A036563 -id:A036563 - cp's OEIS frontend

A051464 Number of divisors of 4*(2^n-1) + 1.

2, 2, 2, 2, 4, 4, 2, 2, 4, 2, 4, 2, 4, 6, 4, 4, 16, 2, 4, 2, 4, 2, 4, 8, 12, 8, 2, 4, 8, 4, 4, 4, 4, 4, 4, 8, 8, 4, 8, 16, 8, 4, 8, 8, 6, 16, 8, 8, 8, 16, 8, 4, 32, 32, 8, 4, 8, 4, 4, 8, 16, 8, 8, 16, 48, 16, 16, 8, 4, 16, 4, 16, 16, 8, 8, 8, 16, 16, 8, 16, 32

Offset: 1

Edwin D. Evans, eevans2(AT)pacbell.net

nonn

Create a table with tau(2^n-1) as the first row (A046801) and tau(m) as the first column (A000005). The second column is tau(A004760) and so on. Rows 2, 3 and 4 are easily described in terms of row 1. This sequence is row 5.

Sean A. Irvine, Table of n, a(n) for n = 1..591

Cf. A000005, A036563.

Array[DivisorSigma[0, 4*(2^# - 1) + 1] &, 81] (* Michael De Vlieger, Sep 15 2021 *)

a(n) = numdiv(4*(2^n-1) + 1); \\ Michel Marcus, Sep 16 2021

a(n) = tau(4*(2^n -1)+1), where d(n) = A000005(n).

a(81) corrected by Sean A. Irvine, Sep 15 2021

A134349 A007318^(-1) * (A007318A124929 + A124929A007318 - A007318).

Original entry on oeis.org

1, 2, 5, 2, 8, 13, 2, 12, 24, 49, 2, 26, 24, 29, 2, 16, 48, 64, 61, 2, 20, 80, 160, 160, 125, 2, 24, 120, 320, 480, 384, 253, 2, 28, 168, 560, 1120, 1344, 896, 509, 2, 32, 224, 896, 2240, 3584, 3584, 2048, 1021

Offset: 0

Gary W. Adamson, Oct 21 2007

Row sums = A134350.

Right border = A036563 starting with "1": (1, 5, 13, 29, 61, 160, ...).

			First few rows of the triangle:
  1;
  2,  5;
  2,  8, 13;
  2, 12, 24,  29;
  2, 16, 48,  64,  61;
  2, 20, 80, 160, 160, 125;
  ...

Cf. A124929, A134350, A036563.

Inverse binomial transform of (A007318*A124929 + A124929*A007318 - A007318).

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.

Original entry on oeis.org

3, 32, 355, 4110, 48887, 588886, 7111107, 85555550, 1022222215, 12111111102, 142222222211, 1655555555542, 19111111111095, 218888888888870, 2488888888888867, 28111111111111086, 315555555555555527, 3522222222222222190, 39111111111111111075, 432222222222222222182

Offset: 0

Ctibor O. Zizka, Mar 06 2008

Sequence generalized: a(n) = a(0)*(B^n) + F(n)* ((B^n)-1)/(B-1); a(0), B integers, F(n) arithmetic function.
Examples:
a(0) = 1, B = 10, F(n) = 1 gives A002275, F(n) = 2 gives A090843, F(n) = 3 gives A097166, F(n) = 4 gives A099914, F(n) = 5 gives A099915.
a(0) = 1, B = 2, F(n) = 1 gives A000225, F(n) = 2 gives A033484, F(n) = 3 gives A036563, F(n) = 4 gives A048487, F(n) = 5 gives A048488, F(n) = 6 gives A048489.
a(0) = 1, B = 3, F(n) = 1 gives A003462, F(n) = 2 gives A048473, F(n) = 3 gives A134931, F(n) = 4 gives A058481, F(n) = 5 gives A116952.
a(0) = 1, B = 4, F(n) = 1 gives A002450, F(n) = 2 gives A020989, F(n) = 3 gives A083420, F(n) = 4 gives A083597, F(n) = 5 gives A083584.
a(0) = 1, B = 5, F(n) = 1 gives A003463, F(n) = 2 gives A057651, F(n) = 3 gives A117617, F(n) = 4 gives A081655.
a(0) = 2, B = 10, F(n) = 1 gives A037559, F(n) = 2 gives A002276.

			a(3) = 3*10^3 + (3*3 + 1)*(10^3 - 1)/9 = 4110.

G. C. Greubel, Table of n, a(n) for n = 0..985
Index entries for linear recurrences with constant coefficients, signature (33,-393,1991,-3930,3300,-1000).

Cf. A002275, A011557.

Table[3*10^n +(n^2 +1)*(10^n -1)/9, {n,0,30}] (* G. C. Greubel, Jan 05 2022 *)

a(n) = 3*(10^n) + (n*n+1)*((10^n)-1)/9; \\ Jinyuan Wang, Feb 27 2020

[3*10^n +(1+n^2)*(10^n -1)/9 for n in (0..30)] # G. C. Greubel, Jan 05 2022

a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

O.g.f.: (3 - 67*x + 478*x^2 - 1002*x^3 + 850*x^4 - 100*x^5)/((1-x)^3 * (1-10*x)^3). - R. J. Mathar, Mar 16 2008

More terms from R. J. Mathar, Mar 16 2008

More terms from Jinyuan Wang, Feb 27 2020

A185647 Expansion of (1+2x)(1+2x^2)/((1-x)(1+x)(1-2*x^2)).

