A244418 -id:A244418 - cp's OEIS frontend

A081436 Fifth subdiagonal in array of n-gonal numbers A081422.

1, 7, 24, 58, 115, 201, 322, 484, 693, 955, 1276, 1662, 2119, 2653, 3270, 3976, 4777, 5679, 6688, 7810, 9051, 10417, 11914, 13548, 15325, 17251, 19332, 21574, 23983, 26565, 29326, 32272, 35409, 38743, 42280, 46026, 49987, 54169, 58578, 63220

Offset: 0

Paul Barry, Mar 21 2003

One of a family of sequences with palindromic generators.
Also as A(n) = (1/6)*(6*n^3 - 3*n^2 + 3*n), n>0: structured pentagonal diamond numbers (vertex structure 5). (Cf. A004068 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers.) - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF4 denominators of A156933. See A157705 for background information. - Johannes W. Meijer, Mar 07 2009
Row 1 of the convolution arrays A213831 and A213833. - Clark Kimberling, Jul 04 2012
Partial sums of A056109. - J. M. Bergot, Jun 22 2013
Number of ordered pairs of intersecting multisets of size 2, each chosen with repetition from {1,...,n}. - Robin Whitty, Feb 12 2014
Row sums of A244418. - L. Edson Jeffery, Jan 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Christian Barrientos, The number of spanning trees of cyclic snakes, Indones. J. Comb. (2025) Vol. 9, No. 1, 21-30. See p. 29.
J. A. Dias da Silva and Pedro J. Freitas, Counting Spectral Radii of Matrices with Positive Entries, arXiv:1305.1139 [math.CO], 2013.
Theorem of the Day, Lovász Local Lemma example involving intersecting pairs of multisets
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081434, A081435, A081437, A156933, A157705, A244418.

List([0..40], n-> (n+1)*(2*(n+1)^2-n)/2); # G. C. Greubel, Aug 14 2019

[(2*n^3+5*n^2+5*n+2)/2: n in [0..40]]; // Vincenzo Librandi, Jul 19 2011

A081436 := proc(n)
    (n+1)*(2*n^2+3*n+2)/2 ;
end proc:
seq(A081436(n),n=0..60) ; # R. J. Mathar, Jun 26 2013

LinearRecurrence[{4, -6, 4, -1}, {1, 7, 24, 58}, 40] (* Jean-François Alcover, Sep 21 2017 *)

a(n)=n^3+5/2*n*(n+1)+1 \\ Charles R Greathouse IV, Jun 20 2013

[(n+1)*(2*(n+1)^2-n)/2 for n in (0..40)] # G. C. Greubel, Aug 14 2019

a(n) = (n+1)*(2*n^2 + 3*n + 2)/2.
G.f.: (1+x)*(1+2*x)/(1-x)^4. (Convolution of A005408 and A016777.)
a(n) = A110449(n, n-1), for n>1.
a(n) = (n+1)*T(n+1) + n*T(n), where T( ) are triangular numbers. Binomial transform of [1, 6, 11, 6, 0, 0, 0, ...]. - Gary W. Adamson, Dec 28 2007
E.g.f.: exp(x)*(2 + 12*x + 11*x^2 + 2*x^3)/2. - Stefano Spezia, Apr 13 2021
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Apr 14 2021

G.f. simplified and crossrefs added by Johannes W. Meijer, Mar 07 2009

A129312 A minimal 2 X 2 subdeterminant array.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 8, 8, 4, 5, 11, 13, 11, 5, 6, 14, 18, 18, 14, 6, 7, 17, 23, 25, 23, 17, 7, 8, 20, 28, 32, 32, 28, 20, 8, 9, 23, 33, 39, 41, 39, 33, 23, 9, 10, 26, 38, 46, 50, 50, 46, 38, 26, 10, 11, 29, 43, 53, 59, 61, 59, 53, 43, 29, 11, 12, 32, 48, 60, 68, 72, 72, 68, 60

Offset: 1

Clark Kimberling, Apr 09 2007

Given that row 1 and column 1 are the sequence (1,2,3,4,...), T is the array of minimal positive subdeterminants in the sense that for each 2 X 2 submatrix
a b
c d,
d is the least integer for which the resulting
determinant is positive; indeed, the determinant is 1.
T(n,n)=A001844(n).
SUM{T(n,k): k=1,2,...,n}=A081436(n).
When T is written as the triangle
1
2 2
3 5 3
4 8 8 4
5 11 13 11 5, etc.,
the row sums are A006527 and the alternating row sums are 1,0,1,0,1,0,1,0,... (A059841).
The underlying function T is the same as in A244418, but this triangle's rows hold n+k constant, while in A244418, n is held constant on each row, and k <= n.
T(n,k) can be interpreted as a figurate number, with an (n-1) x (k-1) rectangle of dots interleaved with an n x k rectangle. The American flag illustrates T(5,6).

