A081659 -id:A081659 - cp's OEIS frontend

A132921 Triangle read by rows: T(n,k) = n + Fibonacci(k) - 1, 1 <= k <= n.

1, 2, 2, 3, 3, 4, 4, 4, 5, 6, 5, 5, 6, 7, 9, 6, 6, 7, 8, 10, 13, 7, 7, 8, 9, 11, 14, 19, 8, 8, 9, 10, 12, 15, 20, 28, 9, 9, 10, 11, 13, 16, 21, 29, 42, 10, 10, 11, 12, 14, 17, 22, 30, 43, 64, 11, 11, 12, 13, 15, 18, 23, 31, 44, 65, 99, 12, 12, 13, 14, 16, 19, 24, 32, 45, 66, 100, 155

Offset: 1

Gary W. Adamson, Sep 05 2007

Right border = A081659, row sums = A132922: (1, 4, 10, 19, 32, ...).

			First few rows of the triangle are:
  1;
  2, 2;
  3, 3, 4;
  4, 4, 5, 6;
  5, 5, 6, 7, 9;
  ...
Column 3 = 4, 5, 6, 7, ...; since A081659(2) = 4.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Row sums are A132922.

Cf. A127648, A127647, A081659.

T[n_,k_]:=n+Fibonacci[k]-1;Table[T[n,k],{n,12},{k,n}]//Flatten (* James C. McMahon, Mar 09 2025 *)

T(n,k)=if(k<=n, n + fibonacci(k) - 1, 0) \\ Andrew Howroyd, Sep 01 2018

Equals (A127648 * A000012 + A000012 * A127647) - A000012 as infinite lower triangular matrices.

Name clarified and terms a(56) and beyond from Andrew Howroyd, Sep 01 2018

A081660 n+A001045(n+1).

Original entry on oeis.org

1, 2, 5, 8, 15, 26, 49, 92, 179, 350, 693, 1376, 2743, 5474, 10937, 21860, 43707, 87398, 174781, 349544, 699071, 1398122, 2796225, 5592428, 11184835, 22369646, 44739269, 89478512, 178956999, 357913970, 715827913, 1431655796, 2863311563

Offset: 0

Paul Barry, Mar 26 2003

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

Cf. A001045, A081659.

[2^(n+1)/3+n+(-1)^n/3: n in [0..40]]; // Vincenzo Librandi, Aug 10 2013

Table[2^(n + 1)/3 + n + (-1)^n/3, {n, 0, 40}] (* Vincenzo Librandi, Aug 10 2013 *)
LinearRecurrence[{3,-1,-3,2},{1,2,5,8},40] (* Harvey P. Dale, Feb 23 2025 *)

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

G.f.: (1-x-2*x^3)/((1+x)*(1-2*x)*(1-x)^2). [Bruno Berselli, Aug 11 2013]

A081662 Partial sums of n + Fibonacci(n+1).

Original entry on oeis.org

1, 3, 7, 13, 22, 35, 54, 82, 124, 188, 287, 442, 687, 1077, 1701, 2703, 4316, 6917, 11116, 17900, 28866, 46598, 75277, 121668, 196717, 318135, 514579, 832417, 1346674, 2178743, 3525042, 5703382, 9227992, 14930912, 24158411, 39088798

Offset: 0

Paul Barry, Mar 26 2003

Harvey P. Dale, Table of n, a(n) for n = 0..1000
W. Kuszmaul, Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations, arXiv preprint arXiv:1509.08216 [cs.DM], 2015-2017.
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A081659, A000045.

Accumulate[Table[Total[{n,Fibonacci[n+1]}],{n,0,40}]] (* or *) LinearRecurrence[ {4,-5,1,2,-1},{1,3,7,13,22},41] (* Harvey P. Dale, Nov 19 2011 *)

a(n) = (1 - 2*sqrt(5)/5)*(sqrt(5)/2 - 1/2)^n*(-1)^n + (sqrt(5)/2 + 1/2)^n*(2*sqrt(5)/5 + 1) + (n^2 + n - 2)/2.
G.f.: (x^3 + x - 1)/((1-x)^3*(x^2 + x - 1)).
a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5); a(0)=1, a(1)=3, a(2)=7, a(3)=13, a(4)=22. - Harvey P. Dale, Nov 19 2011
E.g.f.: exp(x)*(x^2 + 2*x - 2)/2 + 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Feb 13 2023

A081663 a(n) = Fibonacci(2n+1) + n*2^(n-1).

Original entry on oeis.org

1, 3, 9, 25, 66, 169, 425, 1058, 2621, 6485, 16066, 39921, 99601, 249666, 628917, 1592029, 4048866, 10341577, 26517113, 68226722, 176065901, 455514533, 1181040514, 3067684065, 7980068641, 20784441474, 54188706405, 141395801773

Offset: 0

Paul Barry, Mar 26 2003

Binomial transform of n + Fibonacci(n+1), A081659.

