A083064 -id:A083064 - cp's OEIS frontend

A083066 5th row of number array A083064.

1, 5, 29, 173, 1037, 6221, 37325, 223949, 1343693, 8062157, 48372941, 290237645, 1741425869, 10448555213, 62691331277, 376147987661, 2256887925965, 13541327555789, 81247965334733, 487487792008397, 2924926752050381

Offset: 0

Paul Barry, Apr 21 2003

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=8, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)*charpoly(A,2). - Milan Janjic, Feb 21 2010

An Engel expansion of 3/2 to the base b := 6/5 as defined in A181565, with the associated series expansion 3/2 = b + b^2/5 + b^3/(5*29) + b^4/(5*29*173) + .... Cf. A007051. - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mudit Aggarwal and Samrith Ram, Generating Functions for Straight Polyomino Tilings of Narrow Rectangles, J. Int. Seq., Vol. 26 (2023), Article 23.1.4.
R. J. Mathar, Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices, arXiv:1406.7788 [math.CO], 2014, eq (43).
Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. A083064, A083065, A083067, A007051, A007583.

[(4*6^n+1)/5: n in [0..30]]; // Vincenzo Librandi, Nov 06 2011

f[n_]:=6^n; lst={}; Do[a=f[n]; Do[a-=f[m],{m,n-1,1,-1}]; AppendTo[lst,a/6],{n,1,30}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 10 2010 *)

a(n) = (4*6^n+1)/5.
G.f.: (1-2*x)/((1-6*x)*(1-x)).
E.g.f.: (4*exp(6*x)+exp(x))/5.
a(n) = 6*a(n-1)-1 with n>0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = 7*a(n-1)-6*a(n-2). - Vincenzo Librandi, Nov 04 2011
a(n) = 6^n - Sum_{i=0..n-1} 6^i for n>0. - Bruno Berselli, Jun 20 2013

A083065 4th row of number array A083064.

Original entry on oeis.org

1, 4, 19, 94, 469, 2344, 11719, 58594, 292969, 1464844, 7324219, 36621094, 183105469, 915527344, 4577636719, 22888183594, 114440917969, 572204589844, 2861022949219, 14305114746094, 71525573730469, 357627868652344

Offset: 0

Paul Barry, Apr 21 2003

Inverse binomial transform of A090040. [Paul Curtz, Jan 11 2009]
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=7, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)*charpoly(A,2). [Milan Janjic, Feb 21 2010]
For an integer x, consider the sequence P(x) of polynomials p_{1}, p_{2}, p_{3}, . . . defined by p_{1} = x-1, p_{n+1} = x*p_{1} - 1. P(5) = This sequence. P(1), P(2), P(3), P(4) are A000004, A123412, A007051, A007583 resp. [K.V.Iyer, Jun 22 2010]
It appears that if s(n) is a first order rational sequence of the form s(0)=2, s(n)= (3*s(n-1)+2)/(2*s(n-1)+3), n>0, then s(n)=2*a(n)/(2*a(n)-1), n>0.
An Engel expansion of 5/3 to the base b := 5/4 as defined in A181565, with the associated series expansion 5/3 = b + b^2/4 + b^3/(4*19) + b^4/(4*19*94) + .... Cf. A007051. - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A007583, A083066, A007051.

[(3*5^n+1)/4: n in [0..30]]; // Vincenzo Librandi, Nov 04 2011

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]*5-1 od: seq(a[n], n=1..22); # Zerinvary Lajos, Feb 22 2008

CoefficientList[Series[(1-2x)/((1-5x)(1-x)),{x,0,30}],x] (* or *) LinearRecurrence[{6,-5},{1,4},30] (* Harvey P. Dale, Jul 27 2022 *)

a(n) = (3*5^n+1)/4.
G.f.: (1-2*x)/((1-5*x)(1-x)).
E.g.f.: (3*exp(5*x) + exp(x))/4.
a(n) = 5*a(n-1)-1 with n>0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = 6*a(n-1)-5*a(n-2). - Vincenzo Librandi, Nov 04 2011
a(n) = 5^n - Sum_{i=0..n-1} 5^i. - Bruno Berselli, Jun 20 2013

A083068 7th row of number array A083064.

Original entry on oeis.org

1, 7, 55, 439, 3511, 28087, 224695, 1797559, 14380471, 115043767, 920350135, 7362801079, 58902408631, 471219269047, 3769754152375, 30158033218999, 241264265751991, 1930114126015927, 15440913008127415, 123527304065019319, 988218432520154551, 7905747460161236407

Offset: 0

Paul Barry, Apr 21 2003

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=10, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)*charpoly(A,2). - Milan Janjic, Feb 21 2010

Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A083067, A083066, A125837.

f[n_]:=8^n; lst={}; Do[a=f[n]; Do[a-=f[m],{m,n-1,1,-1}]; AppendTo[lst,a/8],{n,1,30}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 10 2010 *)
LinearRecurrence[{9,-8},{1,7},20] (* Harvey P. Dale, Jul 18 2019 *)

a(n)=(6*8^n+1)/7 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (6*8^n+1)/7.
G.f. (1-2*x)/((1-8*x)(1-x)).
E.g.f. (6*exp(8*x)+exp(x))/7.
a(n) = 8*a(n-1)-1 with n>0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = 8^n - sum(8^i, i=0..n-1) for n>0. - Bruno Berselli, Jun 20 2013
a(n) = 1 + A125837(n+1). - Alois P. Heinz, May 20 2023

A083069 Main diagonal of number array A083064.

Original entry on oeis.org

1, 2, 11, 94, 1037, 14006, 224695, 4185098, 88888889, 2122152922, 56288512931, 1642813694534, 52332842192581, 1807181134905134, 67253754435399407, 2683521610790452306, 114296029267382825201, 5175920921241202064498

Offset: 0

Paul Barry, Apr 21 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A083070, A083071.