			Northwest corner:
1 2 3 4 5 6
2 5 8 11 14 17
3 8 13 18 23 28
4 11 18 25 32 39
T(2,2)=5 because 5 is the least positive integer x for which the determinant of the 2 X 2 matrix below is positive:
1 2
2 x

Cf. A001844, A081436, A006527, A059841.

Cf. A244418 (different triangle for the same function T).

(* Array version: *)
Grid[Table[SeriesCoefficient[Series[(n + (n - 1)*x)/(1 - x)^2, {x, 0, 12}], k], {n, 12}, {k, 0, 12}]] (* L. Edson Jeffery, Aug 23 2014 *)
(* Triangle version: *)
Grid[Table[SeriesCoefficient[Series[(n - k + (n - k - 1)*x)/(1 - x)^2, {x, 0, 12}], k], {n, 12}, {k, 0, n - 1}]] (* L. Edson Jeffery, Aug 23 2014 *)

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

Connection to A244418 and interpretation as figurate numbers from Allan C. Wechsler, Nov 18 2018

A243907 Numbers that can be expressed as nm + (n-1)(m-1), n = 2, 3, ... , m = n, n+1, n+2, ... in at least two different ways. Ordered increasingly.

Original entry on oeis.org

23, 32, 38, 41, 50, 53, 59, 68, 74, 77, 83, 86, 88, 95, 98, 104, 113, 116, 122, 123, 128, 131, 137, 138, 140, 143, 149, 158, 163, 167, 173, 176, 179, 182, 185, 188, 193, 194, 200, 203, 212, 213, 215, 218, 221, 228, 230, 233, 238, 239, 242, 248, 254, 257, 263

Offset: 2

Aki Halme, Jun 14 2014

nonn

This sequence was inspired by the flag of the United States. The 50 stars are placed in a rectangular grid with outside dimensions six stars wide by five stars high, but they could also be placed in a grid 17 stars wide by two stars high. This sequence lists, up to 200 stars, all numbers of stars that could be placed in a rectangular field in more than one arrangement.
This is the ordered list of integers that appear several times in A144650.
R(n,m) = n*m + (n-1)*(m-1) = (m-1)*(2*n-1) + n == n (mod (2*n-1)), and also with n interchanged with m. See A244418 for the table a(n,m) = R(n,m) for n >= m >= 1. - Wolfdieter Lang, Jul 10 2014

			23 = 8*2 + 7*1 = 5*3 +4*2.
32 = 11*2 + 10*1 = 5*4 + 4*3.
The first triple solution is 53 = 18*2 + 17*1 = 11*3 + 10*2 = 8*4 + 7*3.

The sequence A186041 lists all possible solutions, including single ones, and has four additional terms at the start. The sequence A140646 also refers to the Stars-and-Stripes, but gives the history, not the geometry of the current arrangement.
Cf. also A144650, with all values organized by rows (but with different offset).
Cf. A053726, A244418.

lista(nn=200) = {v = []; vres = []; for (n=2, nn, for (m=2, n, new = n*m + (n-1)*(m-1); if (! vecsearch(v, new), v = vecsort(concat(v, n*m + (n-1)*(m-1))), if (! vecsearch(vres, new), vres = vecsort(concat(vres, new)));););); for (i=1, min(60, #vres), print1(vres[i], ", "));} \\ Michel Marcus, Jun 29 2014

More terms from Michel Marcus, Jun 29 2014

Edited. Title reformulated. Crossrefs A053726 and A244418 added. - Wolfdieter Lang, Jul 10 2014

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A081436 Fifth subdiagonal in array of n-gonal numbers A081422.

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A129312 A minimal 2 X 2 subdeterminant array.

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A243907 Numbers that can be expressed as nm + (n-1)(m-1), n = 2, 3, ... , m = n, n+1, n+2, ... in at least two different ways. Ordered increasingly.

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cp's OEIS Frontend

A081436 Fifth subdiagonal in array of n-gonal numbers A081422.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A129312 A minimal 2 X 2 subdeterminant array.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A243907 Numbers that can be expressed as n*m + (n-1)*(m-1), n = 2, 3, ... , m = n, n+1, n+2, ... in at least two different ways. Ordered increasingly.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

A243907 Numbers that can be expressed as nm + (n-1)(m-1), n = 2, 3, ... , m = n, n+1, n+2, ... in at least two different ways. Ordered increasingly.