Index entries for linear recurrences with constant coefficients, signature (7,-17,16,-4).

Cf. A000045, A001519, A001787, A081659.

Table[Fibonacci[2n+1]+n 2^(n-1),{n,0,30}] (* or *) LinearRecurrence[{7,-17,16,-4},{1,3,9,25},30] (* Harvey P. Dale, Sep 17 2020 *)

a(n) = A001519(n)+A001787(n).

a(n) = 7*a(n-1)-17*a(n-2)+16*a(n-3)-4*a(n-4). G.f.: -(3*x^3-5*x^2+4*x-1) / ((2*x-1)^2*(x^2-3*x+1)). - Colin Barker, Jun 04 2013

Definition corrected by Matt Lehman, May 21 2010

A132919 Triangle read by rows: T(n,k) = Fibonacci(n) + k - 1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 13, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 34, 35, 36, 37, 38, 39, 40, 41, 42, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

Gary W. Adamson, Sep 05 2007

Left border = Fibonacci numbers, right border = A081659.

Infinite lower triangular matrix by rows: n-th row = n terms of: F(n) followed by (F(n) + 1), (F(n) + 2), (F(n) + 3), ...

			First few rows of the triangle:
  1;
  1,  2;
  2,  3,  4;
  3,  4,  5,  6;
  5,  6,  7,  8,  9;
  8,  9, 10, 11, 12, 13;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A132920.
Cf. A127647, A127648, A081659, A000045.
Cf. A132921, A132923.

T[n_,k_]:=Fibonacci[n]+k-1;Table[T[n,k],{n,11},{k,n}]//Flatten (* James C. McMahon, Mar 09 2025 *)

T(n,k) = if(k<=n, fibonacci(n) + k - 1, 0); \\ Andrew Howroyd, Aug 10 2018

Name changed and terms a(56) and beyond from Andrew Howroyd, Aug 10 2018

A135222 Triangle A049310 + A000012 - I, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 4, 1, 1, 1, 4, 1, 5, 1, 1, 2, 1, 7, 1, 6, 1, 1, 1, 5, 1, 11, 1, 7, 1, 1, 2, 1, 11, 1, 16, 1, 8, 1, 1, 1, 6, 1, 21, 1, 22, 1, 9, 1, 1, 2, 1, 16, 1, 36, 1, 29, 1, 10, 1, 1, 1, 7, 1, 36, 1, 57, 1, 37, 1, 11, 1, 1, 2, 1, 22, 1, 71, 1, 85, 1, 46, 1, 12, 1, 1

Offset: 0

Gary W. Adamson, Nov 23 2007

Row sums = A081659: (1, 2 4, 6, 9, 13, 19, 28, ...).

			First few rows of the triangle:
  1;
  1, 1;
  2, 1,  1;
  1, 3,  1,  1;
  2, 1,  4,  1,  1;
  1, 4,  1,  5,  1, 1;
  2, 1,  7,  1,  6, 1, 1;
  1, 5,  1, 11,  1, 7, 1, 1;
  2, 1, 11,  1, 16, 1, 8, 1, 1;
...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A049310, A081659.

T:= proc(n, k) option remember;
      if k=n then 1
    else 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*Pi/2) )
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..15); # G. C. Greubel, Nov 20 2019

T[n_, k_]:= T[n, k]= If[k==n, 1, 1 + Abs[Simplify[((1+(-1)^(n-k))/2)* Binomial[(n+k)/2, (n-k)/2]*Cos[(n-k)*Pi/2]]] ]; Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

@CachedFunction
def T(n, k):
    if (k==n): return 1
    else: return 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*pi/2) )
[[T(n, k) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Nov 20 2019

T(n,k) = A049310(n,k) + A000012(n,k) - Identity matrix, as infinite lower triangular matrices.

T(n,k) = 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*Pi/2) ), with T(n,n) = 1. - G. C. Greubel, Nov 20 2019

More terms added and offset changed by G. C. Greubel, Nov 20 2019

A210677 a(n) = a(n-1) + a(n-2) + n + 1, a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 5, 10, 20, 36, 63, 107, 179, 296, 486, 794, 1293, 2101, 3409, 5526, 8952, 14496, 23467, 37983, 61471, 99476, 160970, 260470, 421465, 681961, 1103453, 1785442, 2888924, 4674396, 7563351, 12237779, 19801163, 32038976, 51840174, 83879186, 135719397, 219598621, 355318057

Offset: 0

Alex Ratushnyak, May 09 2012

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

Cf. A081659: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=1 (except first 2 terms and sign).
Cf. A001924: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=1 (except first 4 terms).
Cf. A000126: a(n)=a(n-1)+a(n-2)+n-2, a(0)=a(1)=1 (except first term).
Cf. A066982: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=1.
Cf. A030119: a(n)=a(n-1)+a(n-2)+n, a(0)=a(1)=1.
Cf. A210678: a(n)=a(n-1)+a(n-2)+n+2, a(0)=a(1)=1.

