A077802 -id:A077802 - cp's OEIS frontend

A095151 a(n+3) = 3a(n+2) - 2a(n+1) + 1 with a(0)=0, a(1)=2.

0, 2, 7, 18, 41, 88, 183, 374, 757, 1524, 3059, 6130, 12273, 24560, 49135, 98286, 196589, 393196, 786411, 1572842, 3145705, 6291432, 12582887, 25165798, 50331621, 100663268, 201326563, 402653154, 805306337, 1610612704, 3221225439

Offset: 0

Gary W. Adamson, May 30 2004

A sequence generated from a Bell difference row matrix, companion to A095150.
A095150 uses the same recursion rule but the multiplier [1 1 1] instead of [1 0 0].
For n>0, (a(n)) is row 2 of the convolution array A213568. - Clark Kimberling, Jun 20 2012
For n>0, (a(n)) is row 2 of the convolution array A213568. - Clark Kimberling, Jun 20 2012

			a(6) = 183 = 3*88 -2*41 + 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Péter Burcsi, Gabriele Fici, Zsuzsanna Lipták, Rajeev Raman, and Joe Sawada, Generating a Gray code for prefix normal words in amortized polylogarithmic time per word, arXiv:2003.03222 [cs.DS], 2020.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000110, A011971, A095149, A095150.

List([0..30], n-> 3*2^n -(n+3)); # G. C. Greubel, Jul 26 2019

[3*2^n -(n+3): n in [0..30]]; // G. C. Greubel, Jul 26 2019

Table[3*2^n -(n+3), {n,0,30}] (* G. C. Greubel, Jul 26 2019 *)

vector(30, n, n--; 3*2^n -(n+3)) \\ G. C. Greubel, Jul 26 2019

[3*2^n -(n+3) for n in (0..30)] # G. C. Greubel, Jul 26 2019

Let M = a 3 X 3 matrix having Bell triangle difference terms (A095149 is composed of differences of the Bell triangle A011971): (fill in the 3 X 3 matrix with zeros): [1 0 0 / 1 1 0 / 2 1 2] = M. Then M^n * [1 0 0] = [1 n a(n)]. E.g. a(4) = 41 since M^4 * [1 0 0] = [1 4 41].
a(n) = 3*2^n -(n+3) = 2*a(n-1) + n +1 = A000295(n+2) - A000079(n). For n>0, a(n) = A077802(n). - Henry Bottomley, Oct 25 2004
From Colin Barker, Apr 23 2012: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
G.f.: x*(2-x)/((1-x)^2*(1-2*x)). (End)
a(n) = A125128(n) + A000225(n). - Miquel Cerda, Aug 07 2016
a(n) = 2*A125128(n) - A000325(n) + 1. - Miquel Cerda, Aug 12 2016
a(n) = A125128(n) + A000325(n) + n - 1. - Miquel Cerda, Aug 27 2016
E.g.f.: 3*exp(2*x) - (3+x)*exp(x). - G. C. Greubel, Jul 26 2019
Let Prod_{i=0..n-1} (1+x^{2^i}+x^{2*2^i}) = Sum_{j=0..d} b_j x^j, where d=2^{n+1}-2. Then a(n) = Sum_{j=0..d-1} b_j/b_{j+1} (proved). - Richard Stanley, Aug 27 2019

Edited by Robert G. Wilson v, Jun 05 2004

Deleted a comment and file that were unrelated to this sequence. - N. J. A. Sloane, Aug 17 2025

A193911 Sums of the diagonals of the matrix formed by listing the h-Stohr sequences in increasing order.

Original entry on oeis.org

1, 3, 7, 14, 25, 43, 69, 110, 167, 255, 375, 558, 805, 1179, 1681, 2438, 3451, 4975, 7011, 10070, 14153, 20283, 28461, 40734, 57103, 81663, 114415, 163550, 229069, 327355, 458409, 654998, 917123, 1310319, 1834587, 2620998, 3669553, 5242395, 7339525, 10485230

Offset: 1

Jeffrey R. Goodwin, Aug 08 2011

			Portion of the first three rows:
A033627, 2-Stohr  1  2  4  7
A026474, 3-Stohr  1  2  4  8
A051039, 4-Stohr  1  2  4  8
Thus a(1)=1, a(2)=2+1=3, and a(3)=4+2+1=7.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Stöhr Sequence.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,1,4,-2).

