A081422 -id:A081422 - 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

A081437 Diagonal in array of n-gonal numbers A081422.

Original entry on oeis.org

1, 10, 33, 76, 145, 246, 385, 568, 801, 1090, 1441, 1860, 2353, 2926, 3585, 4336, 5185, 6138, 7201, 8380, 9681, 11110, 12673, 14376, 16225, 18226, 20385, 22708, 25201, 27870, 30721, 33760, 36993, 40426, 44065, 47916, 51985, 56278, 60801, 65560

Offset: 0

Paul Barry, Mar 21 2003

One of a family of sequences with palindromic generators.
For q a prime power, a(q-1) = q^3 + q^2 - q is the number of pairs of commuting nilpotent 2*2 matrices with coefficients in GF(q). (Proof: the zero matrix commutes with all q^2 nilpotent matrices, there are q^2-1 nonzero nilpotent matrices, all conjugate, each commuting with q nilpotent matrices.) - Mark Wildon, Jun 20 2017
Also the cyclomatic number (= circuit rank) of the n+1 X n+1 rook graph. - Eric W. Weisstein, Jun 20 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Circuit Rank
Eric Weisstein's World of Mathematics, Rook Graph
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081435, A081436, A081437, A081438.

Equals A027620(n-1) + 1.

List([0..40], n-> (n+1)^3+n*(n+1)); # G. C. Greubel, Aug 14 2019

[n^3+4*n^2+4*n+1: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013

a:=n->sum(n*k, k=0..n):seq(a(n)+sum(n*k, k=2..n), n=1..40); # Zerinvary Lajos, Jun 10 2008
a:=n->sum(-2+sum(2+sum(2, j=1..n),j=1..n),j=1..n):seq(a(n)/2,n=1..40); # Zerinvary Lajos, Dec 06 2008

Table[n^3 + 4 n^2 + 4n + 1, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 10, 33, 76}, 40] (* Harvey P. Dale, Jan 24 2012 *)
CoefficientList[Series[(1 + 5 x - 7 x^2 + x^3)/(1 - x)^5, {x, 0, 60}], x] (* Vincenzo Librandi, Aug 08 2013 *)

vector(40, n, n--; (n+1)^3+n*(n+1)) \\ G. C. Greubel, Aug 14 2019

[(n+1)^3+n*(n+1) for n in (0..40)] # G. C. Greubel, Aug 14 2019

a(n) = n^3 + 4*n^2 + 4*n + 1.
G.f.: (1 +5*x -7*x^2 +x^3)/(1-x)^5.
a(0)=1, a(1)=10, a(2)=33, a(3)=76; for n>3, a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Harvey P. Dale, Jan 24 2012
E.g.f.: (1 +9*x +7*x^2 +x^3)*exp(x). - G. C. Greubel, Aug 14 2019

A081435 Diagonal in array of n-gonal numbers A081422.

Original entry on oeis.org

1, 5, 18, 46, 95, 171, 280, 428, 621, 865, 1166, 1530, 1963, 2471, 3060, 3736, 4505, 5373, 6346, 7430, 8631, 9955, 11408, 12996, 14725, 16601, 18630, 20818, 23171, 25695, 28396, 31280, 34353, 37621, 41090, 44766, 48655, 52763, 57096, 61660

Offset: 0

Paul Barry, Mar 21 2003

One of a family of sequences with palindromic generators.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081434, A081436, A081437.

List([0..40], n-> (n+1)*(2*(n+1)^2-3*n)/2); # G. C. Greubel, Aug 14 2019

[(2*n^3+3*n^2+3*n+2)/2: n in [0..40]]; // Vincenzo Librandi, Aug 08 2013

a := n-> (n+1)*(2*(n+1)^2-3*n)/2; seq(a(n), n = 0..40); # G. C. Greubel, Aug 14 2019