LinearRecurrence[{3, -2, -1, 1}, {1, 1, 5, 10}, 39] (* Jean-François Alcover, Oct 05 2017 *)

From Colin Barker, Jun 30 2012: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: (1 - 2*x + 4*x^2 - 2*x^3)/((1 - x)^2*(1 - x - x^2)). (End)
E.g.f.: exp(x/2)*(25*cosh(sqrt(5)*x/2) + 7*sqrt(5)*sinh(sqrt(5)*x/2))/5 - exp(x)*(4 + x). - Stefano Spezia, Feb 24 2023

A210678 a(n) = a(n-1)+a(n-2)+n+2, a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 6, 12, 24, 43, 75, 127, 212, 350, 574, 937, 1525, 2477, 4018, 6512, 10548, 17079, 27647, 44747, 72416, 117186, 189626, 306837, 496489, 803353, 1299870, 2103252, 3403152, 5506435, 8909619, 14416087, 23325740, 37741862, 61067638, 98809537, 159877213, 258686789, 418564042

Offset: 0

Alex Ratushnyak, May 09 2012

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

Cf. A081659: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=1 (except first 2 terms and sign).
Cf. A001924: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=1 (except first 4 terms).
Cf. A000126: a(n)=a(n-1)+a(n-2)+n-2, a(0)=a(1)=1 (except first term).
Cf. A066982: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=1.
Cf. A030119: a(n)=a(n-1)+a(n-2)+n,   a(0)=a(1)=1.
Cf. A210677: a(n)=a(n-1)+a(n-2)+n+1, a(0)=a(1)=1.

LinearRecurrence[{3,-2,-1,1},{1,1,6,12},40] (* Harvey P. Dale, Dec 10 2014 *)
nxt[{n_,a_,b_}]:={n+1,b,a+b+n+3}; NestList[nxt,{1,1,1},40][[;;,2]] (* Harvey P. Dale, Mar 19 2023 *)

From Colin Barker, Jun 30 2012: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: (1 -2*x + 5*x^2 - 3*x^3)/((1 - x)^2*(1 - x - x^2)). (End)

A125100 Triangle read by rows: T(n,k) = Fibonacci(k+1)binomial(n,k) + (k+1)binomial(n,k+1) (0 <= k <= n).

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 9, 9, 3, 5, 16, 24, 16, 5, 6, 25, 50, 50, 30, 8, 7, 36, 90, 120, 105, 54, 13, 8, 49, 147, 245, 280, 210, 98, 21, 9, 64, 224, 448, 630, 616, 420, 176, 34, 10, 81, 324, 756, 1260, 1512, 1344, 828, 315, 55, 11, 100, 450, 1200, 2310, 3276, 3570, 2880, 1620

Offset: 0

Gary W. Adamson, Nov 20 2006

Binomial transform of the bidiagonal matrix with the Fibonacci numbers (1, 1, 2, 3, 5, 8, ...) in the main diagonal and (1, 2, 3, ...) in the subdiagonal.

Sum of terms in row n = n*2^(n-1) + Fibonacci(2n+1) (A081663).

			First few rows of the triangle:
  1;
  2,   1;
  3,   4,   2;
  4,   9,   9,   3;
  5,  16,  24,  16,   5;
  6,  25,  50,  50,  30,   8;
  7,  36,  90, 120, 105,  54,  13;
  8,  49, 147, 245, 280, 210,  98,  21;
  ...

Cf. A081663, A081659.

Cf. A000045.

with(combinat): T:=(n,k)->binomial(n,k)*fibonacci(k+1)+(k+1)*binomial(n,k+1): for n from 0 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Edited by N. J. A. Sloane, Nov 29 2006

cp's OEIS Frontend

A132921 Triangle read by rows: T(n,k) = n + Fibonacci(k) - 1, 1 <= k <= n.

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A081660 n+A001045(n+1).

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Magma

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A081662 Partial sums of n + Fibonacci(n+1).

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A081663 a(n) = Fibonacci(2n+1) + n*2^(n-1).

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A132919 Triangle read by rows: T(n,k) = Fibonacci(n) + k - 1.

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A135222 Triangle A049310 + A000012 - I, read by rows.

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A210677 a(n) = a(n-1) + a(n-2) + n + 1, a(0) = a(1) = 1.

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A210678 a(n) = a(n-1)+a(n-2)+n+2, a(0)=a(1)=1.

A125100 Triangle read by rows: T(n,k) = Fibonacci(k+1)binomial(n,k) + (k+1)binomial(n,k+1) (0 <= k <= n).