A023548 -id:A023548 - cp's OEIS frontend

A210730 a(n) = a(n-1) + a(n-2) + n + 2 with n>1, a(0)=a(1)=0.

0, 0, 4, 9, 19, 35, 62, 106, 178, 295, 485, 793, 1292, 2100, 3408, 5525, 8951, 14495, 23466, 37982, 61470, 99475, 160969, 260469, 421464, 681960, 1103452, 1785441, 2888923, 4674395, 7563350, 12237778, 19801162, 32038975, 51840173, 83879185, 135719396

Offset: 0

Alex Ratushnyak, May 10 2012

Deleting the 0's leaves row 4 of the convolution array A213579. - Clark Kimberling, Jun 20 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A033818: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=0 (except first 2 terms and sign).
Cf. A002062: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=0 (except the first term and sign).
Cf. A065220: a(n)=a(n-1)+a(n-2)+n-3, a(0)=a(1)=0.
Cf. A001924: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=0 (except the first term).
Cf. A023548: a(n)=a(n-1)+a(n-2)+n,   a(0)=a(1)=0 (except first 2 terms).
Cf. A023552: a(n)=a(n-1)+a(n-2)+n+1, a(0)=a(1)=0 (except first 2 terms).
Cf. A210731: a(n)=a(n-1)+a(n-2)+n+3, a(0)=a(1)=0.
Cf. A000045, A213579.

F:=Fibonacci;; List([0..40], n-> F(n+3)+3*F(n+1)-n-5); # G. C. Greubel, Jul 08 2019

I:=[0, 0, 4, 9]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)-Self(n-3)+Self(n-4): n in [1..37]]; // Bruno Berselli, May 10 2012

F:=Fibonacci; [F(n+3)+3*F(n+1)-n-5: n in [0..40]]; // G. C. Greubel, Jul 08 2019

RecurrenceTable[{a[0]==a[1]==0, a[n]==a[n-1] +a[n-2] +n+2}, a, {n, 40}] (* Bruno Berselli, May 10 2012 *)
LinearRecurrence[{3,-2,-1,1},{0,0,4,9},40] (* Harvey P. Dale, Jul 24 2013 *)
With[{F=Fibonacci}, Table[F[n+3]+2*F[n+1]-n-5, {n, 40}]] (* G. C. Greubel, Jul 08 2019 *)

concat(vector(2), Vec(x^2*(4-3*x)/((1-x)^2*(1-x-x^2)) + O(x^50))) \\ Colin Barker, Mar 11 2017

vector(40, n, n--; f=fibonacci; f(n+3)+3*f(n+1)-n-5) \\ G. C. Greubel, Jul 08 2019

f=fibonacci; [f(n+3)+3*f(n+1)-n-5 for n in (0..40)] # G. C. Greubel, Jul 08 2019

G.f.: x^2*(4-3*x)/((1-x)^2*(1-x-x^2)). - Bruno Berselli, May 10 2012
a(n) = A210677(n)-1. - Bruno Berselli, May 10 2012
a(0)=0, a(1)=0, a(2)=4, a(3)=9, a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). - Harvey P. Dale, Jul 24 2013
a(n) = -5 + (2^(-1-n)*((1-sqrt(5))^n*(-7+5*sqrt(5)) + (1+sqrt(5))^n*(7+5*sqrt(5)))) / sqrt(5) - n. - Colin Barker, Mar 11 2017
a(n) = Fibonacci(n+3) + 3*Fibonacci(n+1) - n - 5. - G. C. Greubel, Jul 08 2019

A210731 a(n) = a(n-1) + a(n-2) + n + 3 with n>1, a(0) = a(1) = 0.

Original entry on oeis.org

0, 0, 5, 11, 23, 42, 74, 126, 211, 349, 573, 936, 1524, 2476, 4017, 6511, 10547, 17078, 27646, 44746, 72415, 117185, 189625, 306836, 496488, 803352, 1299869, 2103251, 3403151, 5506434, 8909618, 14416086, 23325739, 37741861, 61067637, 98809536

Offset: 0

Alex Ratushnyak, May 10 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A033818: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=0 (except first 2 terms and sign).
Cf. A002062: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=0 (except the first term and sign).
Cf. A065220: a(n)=a(n-1)+a(n-2)+n-3, a(0)=a(1)=0.
Cf. A001924: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=0 (except the first term).
Cf. A023548: a(n)=a(n-1)+a(n-2)+n,   a(0)=a(1)=0 (except first 2 terms).
Cf. A023552: a(n)=a(n-1)+a(n-2)+n+1, a(0)=a(1)=0 (except first 2 terms).
Cf. A210730: a(n)=a(n-1)+a(n-2)+n+2, a(0)=a(1)=0.