Table[(n^3 +(n+1)^3 -1)/2 +1, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, May 04 2011 *)
CoefficientList[Series[(1 +3x^2 -4x^3)/(1-x)^5, {x,0,40}], x] (* Vincenzo Librandi, Aug 08 2013 *)
LinearRecurrence[{4,-6,4,-1},{1,5,18,46},40] (* Harvey P. Dale, Dec 28 2024 *)

vector(40, n, n--; (n+1)*(2*(n+1)^2-3*n)/2) \\ G. C. Greubel, Aug 14 2019

[(n+1)*(2*(n+1)^2-3*n)/2 for n in (0..40)] # G. C. Greubel, Aug 14 2019

a(n) = (2*n^3 +3*n^2 +3*n +2)/2.
G.f.: (1 +3*x^2 -4*x^3)/(1-x)^5.
E.g.f.: (2 +8*x +9*x^2 +2*x^3)*exp(x)/2. - G. C. Greubel, Aug 14 2019

A081438 Diagonal in array of n-gonal numbers A081422.

Original entry on oeis.org

1, 11, 36, 82, 155, 261, 406, 596, 837, 1135, 1496, 1926, 2431, 3017, 3690, 4456, 5321, 6291, 7372, 8570, 9891, 11341, 12926, 14652, 16525, 18551, 20736, 23086, 25607, 28305, 31186, 34256, 37521, 40987, 44660, 48546, 52651, 56981, 61542, 66340

Offset: 0

Paul Barry, Mar 21 2003

One of a family of sequences with palindromic generators.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081436, A081437, A081441, A177058.

List([0..45], n-> (2*n^3+9*n^2+9*n+2)/2); # G. C. Greubel, Aug 14 2019

[(2*n^3+9*n^2+9*n+2)/2: n in [0..45]]; // Vincenzo Librandi, Aug 08 2013

seq((2*n^3+9*n^2+9*n+2)/2, n=0..45); # G. C. Greubel, Aug 14 2019

CoefficientList[Series[(1 +6x -9x^2 +2x^3)/(1-x)^5, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 08 2013 *)
LinearRecurrence[{4,-6,4,-1},{1,11,36,82},50] (* Harvey P. Dale, Jan 20 2022 *)

vector(45, n, n--; (2*n^3+9*n^2+9*n+2)/2) \\ G. C. Greubel, Aug 14 2019

[(2*n^3+9*n^2+9*n+2)/2 for n in (0..45)] # G. C. Greubel, Aug 14 2019

a(n) = (2*n^3+9*n^2+9*n+2)/2.
G.f.: (1+6*x-9*x^2+2*x^3)/(1-x)^5.
From Bruno Berselli, Jun 04 2010: (Start)
G.f.: (1+7*x-2*x^2)/(1-x)^4 (simplified).
a(n) = (n+1)*(2*n^2+7*n+2)/2.
a(n) -4*a(n-1) +6*a(n-2) -4*a(n-3) +a(n-4) = 0, with n>3.
a(n) = (A177058(n+3) + A177058(n+2))/2. (End)
E.g.f.: (1/2)*exp(x)*(2 +20*x + 15*x^2 + 2*x^3). - Stefano Spezia, Aug 15 2019

A081423 Subdiagonal of array of n-gonal numbers A081422.

Original entry on oeis.org

1, 3, 12, 34, 75, 141, 238, 372, 549, 775, 1056, 1398, 1807, 2289, 2850, 3496, 4233, 5067, 6004, 7050, 8211, 9493, 10902, 12444, 14125, 15951, 17928, 20062, 22359, 24825, 27466, 30288, 33297, 36499, 39900, 43506, 47323, 51357, 55614, 60100

Offset: 0

Paul Barry, Mar 21 2003

One of a family of sequences with palindromic generators.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081435, A081436, A081437.

