Magma
```
[n*(16*n^2-21*n+7)/2: n in [0..40]];
```

Table[n (16 n^2 - 21 n + 7)/2, {n, 0, 40}]
LinearRecurrence[{4,-6,4,-1},{0,1,29,132},40] (* Harvey P. Dale, May 08 2025 *)

vector(40, n, n--; n*(16*n^2-21*n+7)/2)

Sage
```
[n*(16*n^2-21*n+7)/2 for n in (0..40)]
```

G.f.: x*(1 + 25*x + 22*x^2)/(1 - x)^4. [corrected by Georg Fischer, May 10 2019]
a(n) = A000217(n)*A001107(n) - A000217(n-1)*A001107(n-1), with A000217(-1) = 0.
a(n) = A000292(n) + 25*A000292(n-1) + 22*A000292(n-2), with A000292(-2) = A000292(-1) = 0.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - Wesley Ivan Hurt, Dec 18 2020
E.g.f.: exp(x)*x*(2 + 27*x + 16*x^2)/2. - Elmo R. Oliveira, Aug 08 2025

A100164 Structured rhombic triacontahedral numbers (vertex structure 11).

Original entry on oeis.org

1, 32, 143, 384, 805, 1456, 2387, 3648, 5289, 7360, 9911, 12992, 16653, 20944, 25915, 31616, 38097, 45408, 53599, 62720, 72821, 83952, 96163, 109504, 124025, 139776, 156807, 175168, 194909, 216080, 238731, 262912, 288673, 316064, 345135, 375936, 408517, 442928

Offset: 1

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Also structured triakis icosahedral numbers (vertex structure 11) (cf. A100172 = alternate vertex).

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A100165 (alternate vertex), A100145 for more on structured polyhedral numbers.

Cf. A001622, A100172.

[(1/6)*(50*n^3-60*n^2+16*n): n in [1..40]]; // Vincenzo Librandi, Jul 25 2011

a[n_] := (n*(5*n - 2)*(5*n - 4))/3; Array[a, 30] (* Amiram Eldar, Sep 20 2022 *)

a(n) = (1/6)*(50*n^3 - 60*n^2 + 16*n) = (1/3)*n*(5*n-2)*(5*n-4).
From Jaume Oliver Lafont, Sep 08 2009: (Start)
a(n) = (5*(n-1) + 1)*(5*(n-1) + 3)*(5*(n-1) + 5)/15.
G.f.: x*(1 + 28*x + 21*x^2)/(1-x)^4. (End)
Sum_{n>=1} 1/a(n) = 3*sqrt((25-2*sqrt(5))/5)*Pi/16 + 9*sqrt(5)*log(phi)/16 - 15*log(5)/32, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 20 2022

A268759 Triangle T(n,k) read by rows: T(n,k) = (1/4)(1 + k)(2 + k)(k - n)(1 + k - n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 9, 6, 0, 0, 10, 18, 18, 10, 0, 0, 15, 30, 36, 30, 15, 0, 0, 21, 45, 60, 60, 45, 21, 0, 0, 28, 63, 90, 100, 90, 63, 28, 0, 0, 36, 84, 126, 150, 150, 126, 84, 36, 0, 0, 45, 108, 168, 210, 225, 210, 168, 108, 45, 0, 0, 55, 135, 216, 280, 315

Offset: 0

Bradley Klee, Feb 20 2016

Off-diagonal elements of angular momentum matrices J_1^2 and J_2^2.
Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in the block-diagonal, Jordan-Schwinger form (cf. Harter, Klee, Schwinger). The triangle terms T(n,k) satisfy:(1/2)T(n,k)^(1/2) =  =  = -  = - . In the Dirac notation, we write elements m_{ij} of matrix M as =m_{ij}. Matrices for J_1^2 and J_2^2 are sparse. These equalities and the central-diagonal equalities of A141387 determine the only nonzero entries.
Notice that a(n) = T(n,k) is always a multiple of the triangular numbers, up to an offset. Conjecture: the triangle tabulating matrix elements  is determined entirely by the coefficients: binomial(n,p) (cf. A094053). Various sequences along the diagonals of matrix J_1^p lead to other numbers with geometric interpretations (Cf. A000567, A100165).

			0;
0,  0;
1,  0,  0;
3,  3,  0,  0;
6,  9,  6,  0,  0;
10, 18, 18, 10, 0,  0;
15, 30, 36, 30, 15, 0, 0;
...

W. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-346.
B. Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016.
J. Schwinger, On Angular Momentum , Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.

Cf. A114327, A094053, A141387.

Flatten[Table[(1/4) (1 + k) (2 + k) (k - n) (1 + k - n), {n, 0, 10, 1}, {k, 0, n, 1}]]

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

G.f.: x^2/((1-x)^3(1-x*y)^3)

A100173 Structured pentakis dodecahedral numbers (vertex structure 6).

Original entry on oeis.org

1, 32, 148, 404, 855, 1556, 2562, 3928, 5709, 7960, 10736, 14092, 18083, 22764, 28190, 34416, 41497, 49488, 58444, 68420, 79471, 91652, 105018, 119624, 135525, 152776, 171432, 191548, 213179, 236380, 261206, 287712

Offset: 1

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A100165 = alternate vertex; A100145 for more on structured numbers.

[(1/6)*(55*n^3-75*n^2+26*n): n in [1..40]]; // Vincenzo Librandi, Jul 26 2011

LinearRecurrence[{4,-6,4,-1},{1,32,148,404},40] (* Harvey P. Dale, Aug 04 2022 *)

a(n) = (1/6)*(55*n^3 - 75*n^2 + 26*n).

G.f.: x*(1 + 28*x + 26*x^2)/(1-x)^4. - Colin Barker, May 29 2012

cp's OEIS Frontend

A260260 a(n) = n*(16*n^2 - 21*n + 7)/2.

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A100164 Structured rhombic triacontahedral numbers (vertex structure 11).

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A268759 Triangle T(n,k) read by rows: T(n,k) = (1/4)*(1 + k)*(2 + k)*(k - n)*(1 + k - n).

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A100173 Structured pentakis dodecahedral numbers (vertex structure 6).

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Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A260260 a(n) = n(16n^2 - 21*n + 7)/2.

A268759 Triangle T(n,k) read by rows: T(n,k) = (1/4)(1 + k)(2 + k)(k - n)(1 + k - n).