A053219 -id:A053219 - cp's OEIS frontend

A053220 a(n) = (3n-1) 2^(n-2).

1, 5, 16, 44, 112, 272, 640, 1472, 3328, 7424, 16384, 35840, 77824, 167936, 360448, 770048, 1638400, 3473408, 7340032, 15466496, 32505856, 68157440, 142606336, 297795584, 620756992, 1291845632, 2684354560, 5570035712, 11542724608, 23890755584, 49392123904

Offset: 1

Asher Auel, Jan 01 2000

Coefficients in the hypergeometric series identity 1 - 5*x/(x + 4) + 16*x*(x - 1)/((x + 4)*(x + 6)) - 44*x*(x - 1)*(x - 2)/((x + 4)*(x + 6)*(x + 8)) + ... = 0, valid in the half-plane Re(x) > 0. Cf. A276289. - Peter Bala, May 30 2019

For n>=2, a(n) is the total number of ones in runs of ones of length >=5 over all binary strings of length n+3. - Félix Balado, Aug 06 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Marcella Anselmo, Giuseppa Castiglione, Manuela Flores, Dora Giammarresi, Maria Madonia, and Sabrina Mantaci, Hypercubes and Isometric Words based on Swap and Mismatch Distance, arXiv:2303.09898 [math.CO], 2023.
F. K. Hwang and C. L. Mallows, Enumerating nested and consecutive partitions, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A053219, A053221, A132776, A276289.

Center elements from triangle A053218. Also a diagonal of triangle A056242.

a053220 n = a056242 (n + 1) n  -- Reinhard Zumkeller, May 08 2014

[(3*n-1)*2^(n-2): n in [1..50]]; // Vincenzo Librandi, May 09 2011

ListCorrelate[{1, 1}, Table[n 2^(n - 1), {n, 0, 28}]] (* or *) ListConvolve[{1, 1}, Table[n 2^(n - 1), {n, 0, 28}]] (* Ross La Haye, Feb 24 2007 *)
LinearRecurrence[{4, -4}, {1, 5}, 35] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)
Array[(3# - 1) 2^(# - 2) &, 35] (* Alonso del Arte, Sep 04 2018 *)
CoefficientList[Series[(1 + x)/(1 - 2 * x)^2, {x, 0, 50}], x] (* Stefano Spezia, Sep 04 2018 *)

PARI
```
a(n)=if(n<1,0,(3*n-1)*2^(n-2))
```

a(n)=(3*n-1)<<(n-2) \\ Charles R Greathouse IV, Apr 17 2012

G.f.: x*(1+x)/(1-2*x)^2.
a(n) = (3*n-1) * 2^(n-2).
E.g.f.: exp(2*x)*(1+3*x). The sequence 0, 1, 5, 16, ... has a(n) = ((3n-1)*2^n + 0^n)/4 (offset 0). It is the binomial transform of A032766. The sequence 1, 5, 16, ... has a(n) = (2+3n)*2^(n-1) (offset 0). It is the binomial transform of A016777. - Paul Barry, Jul 23 2003
Row sums of A132776(n-1). - Gary W. Adamson, Aug 29 2007
a(n+1) = det(f(i-j+1)){1 <= i, j <= n}, where f(0) = 1, f(1) = 5 and for k > 0, we have f(k+1) = 9 and f(-k) = 0. - _Mircea Merca, Jun 23 2012

A053218 Triangle read by rows: T(n,k) = T(n,k-1) + T(n-1,k-1) for k >= 2 with T(n,1) = n.

Original entry on oeis.org

1, 2, 3, 3, 5, 8, 4, 7, 12, 20, 5, 9, 16, 28, 48, 6, 11, 20, 36, 64, 112, 7, 13, 24, 44, 80, 144, 256, 8, 15, 28, 52, 96, 176, 320, 576, 9, 17, 32, 60, 112, 208, 384, 704, 1280, 10, 19, 36, 68, 128, 240, 448, 832, 1536, 2816, 11, 21, 40, 76, 144, 272, 512, 960, 1792, 3328

Offset: 1

Asher Auel, Jan 01 2000

Last term in each row gives A001792. Difference between center term of row 2n-1 and row sum of row n, (A053220(n+4) - A053221(n+4)) gives A045618(n).
For all integers k >= 2, if a sequence k,k-1,k+2,k-3,k+4,...,2,2k-2,1,2k-1, b0(n) with offset 1, is written, the sequence b0(2)-b0(1), b0(3)-b0(2), b0(4)-b0(3), ..., b0(2k-1)-b0(2k-2), b1(n) with offset 1, is written under it, the sequence b1(2)-b1(1), b1(3)-b1(2), b1(4)-b1(3), ..., b1(2k-2)-b1(2k-3), b2(n) with offset 1, is written under this, and so on until the sequence b(2k-3)(2)-b(2k-3)(1), b(2k-2)(n) with offset 1 (which will contain only one term), is written, and then the sequence b1(1); b1(2),b2(1); b1(3),b2(2),b3(1); ...; b1(2k-2), b2(2k-3), b3(2k-4), ..., b(2k-2)(1) is obtained, then this sequence will be identical to the first 2k^2-3k+1 terms of a(n), except that the first term of this sequence will be negative, the next two terms will be positive, the next three will be negative, the next four positive, and so on.
Subtriangle of triangle in A152920. - Philippe Deléham, Nov 21 2011