List([0..40], n-> (2*n^3+n^2+n+2)/2); # G. C. Greubel, Aug 14 2019

[(2*n^3+n^2+n+2)/2: n in [0..40]]; // Vincenzo Librandi, Aug 08 2013

a := n-> (2*n^3+n^2+n+2)/2; seq(a(n), n = 0..40); # G. C. Greubel, Aug 14 2019

CoefficientList[Series[(1 -2x +7x^2 -6x^3)/(1-x)^5, {x,0,40}], x] (* Vincenzo Librandi, Aug 08 2013 *)

vector(40, n, n--; (2*n^3+n^2+n+2)/2) \\ G. C. Greubel, Aug 14 2019

[(2*n^3+n^2+n+2)/2 for n in (0..40)] # G. C. Greubel, Aug 14 2019

a(n) = (2*n^3 + n^2 + n + 2)/2.
G.f.: (1 -2*x +7*x^2 -6*x^3)/(1-x)^5.
E.g.f.: (2 +4*x +7*x^2 +2*x^3)*exp(x)/2. - G. C. Greubel, Aug 14 2019

A081441 a(n) = (2*n^3 - n^2 - n + 2)/2.

Original entry on oeis.org

1, 1, 6, 22, 55, 111, 196, 316, 477, 685, 946, 1266, 1651, 2107, 2640, 3256, 3961, 4761, 5662, 6670, 7791, 9031, 10396, 11892, 13525, 15301, 17226, 19306, 21547, 23955, 26536, 29296, 32241, 35377, 38710, 42246, 45991, 49951, 54132, 58540, 63181

Offset: 0

Paul Barry, Mar 21 2003

Diagonal in array of n-gonal numbers A081422.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005564, A081422, A081436, A081437, A081438.

List([0..40], n-> (2*n^3-n^2-n+2)/2); # G. C. Greubel, Aug 14 2019

[(2*n^3-n^2-n+2)/2: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013

a:= n-> (2*n^3-n^2-n+2)/2: seq(a(n), n=0..50); # Zerinvary Lajos, Sep 13 2006

Table[(2n^3-n^2-n+2)/2,{n,0,40}] (* Harvey P. Dale, May 29 2012 *)
CoefficientList[Series[(1 - 4 x + 11 x^2 - 8 x^3) / (1 - x)^5, {x, 0, 50}],x] (* Vincenzo Librandi, Aug 08 2013 *)

vector(40, n, n--; (2*n^3-n^2-n+2)/2) \\ G. C. Greubel, Aug 14 2019

[(2*n^3-n^2-n+2)/2 for n in (0..40)] # G. C. Greubel, Aug 14 2019

G.f.: (1 -4*x +11*x^2 -8*x^3)/(1-x)^5.
a(n) = (n + 1)*(2*n^2 - 3*n + 2)/2 = (n-1)*A005564(n+1) - n*A005564(n), where A005564(0..2) = 0, -1, 0. - Bruno Berselli, May 19 2015
E.g.f.: (2 + 5*x^2 + 2*x^3)*exp(x)/2. - G. C. Greubel, Aug 14 2019

A095311 47-gonal numbers.

Original entry on oeis.org

1, 47, 138, 274, 455, 681, 952, 1268, 1629, 2035, 2486, 2982, 3523, 4109, 4740, 5416, 6137, 6903, 7714, 8570, 9471, 10417, 11408, 12444, 13525, 14651, 15822, 17038, 18299, 19605, 20956, 22352, 23793, 25279, 26810, 28386, 30007, 31673, 33384

Offset: 1

Gary W. Adamson, Jun 02 2004

			a(6) = 681 = 3*a(5) - 3*a(4) + a(3) = 3*455 - 3*274 + 138.
a(37) = 30007 since M^37 * [1 0 0] = [1 37 30007].

Albert H. Beiler, "Recreations in the Theory of Numbers", Dover, 1966, pp. 185-194.

Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A081422, A000326, A000384, A000566, A000567, ... (all polygonal sequences).