A193911=t={0,1}; Do[AppendTo[t,t[[-2]]+t[[-1]]]; AppendTo[t,2*t[[-2]]],{n,41}]; Drop[Nest[Accumulate,t,2],1] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2012 *)
LinearRecurrence[{2,2,-6,1,4,-2},{1,3,7,14,25,43},40] (* Harvey P. Dale, Jun 20 2015 *)

All h-Stohr sequences have formula: h terms 1,2,..,2^(n-1),..,2^(h-1) and then continue (2^h-1)(n-h)+1. - Henry Bottomley, Feb 04 2000
So we get the sums from the piecewise function:
for odd n>=1, a(n)=2^((n+1)/2)-n+((n+1)/2)-2+Sum_{i=0..((n+1)/2)-1}(2*i+1)*(2^(((n+1)/2)-i) -1);
for even n>=2, a(n)=2^((n/2)+2)-n-4+Sum_{i=0..(n/2)-1}(2*i+1)*(2^((n/2)-i) -1). - Jeffrey R. Goodwin, Aug 09 2011
Let odd m>=3, then a(n)=a(m)-A000295(((m+1)/2)+1), where n>=2 is even. - Jeffrey R. Goodwin, Aug 09 2011
Let even m>=2, then a(n)=a(m)-A077802(m/2)=a(m)-A095151(m/2), where n>=1 is odd. - Jeffrey R. Goodwin, Aug 09 2011
From Alexander R. Povolotsky, Aug 09 2011: (Start)
G.f.: x*(1 + x - x^2)/((-1 + x)^3*(-1 - x + 2*x^2 + 2*x^3)).
a(n+4) = -2*a(n)+3*a(n+2)+n+5.
a(n) = 1/8*(2^(n/2+2)*((10-7*sqrt(2))*(-1)^n+10+7*sqrt(2))-(-1)^n-2*n*(n+12)-79). (End)

A078436 Triangle read by rows in which n-th row counts multisets associated with hook partitions.

Original entry on oeis.org

1, 2, 0, 3, 4, 0, 4, 6, 8, 0, 5, 8, 12, 16, 0, 6, 10, 16, 24, 32, 0, 7, 12, 20, 32, 48, 64, 0, 8, 14, 24, 40, 64, 96, 128, 0, 9, 16, 28, 48, 80, 128, 192, 256, 0, 10, 18, 32, 56, 96, 160, 256, 384, 512, 0, 11, 20, 36, 64, 112, 192, 320, 512, 768, 1024, 0, 12, 22, 40, 72, 128

Offset: 1

Alford Arnold, Dec 30 2002

Row sums appear to be A077802. When more general partition types are included, such as 22^(n-4) yielding 9 18 36 72 ..., the array row sums becomes 1,2,7,18,50,118,301,... in agreement with A074141.

			Triangle begins 1; 2,0; 3,4,0; 4,6,8,0; 5,8,12,16,0; ...
a(13) = 12 because we find 1 + 3 + 4 + 3 + 1 multisets of type 21^(n-2): they are 4; 14,24,34; 114,124,134,234; 1124,1134,1234; and 11234

Cf. A077802, A074139, A074141.

G.f.: x*y*(2-x)/(1-2*x*y)/(1-x)^2. - Vladeta Jovovic, Dec 31 2002

More terms from Vladeta Jovovic, Dec 31 2002

A132048 3*A007318 - A103451 - A000012.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 8, 8, 1, 1, 11, 17, 11, 1, 1, 14, 29, 29, 14, 1, 1, 17, 44, 59, 44, 17, 1, 1, 20, 62, 104, 104, 62, 20, 1, 1, 23, 83, 167, 209, 167, 83, 23, 1, 1, 26, 107, 251, 377, 377, 251, 107, 26, 1

Offset: 0

Gary W. Adamson, Aug 08 2007

Row sums = A077802: (1, 2, 7, 18, 41, 88, ...).

			First few rows of the triangle:
  1;
  1,  1;
  1,  5,  1;
  1,  8,  8,  1;
  1, 11, 17, 11,  1;
  1, 14, 29, 29, 14,  1;
  1, 17, 44, 59, 44, 17,  1;
  ...

Cf. A007318, A103451, A000012, A077802.

3*A007318 - A103451 - A000012 as infinite lower triangular matrices.

A288870 Triangle T from array A(k,n) = (2k+1)2^n + 1, k >=0, n >= 0 read by downwards antidiagonals.

Original entry on oeis.org

2, 3, 4, 5, 7, 6, 9, 13, 11, 8, 17, 25, 21, 15, 10, 33, 49, 41, 29, 19, 12, 65, 97, 81, 57, 37, 23, 14, 129, 193, 161, 113, 73, 45, 27, 16, 257, 385, 321, 225, 145, 89, 53, 31, 18, 513, 769, 641, 449, 289, 177, 105, 61, 35, 20, 1025, 1537, 1281, 897, 577, 353, 209, 121, 69, 39, 22

Offset: 0

Wolfdieter Lang, Jun 21 2017

This entry was motivated by a class work of Ferran D.

