A153792 -id:A153792 - cp's OEIS frontend

A153793 13 times pentagonal numbers: a(n) = 13n(3*n-1)/2.

0, 13, 65, 156, 286, 455, 663, 910, 1196, 1521, 1885, 2288, 2730, 3211, 3731, 4290, 4888, 5525, 6201, 6916, 7670, 8463, 9295, 10166, 11076, 12025, 13013, 14040, 15106, 16211, 17355, 18538, 19760, 21021, 22321, 23660, 25038, 26455

Offset: 0

Omar E. Pol, Jan 01 2009

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A153792.

[13*n*(3*n-1)/2: n in [0..60]]; // Wesley Ivan Hurt, Aug 29 2016

A153793:=n->13*n*(3*n-1)/2: seq(A153793(n), n=0..60); # Wesley Ivan Hurt, Aug 29 2016

Table[13*n*(3*n-1)/2, {n,0,25}] (* or *) LinearRecurrence[{3,-3,1}, {0,13,65}, 25] (* G. C. Greubel, Aug 29 2016 *)
13*PolygonalNumber[5,Range[0,40]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 16 2016 *)

a(n) = (39*n^2 - 13*n)/2; \\ Altug Alkan, Aug 29 2016

a(n) = (39*n^2 - 13*n)/2 = 13*A000326(n).
a(n) = 39*n + a(n-1) - 26 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
G.f.: 13*x*(1 + 2*x)/(1-x)^3. - Colin Barker, Feb 14 2012
From G. C. Greubel, Aug 29 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
E.g.f.: (13/2)*x*(2+3*x)*exp(x). (End)

A361521 Array read by descending antidiagonals. A(n, k) is the number of the nonempty multiset combinations of {0, 1} as defined in A361682.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 5, 4, 0, 0, 9, 12, 6, 0, 0, 14, 24, 21, 8, 0, 0, 20, 40, 45, 32, 10, 0, 0, 27, 60, 78, 72, 45, 12, 0, 0, 35, 84, 120, 128, 105, 60, 14, 0, 0, 44, 112, 171, 200, 190, 144, 77, 16, 0, 0, 54, 144, 231, 288, 300, 264, 189, 96, 18, 0

Offset: 0

Peter Luschny, Mar 22 2023

A detailed combinatorial interpretation can be found in A361682.

			[0] 0,  0,  0,   0,   0,   0,   0,    0, ...  A000004
[1] 0,  2,  5,   9,  14,  20,  27,   35, ...  A000096
[2] 0,  4, 12,  24,  40,  60,  84,  112, ...  A046092
[3] 0,  6, 21,  45,  78, 120, 171,  231, ...  A081266
[4] 0,  8, 32,  72, 128, 200, 288,  392, ...  A139098
[5] 0, 10, 45, 105, 190, 300, 435,  595, ...
[6] 0, 12, 60, 144, 264, 420, 612,  840, ...  A153792
[7] 0, 14, 77, 189, 350, 560, 819, 1127, ...
       | A028347 |     A163761
     A005843  A067725
.
[0] 0;
[1] 0,  0;
[2] 0,  2,   0;
[3] 0,  5,   4,   0;
[4] 0,  9,  12,   6,   0;
[5] 0, 14,  24,  21,   8,   0;
[6] 0, 20,  40,  45,  32,  10,   0;
[7] 0, 27,  60,  78,  72,  45,  12,  0;
[8] 0, 35,  84, 120, 128, 105,  60, 14,  0;
[9] 0, 44, 112, 171, 200, 190, 144, 77, 16, 0;

Rows: A000004, A000096, A046092, A081266, A139098, A153792.
Columns: A000004, A005843, A028347, A067725, A163761, A068380.
Cf. A361682.

A := (n, k) -> n*k*(4 + n*(k - 1))/2:
for n from 0 to 7 do seq(A(n, k), k = 0..7) od;

A(n, k) = n*k*(4 + n*(k - 1))/2.
T(n, k) = k*(n - k)*(4 + k*(n - k - 1))/2.
A(n, k) = A361682(n, k) - 1.

A277983 a(n) = 54n^2 - 78n + 36.

Original entry on oeis.org

36, 12, 96, 288, 588, 996, 1512, 2136, 2868, 3708, 4656, 5712, 6876, 8148, 9528, 11016, 12612, 14316, 16128, 18048, 20076, 22212, 24456, 26808, 29268, 31836, 34512, 37296, 40188, 43188, 46296, 49512, 52836, 56268, 59808, 63456, 67212, 71076, 75048, 79128, 83316

Offset: 0

Emeric Deutsch, Nov 11 2016

For n>=1, a(n) is the second Zagreb index of the triangular grid graph T[n] (see the West reference, p. 390). The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph.

The M-polynomial of the triangular grid graph T[n] is M(T[n], x, y) = 6*x^2*y^4 + 3*(n-1)*x^4*y^4 +6*(n-2)*x^4*y^6+3*(n-2)*(n-3)*x^6*y^6/2.

D. B. West, Introduction to Graph Theory, 2nd edition, Prentice-Hall, 2001.

E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
Eric Weisstein's World of Mathematics, .html">Triangular Grid Graph
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A153792.

[54*n^2-78*n+36: n in [0..50]]; // Bruno Berselli, Nov 11 2016

Maple
```
seq(54*n^2-78*n+36, n=0..40);
```

Table[54 n^2 - 78 n + 36, {n, 0, 50}] (* Bruno Berselli, Nov 11 2016 *)

a(n)=54*n^2-78*n+36 \\ Charles R Greathouse IV, Jun 17 2017

[54*n^2-78*n+36 for n in range(50)] # Bruno Berselli, Nov 11 2016

O.g.f.: 12*(14*x^2 - 8*x + 3)/(1 - x)^3.
E.g.f.: 6*(9*x^2 - 4*x + 6)*exp(x). - Bruno Berselli, Nov 11 2016
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Wesley Ivan Hurt, Jan 15 2022
a(n) = 12*A064225(n-1). - R. J. Mathar, Jul 22 2022

cp's OEIS Frontend

A153793 13 times pentagonal numbers: a(n) = 13n(3*n-1)/2.

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A361521 Array read by descending antidiagonals. A(n, k) is the number of the nonempty multiset combinations of {0, 1} as defined in A361682.

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A277983 a(n) = 54n^2 - 78n + 36.

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cp's OEIS Frontend

A153793 13 times pentagonal numbers: a(n) = 13*n*(3*n-1)/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A361521 Array read by descending antidiagonals. A(n, k) is the number of the nonempty multiset combinations of {0, 1} as defined in A361682.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A277983 a(n) = 54*n^2 - 78*n + 36.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A153793 13 times pentagonal numbers: a(n) = 13n(3*n-1)/2.

A277983 a(n) = 54n^2 - 78n + 36.