F:=Fibonacci;; List([0..40], n-> F(n+3)+4*F(n+1)-n-6); # G. C. Greubel, Jul 09 2019

F:=Fibonacci; [F(n+3)+4*F(n+1)-n-6: n in [0..40]]; // G. C. Greubel, Jul 09 2019

With[{F = Fibonacci}, Table[F[n+3]+4*F[n+1]-n-6, {n,0,40}]] (* G. C. Greubel, Jul 09 2019 *)
nxt[{n_,a_,b_}]:={n+1,b,a+b+n+4}; NestList[nxt,{1,0,0},40][[;;,2]] (* or *) LinearRecurrence[{3,-2,-1,1},{0,0,5,11},40] (* Harvey P. Dale, Dec 30 2024 *)

vector(40, n, n--; f=fibonacci; f(n+3)+4*f(n+1)-n-6) \\ G. C. Greubel, Jul 09 2019

f=fibonacci; [f(n+3)+4*f(n+1)-n-6 for n in (0..40)] # G. C. Greubel, Jul 09 2019

From Colin Barker, Jun 29 2012: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: x^2*(5-4*x)/((1-x)^2*(1-x-x^2)). (End)
a(n) = Fibonacci(n+3) + 4*Fibonacci(n+1) - (n+6). - G. C. Greubel, Jul 09 2019

A033814 Convolution of positive integers n with Lucas numbers L(k)(A000032) for k >= 4.

Original entry on oeis.org

7, 25, 61, 126, 238, 426, 737, 1247, 2079, 3432, 5628, 9188, 14955, 24293, 39409, 63874, 103466, 167534, 271205, 438955, 710387, 1149580, 1860216, 3010056, 4870543, 7880881, 12751717, 20632902, 33384934, 54018162, 87403433, 141421943, 228825735, 370248048

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000032, A000045, A023548, A023537.

List([1..40], n-> Lucas(1, -1, n+7)[2] -11*n-29 ) # G. C. Greubel, Jun 01 2019

[Lucas(n+7) - 11*n - 29 : n in [1..40]]; // G. C. Greubel, Jun 01 2019

Table[LucasL[n+7] -11*n-29, {n,1,40}] (* G. C. Greubel, Jun 01 2019 *)

vector(40, n, fibonacci(n+8) + fibonacci(n+6) -11*n-29) \\ G. C. Greubel, Jun 01 2019

from sympy import lucas
def a(n): return lucas(n+7) - 11*n - 29
print([a(n) for n in range(1, 35)]) # Michael S. Branicky, Jul 25 2021

[lucas_number2(n+7,1,-1) -11*n-29 for n in (1..40)] # G. C. Greubel, Jun 01 2019

a(n) = L(7)*(F(n+1)-1) + L(6)*F(n) - L(5)*n, F(n): Fibonacci (A000045) and L(n): Lucas (A000032).
G.f.: x*(7+4*x)/((1-x-x^2)*(1-x)^2).
a(n) = A000032(n+7) - 11*n - 29. - G. C. Greubel, Jun 01 2019

A033817 Convolution of natural numbers n >= 1 with Lucas numbers L(k) for k >= -4.

Original entry on oeis.org

7, 10, 16, 21, 28, 36, 47, 62, 84, 117, 168, 248, 375, 578, 904, 1429, 2276, 3644, 5855, 9430, 15212, 24565, 39696, 64176, 103783, 167866, 271552, 439317, 710764, 1149972, 1860623, 3010478, 4870980, 7881333, 12752184, 20633384, 33385431, 54018674, 87403960, 141422485

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3, -2, -1, 1).

Cf. A000032, A000045, A023548, A023537, A033814.