			The array A begins:
k\n  0  1  2   3   4   5    6    7    8    9    10 ...
0:   2  3  5   9  17  33   65  129  257  513  1025
1:   4  7 13  25  49  97  193  385  769 1537  3073
2:   6 11 21  41  81 161  321  641 1281 2561  5121
3:   8 15 29  57 113 225  449  897 1793 3585  7169
4:  10 19 37  73 145 289  577 1153 2305 4609  9217
5:  12 23 45  89 177 353  705 1409 2817 5633 11265
6:  14 27 53 105 209 417  833 1665 3329 6657 13313
7:  16 31 61 121 241 481  961 1921 3841 7681 15361
8:  18 35 69 137 273 545 1089 2177 4353 8705 17409
9:  20 39 77 153 305 609 1217 2433 4865 9729 19457
...
The triangle T begins:
m\k    0    1    2   3   4   5   6   7  8  9 10 ...
0:     2
1:     3    4
2:     5    7    6
3:     9   13   11   8
4:    17   25   21  15  10
5:    33   49   41  29  19  12
6:    65   97   81  57  37  23  14
7:   129  193  161 113  73  45  27 16
8:   257  385  321 225 145  89  53 31 18
9:   513  769  641 449 289 177 105 61 35 20
10: 1025 1537 1281 897 577 353 209 121 69 39 22
...

Cf. A288871. Columns of T (no 0's, or rows of A): A000051, A181565, A083575, A083686, A083705, A083683, A168596.

Row sums give A077802(n+1) or A095151(n+1).

Table[(2 k + 1)*2^(m - k) + 1, {m, 0, 10}, {k, 0, m}] // Flatten (* Michael De Vlieger, Jun 25 2017 *)

A(n, k) = (2*n + 1)*2^k + 1;
for(n=0, 10, for(k=0, n, print1(A(k, n - k),", "))) \\ Indranil Ghosh, Jun 22 2017

Array A(k, n) = (2*k+1)*2^n + 1 for k >= 0 and n >= 0.
Triangle T(m, k) = A(k, m-k) = (2*k+1)*2^(m-k) + 1, k >= m >= 0, otherwise T(m, k) = 0.
O.g.f. for column k of T: x^k*(2*(k+1) - (2*k+3)*x)/((1-2*x)*(1-x)), k >= 0.
E.g.f. for column k of T (without leading 0's): (2*k+1)*exp(2*x) + exp(x), k>=0.
E.g.f. for column k of T: 2^(-k)*(2*k+1)*exp(2*x) + exp(x) - S(k,x), with S(k, x) = 2^(-k)* Sum_{m=1..k} A288871(k,m)*x^(m-1)/(m-1)! if k >=1 and S(0,x) = 0.

A131398 3*A007318 - A097806 - A000012.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 2, 8, 7, 1, 2, 11, 17, 10, 1, 2, 14, 29, 29, 13, 1, 2, 17, 44, 59, 44, 16, 1, 2, 20, 62, 104, 104, 62, 19, 1, 2, 23, 83, 167, 209, 167, 83, 22, 1, 2, 26, 107, 251, 377, 377, 251, 107, 25, 1

Offset: 0

Gary W. Adamson, Jul 05 2007

Row sums = A077802, (1, 2, 7, 18, 41, 88,...).

			First few rows of the triangle are:
1;
1, 1;
2, 4, 1;
2, 8, 7, 1;
2, 11, 17, 10, 1;
2, 14, 29, 29, 13, 1;
...

Cf. A007318, A097806, A000012, A077802.

3*A007318 - A097806 - A000012 as infinite lower triangular matrices.

cp's OEIS Frontend

A095151 a(n+3) = 3*a(n+2) - 2*a(n+1) + 1 with a(0)=0, a(1)=2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

Extensions

A193911 Sums of the diagonals of the matrix formed by listing the h-Stohr sequences in increasing order.

Views

Author

Keywords

Examples

Links

Programs

Mathematica

Formula

A078436 Triangle read by rows in which n-th row counts multisets associated with hook partitions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A132048 3*A007318 - A103451 - A000012.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A288870 Triangle T from array A(k,n) = (2*k+1)*2^n + 1, k >=0, n >= 0 read by downwards antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A131398 3*A007318 - A097806 - A000012.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A095151 a(n+3) = 3a(n+2) - 2a(n+1) + 1 with a(0)=0, a(1)=2.

A288870 Triangle T from array A(k,n) = (2k+1)2^n + 1, k >=0, n >= 0 read by downwards antidiagonals.