Table[(n (n + 2)^n + 1) / (n + 1), {n, 0, 20}] (* Vincenzo Librandi, Jun 21 2013 *)

a(n) = (n*(n+2)^n+1)/(n+1).

A083067 6th row of number array A083064.

Original entry on oeis.org

1, 6, 41, 286, 2001, 14006, 98041, 686286, 4804001, 33628006, 235396041, 1647772286, 11534406001, 80740842006, 565185894041, 3956301258286, 27694108808001, 193858761656006, 1357011331592041, 9499079321144286

Offset: 0

Paul Barry, Apr 21 2003

Binomial transform of A052934 - Paul Barry, Apr 30 2003

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=9, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)*charpoly(A,2). [From Milan Janjic, Feb 21 2010]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A083066, A083068.

[(5*7^n+1)/6: n in [0..30]]; // Vincenzo Librandi, Nov 06 2011

f[n_]:=7^n; lst={}; Do[a=f[n]; Do[a-=f[m],{m,n-1,1,-1}]; AppendTo[lst,a/7],{n,1,30}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 10 2010 *)

a(n) = (5*7^n+1)/6.
G.f.: (1-2*x)/((1-7*x)*(1-x)).
E.g.f.: (5*exp(7*x)+exp(x))/6.
a(n) = 7*a(n-1)-1 with a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = 8*a(n-1)-7*a(n-2). - Vincenzo Librandi, Nov 06 2011
a(n) = 7^n - sum(7^i, i=0..n-1) for n>0. [Bruno Berselli, Jun 20 2013]

A083071 First subdiagonal of number array A083064.

Original entry on oeis.org

1, 3, 19, 173, 2001, 28087, 465011, 8888889, 192922993, 4690709411, 126370284195, 3738060156613, 120478742327009, 4203359652212463, 157854212399438371, 6349779403743490289, 272416890591642213921, 12417347368421052631579

Offset: 0

Paul Barry, Apr 21 2003

Cf. A083069, A083070.

a(n) = ((n+1)*(n+3)^n + 1)/(n+2).

A083070 First superdiagonal of number array A083064.

Original entry on oeis.org

1, 5, 43, 469, 6221, 98041, 1797559, 37665881, 888888889, 23343682141, 675462155171, 21356578028941, 732659790696133, 27107717023577009, 1076060070966390511, 45619867383437689201, 2057328526812890853617

Offset: 1

Paul Barry, Apr 21 2003

Cf. A083069, A083071.

a(n) = ((n-1)*(n+1)^n + 1)/n.

A083072 A subdiagonal of number array A083064.

Original entry on oeis.org

1, 4, 29, 286, 3511, 51668, 888889, 17538454, 390892451, 9720791092, 267004296901, 8031916155134, 262709978263279, 9285541905849316, 352765522430193905, 14337731083770642838, 620867368421052631579

Offset: 0

Paul Barry, Apr 21 2003

Cf. A083071, A083069.

a(n) = ((n+2)*(n+4)^n + 1)/(n+3).

A083073 A subdiagonal of number array A083064.

Original entry on oeis.org

1, 5, 41, 439, 5741, 88889, 1594405, 32574371, 747753161, 19071735493, 535461077009, 16419373641455, 546208347402901, 19598084579455217, 754617425461612781, 31043368421052631579, 1359040255729886061905

Offset: 0

Paul Barry, Apr 21 2003

Cf. A083072, A083071, A083069.

a(n) = ((n+3)*(n+5)^n + 1)/(n+4).

A083074 a(n) = n^3 - n^2 - n - 1.

Original entry on oeis.org

-1, -2, 1, 14, 43, 94, 173, 286, 439, 638, 889, 1198, 1571, 2014, 2533, 3134, 3823, 4606, 5489, 6478, 7579, 8798, 10141, 11614, 13223, 14974, 16873, 18926, 21139, 23518, 26069, 28798, 31711, 34814, 38113, 41614, 45323, 49246, 53389, 57758, 62359, 67198, 72281

Offset: 0

Paul Barry, Apr 21 2003

Values of tribonacci polynomial n^3 - n^2 - n - 1 for n >= 0. - Artur Jasinski, Nov 19 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..780
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Apart from initial terms, a column of A083064.

Cf. A000027, A002378, A028387.

[n^3 - n^2 - n - 1: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

Table[n^3 - n^2 - n - 1, {n, 0, 41}] (* Artur Jasinski, Nov 19 2006 *)
LinearRecurrence[{4,-6,4,-1},{-1,-2,1,14},50] (* Harvey P. Dale, Oct 11 2020 *)

a(n)=n^3-n^2-n-1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = n^3 + 5*n^2 + 7*n + 1 = (n*(n + 2)^3 + 1)/(n + 1) [with a different offset].
G.f.: (2*x^3+3*x^2+2*x-1)/(x-1)^4. - Alois P. Heinz, Jan 25 2023
From Elmo R. Oliveira, Aug 08 2025: (Start)
E.g.f.: exp(x)*(-1 - x + 2*x^2 + x^3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

Simpler definition from Alonso del Arte, Sep 16 2004

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A083066 5th row of number array A083064.

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A083065 4th row of number array A083064.

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A083068 7th row of number array A083064.

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A083069 Main diagonal of number array A083064.

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A083067 6th row of number array A083064.

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A083071 First subdiagonal of number array A083064.

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A083070 First superdiagonal of number array A083064.

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A083072 A subdiagonal of number array A083064.

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A083073 A subdiagonal of number array A083064.

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A083074 a(n) = n^3 - n^2 - n - 1.