			Triangle T(n,k) begins:
  1;
  2,  3;
  3,  5,  8;
  4,  7, 12, 20;
  5,  9, 16, 28, 48;
  6, 11, 20, 36, 64, 112;
  7, 13, 24, 44, 80, 144, 256;
  ...

Cf. A053219 (reverse of this triangle), A053220 (center elements), A053221 (row sums), A001792, A045618, A152920.

NestList[FoldList[Plus, #[[1]] + 1, #] &, {1}, 10] // Grid (* Geoffrey Critzer, Jun 27 2013 *)

T(n, k) = n*2^(k-1) - (k-1)*2^(k-2). - Ya-Ping Lu, Mar 24 2023

A053221 Row sums of triangle A053218.

Original entry on oeis.org

1, 5, 16, 43, 106, 249, 568, 1271, 2806, 6133, 13300, 28659, 61426, 131057, 278512, 589807, 1245166, 2621421, 5505004, 11534315, 24117226, 50331625, 104857576, 218103783, 452984806, 939524069, 1946157028, 4026531811, 8321499106

Offset: 1

Asher Auel, Jan 01 2000

Considered as a vector, the sequence = A074909 * [1, 2, 3, ...], where A074909 is the beheaded Pascal's triangle as a matrix. - Gary W. Adamson, Mar 06 2012
a(n) is the sum of the upper left n X n subarray of A052509 (viewed as an infinite square array). For example (1+1+1) + (1+2+2) + (1+3+4) = 16. - J. M. Bergot, Nov 06 2012
Number of ternary strings of length n that contain at least one 2 and at most one 0. For example, a(3) = 16 since the strings are the 6 permutations of 201, the 3 permutations of 211, the 3 permutations of 220, the 3 permutations of 221, and 222. - Enrique Navarrete, Jul 25 2021

			a(4) = 4 + 7 + 12 + 20 = 43.

Michael De Vlieger, Table of n, a(n) for n = 1..3311
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A053218, A053219, A053220.

Cf. A006127, A074909.

[(n+2)*2^(n-1)-n-1: n in [1..50]]; // G. C. Greubel, Sep 03 2018

A053221 := proc(n) (n+2)*2^(n-1)-n-1 ; end proc: # R. J. Mathar, Sep 02 2011

Table[(n + 2)*2^(n - 1) - n - 1, {n, 29}] (* or *)
Rest@ CoefficientList[Series[-x (-1 + x + x^2)/((2 x - 1)^2*(x - 1)^2), {x, 0, 29}], x] (* Michael De Vlieger, Sep 22 2017 *)
LinearRecurrence[{6,-13,12,-4},{1,5,16,43},30] (* Harvey P. Dale, Jun 28 2021 *)

vector(50,n, (n+2)*2^(n-1)-n-1) \\ G. C. Greubel, Sep 03 2018

a(n) = (n+2)*2^(n-1)-n-1. - Vladeta Jovovic, Feb 28 2003
G.f.: -x*(-1+x+x^2) / ( (2*x-1)^2*(x-1)^2 ). - R. J. Mathar, Sep 02 2011
a(n) = (1/2) * Sum_{k=1..n} Sum_{i=1..n} C(k,i) + C(n,k). - Wesley Ivan Hurt, Sep 22 2017
E.g.f.: exp(x)*(exp(x)-1)*(1+x). - Enrique Navarrete, Jul 25 2021
a(n+1) = 2*a(n) + A006127(n). - Ya-Ping Lu, Jan 01 2024

cp's OEIS Frontend

A053220 a(n) = (3n-1) 2^(n-2).

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A053218 Triangle read by rows: T(n,k) = T(n,k-1) + T(n-1,k-1) for k >= 2 with T(n,1) = n.

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A053221 Row sums of triangle A053218.

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cp's OEIS Frontend

A053220 a(n) = (3*n-1) * 2^(n-2).

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Magma

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Formula

A053218 Triangle read by rows: T(n,k) = T(n,k-1) + T(n-1,k-1) for k >= 2 with T(n,1) = n.

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A053221 Row sums of triangle A053218.

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Magma

Maple

Mathematica

PARI

Formula

A053220 a(n) = (3n-1) 2^(n-2).