I:=[1,47,138]; [n le 3 select I[n]  else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jul 25 2015

a[n_] := (MatrixPower[{{1, 0, 0}, {1, 1, 0}, {1, 45, 1}}, n].{{1}, {0}, {0}})[[3, 1]]; Table[ a[n], {n, 40}] (* Robert G. Wilson v, Jun 05 2004 *)
LinearRecurrence[{3, -3, 1}, {1, 47, 138}, 40] (* Vincenzo Librandi, Jul 25 2015 *)
PolygonalNumber[47,Range[40]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 01 2016 *)

a(n)=([0,1,0; 0,0,1; 1,-3,3]^(n-1)*[1;47;138])[1,1] \\ Charles R Greathouse IV, Jun 17 2017

a(n+3) = 3*a(n+2) - 3*a(n+1) - a(n); a(1) = 1, a(2) = 47, a(3) = 138.
Let M = the 3 X 3 matrix [1 0 0 / 1 1 0 / 1 45 1]. Then M^n * [1 0 0] = [1 n a(n)].
From Colin Barker, Jul 27 2013: (Start)
a(n) = (n*(45*n-43))/2.
G.f.: -x*(44*x+1) / (x-1)^3. (End)
E.g.f.: exp(x)*(x + 45*x^2/2). - Nikolaos Pantelidis, Feb 10 2023

Edited by N. J. A. Sloane and Robert G. Wilson v, Jun 05 2004

A136311 Array read by antidiagonals: a(1) = 1; a(n+1) = a(n) + (largest k-gonal number <= a(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 6, 1, 2, 3, 4, 12, 1, 2, 3, 4, 8, 22, 1, 2, 3, 4, 5, 12, 43, 1, 2, 3, 4, 5, 10, 21, 79, 1, 2, 3, 4, 5, 6, 15, 37, 157, 1, 2, 3, 4, 5, 6, 12, 27, 73, 310, 1, 2, 3, 4, 5, 6, 7, 18, 49, 137, 610, 1, 2, 3, 4, 5, 6, 7, 14, 33, 84, 258, 1205, 1, 2, 3, 4, 5, 6, 7, 8, 21, 61

Offset: 1

Jonathan Vos Post, Mar 22 2008

			The array begins:
==================================================================
n=..|.1.|.2.|.3.|.4.|..5.|..6.|..7.|..8.|...9.|..10.|..11.|...12.|
==================================================================
k=3.|.1.|.2.|.3.|.6.|.12.|.22.|.43.|.79.|.157.|.310.|.610.|.1205.|.A060985
k=4.|.1.|.2.|.3.|.4.|..8.|.12.|.21.|.37.|..73.|.137.|.258.|..514.|.A060984
k=5.|.1.|.2.|.3.|.4.|..5.|.10.|.15.|.27.|..49.|..84.|.154.|..299.|
k=6.|.1.|.2.|.3.|.4.|..5.|..6.|.12.|.18.|..33.|..61.|.106.|..197.|
k=7.|.1.|.2.|.3.|.4.|..5.|..6.|..7.|.14.|..21.|..39.|..73.|..128.|
k=8.|.1.|.2.|.3.|.4.|..5.|..6.|..7.|..8.|..16.|..24.|..45.|...85.|
k=9.|.1.|.2.|.3.|.4.|..5.|..6.|..7.|..8.|...9.|..18.|..27.|...51.|
==================================================================

Cf. A060984, A060985.

A081422 := proc(k,n) n/2*((k-2)*n-k+4) ; end: A136311 := proc(k,n) option remember ; local aprev,n2 ; if n = 1 then RETURN(1) ; else aprev := A136311(k,n-1) ; for n2 from 0 do if A081422(k,n2) > aprev then RETURN( aprev+A081422(k,n2-1)); fi; od: fi ; end: for d from 4 to 20 do for n from 1 to d-3 do printf("%d,", A136311(d-n,n)) ; od: od: # R. J. Mathar, Jun 12 2008

More terms from R. J. Mathar, Jun 12 2008

cp's OEIS Frontend

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

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A081437 Diagonal in array of n-gonal numbers A081422.

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A081435 Diagonal in array of n-gonal numbers A081422.

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A081438 Diagonal in array of n-gonal numbers A081422.

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A081423 Subdiagonal of array of n-gonal numbers A081422.

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A081441 a(n) = (2*n^3 - n^2 - n + 2)/2.

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