List([1..40], n-> Lucas(1, -1, n-1)[2] +4*n+1 ) # G. C. Greubel, Jun 01 2019

[Lucas(n-1) + 4*n + 1 : n in [1..40]]; // G. C. Greubel, Jun 01 2019

Table[LucasL[n-1] +4*n+1, {n,1,40}] (* G. C. Greubel, Jun 01 2019 *)

vector(40, n, fibonacci(n) + fibonacci(n-2) +4*n+1) \\ G. C. Greubel, Jun 01 2019

[lucas_number2(n-1,1,-1) +4*n+1 for n in (1..40)] # G. C. Greubel, Jun 01 2019

a(n) = L(-1)*(F(n+1)-1) + L(-2)*F(n) - L(-3)*n, F(n): Fibonacci (A000045), L(n): Lucas (A000032) with L(-n)=(-1)^n*L(n)
G.f.: x*(7-11*x)/((1-x-x^2)*(1-x)^2).
a(n) = Lucas(n-1) + 4*n + 1. - G. C. Greubel, Jun 01 2019

Terms a(31) onward added by G. C. Greubel, Jun 01 2019

A163704 Number of n X 2 binary arrays with all 1s connected, a path of 1s from left column to lower right corner, and no 1 having more than two 1s adjacent.

Original entry on oeis.org

1, 5, 11, 21, 38, 66, 112, 187, 309, 507, 828, 1348, 2190, 3553, 5759, 9329, 15106, 24454, 39580, 64055, 103657, 167735, 271416, 439176, 710618, 1149821, 1860467, 3010317, 4870814, 7881162, 12752008, 20633203, 33385245, 54018483, 87403764

Offset: 1

R. H. Hardin, Aug 03 2009

nonn

			All solutions for n=3:
  0 0   0 0   0 0   0 0   1 0   0 1   1 1   0 0   1 0   1 1   1 1
  0 0   0 1   1 0   1 1   1 0   0 1   1 0   1 1   1 1   0 1   0 1
  1 1   1 1   1 1   1 1   1 1   1 1   1 1   0 1   0 1   0 1   1 1

R. H. Hardin, Table of n, a(n) for n = 1..100

Cf. A023548. - R. J. Mathar, Aug 06 2009

Empirical: a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n >= 6.
Conjectures from Colin Barker, Mar 25 2018: (Start)
G.f.: x*(1 + 2*x - 2*x^2 - x^3 + x^4) / ((1 - x)^2*(1 - x - x^2)).
a(n) = -4 + (2^(-n)*((1-sqrt(5))^n*(-5+2*sqrt(5)) + (1+sqrt(5))^n*(5+2*sqrt(5)))) / sqrt(5) - n for n>1.
(End)

A033813 Convolution of natural numbers n >= 1 with Lucas numbers L(k) (A000032) for k >= 3.

Original entry on oeis.org

4, 15, 37, 77, 146, 262, 454, 769, 1283, 2119, 3476, 5676, 9240, 15011, 24353, 39473, 63942, 103538, 167610, 271285, 439039, 710475, 1149672, 1860312, 3010156, 4870647, 7880989, 12751829, 20633018, 33385054, 54018286, 87403561, 141422075, 228825871, 370248188

Offset: 1

Wolfdieter Lang

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000032, A000045, A023548, A023553.

LinearRecurrence[{3,-2,-1,1},{4,15,37,77},40] (* or *) Rest[ CoefficientList[ Series[x (4+3x)/((1-x-x^2)(1-x)^2),{x,0,40}],x]] (* Harvey P. Dale, May 23 2011 *)

a(n) = L(6)*(F(n+1)-1)+L(5)*F(n)-L(4)*n, F(n): Fibonacci (A000045).
G.f.: x*(4+3*x)/((1-x-x^2)*(1-x)^2).
a(0)=4, a(1)=15, a(2)=37, a(3)=77, a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). - Harvey P. Dale, May 23 2011

More terms from Jason Yuen, Aug 27 2025

cp's OEIS Frontend

A210730 a(n) = a(n-1) + a(n-2) + n + 2 with n>1, a(0)=a(1)=0.

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A210731 a(n) = a(n-1) + a(n-2) + n + 3 with n>1, a(0) = a(1) = 0.

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A033814 Convolution of positive integers n with Lucas numbers L(k)(A000032) for k >= 4.

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A033817 Convolution of natural numbers n >= 1 with Lucas numbers L(k) for k >= -4.

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A163704 Number of n X 2 binary arrays with all 1s connected, a path of 1s from left column to lower right corner, and no 1 having more than two 1s adjacent.

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A033813 Convolution of natural numbers n >= 1 with Lucas numbers L(k) (A000032) for k >= 3.

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