Original entry on oeis.org

1, 2, 5, 10, 13, 26, 29, 58, 61, 122, 125, 250, 253, 506, 509, 1018, 1021, 2042, 2045, 4090, 4093, 8186, 8189, 16378, 16381, 32762, 32765, 65530, 65533, 131066, 131069, 262138, 262141, 524282, 524285, 1048570, 1048573, 2097146, 2097149, 4194298, 4194301

Offset: 0

Philippe Deléham, Apr 23 2013

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Cf. A036563, A131130

LinearRecurrence[{0, 3, 0, -2}, {1, 2, 5, 10}, 50] (* G. C. Greubel, Jul 09 2017 *)

x='x+O('x^50); Vec((1+2*x)*(1+2*x^2)/((1-x)*(1+x)*(1-2*x^2))) \\ G. C. Greubel, Jul 09 2017

a(n) = a(n-1)*2 if n odd.
a(n) = a(n-1)+3 if n even.
a(2n) = 2^(n+2)-3 = A036563(n+2).
a(2n+1) = 2^(n+3)-6 = A131130(n+1).
a(n) = 3*a(n-2) - 2*a(n-4) with a(0)=1, a(1)=2, a(2)=5, a(3)=10.

A220088 a(n) = 2^n - 81.

Original entry on oeis.org

-80, -79, -77, -73, -65, -49, -17, 47, 175, 431, 943, 1967, 4015, 8111, 16303, 32687, 65455, 130991, 262063, 524207, 1048495, 2097071, 4194223, 8388527, 16777135, 33554351, 67108783, 134217647, 268435375, 536870831, 1073741743, 2147483567, 4294967215

Offset: 0

Andreas Rieber, Dec 04 2012

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

Cf. A000225, A036563, A185346, A220087.

Table[2^n - 81, {n, 0, 40}] (* T. D. Noe, Dec 04 2012 *)

From Chai Wah Wu, Jan 17 2020: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1.
G.f.: (161*x - 80)/((x - 1)*(2*x - 1)). (End)
From Elmo R. Oliveira, Nov 11 2023: (Start)
a(n) = 2*a(n-1) + 81 with a(0) = -80.
E.g.f.: exp(2*x) - 81*exp(x). (End)

A225209 a(n) = (39216^n -16208^n +1890*4^n -767)/105.

Original entry on oeis.org

1, 249, 8537, 186073, 3427545, 58664153, 970097881, 15776875737, 254486643929, 4088295982297, 65545039643865, 1049779971687641, 16804957869966553, 268947166998693081, 4303697458594972889, 68863501862374868185

Offset: 1

J. M. Bergot, May 01 2013

Starting at n=1, a cube has an edge=2^(n+1)-3. The beginning cube has a value of 1 and is surrounded by 2^n layers of cubes each valued at 2^n. The sum of all cubes with values of 2^n is a(n).

Indices of primes in this sequence: 3, 10, 12, 21, 37, 70, 102, 201, 961, 1854, ....

			The first cubes has value 1=a(1).  The second cube has 2 layers of cubes each valued at 2 surrounding the cube of value 1 to give (5^3-1)*2+1=249=a(2).  Next surround by 2^2 layers of cubes each valued at 2^2: (13^3-5^3)*4+249=8537=a(3).  Finally, surround by 2^3 layers of cubes each of value 2^3 to get (29^3-13^3)*8 + 8537 = 186073 = a(4).

G. C. Greubel, Table of n, a(n) for n = 1..825
Index entries for linear recurrences with constant coefficients, signature (29,-252,736,-512).

List([1..20], n-> (392*2^(4*n) -1620*2^(3*n) +1890*2^(2*n) -767)/105); # G. C. Greubel, Dec 31 2019

[(392*2^(4*n) -1620*2^(3*n) +1890*2^(2*n) -767)/105: n in [1..20]]; // G. C. Greubel, Dec 31 2019

seq( (392*2^(4*n) -1620*2^(3*n) +1890*2^(2*n) -767)/105, n=1..20); # G. C. Greubel, Dec 31 2019

LinearRecurrence[{29,-252,736,-512},{1,249,8537,186073},20] (* Harvey P. Dale, Apr 22 2018 *)

vector(20, n, (392*2^(4*n) -1620*2^(3*n) +1890*2^(2*n) -767)/105) \\ G. C. Greubel, Dec 31 2019

[(392*2^(4*n) -1620*2^(3*n) +1890*2^(2*n) -767)/105 for n in (1..20)] # G. C. Greubel, Dec 31 2019

a(n) = 29*a(n-1) - 252*a(n-2) + 736*a(n-3) - 512*a(n-4).
a(n) = a(n-1) + 7*2^(4*n-1) - 27*2^(3*n-1) + 27*2^(2*n-1), for n>0.
G.f. x*(1 +220*x +1568*x^2 +512*x^3)/( (1-x)*(1-4*x)*(1-8*x)*(1-16*x) ). - R. J. Mathar, May 09 2013
a(n) = a(n-1) +2^(n-1)*(A036563(n+1)^3 -A036563(n)^3). - R. J. Mathar, May 18 2013

A334164 a(n) is the number of ON-cells in the completed n-th level of a triangular wedge in the hexagonal grid of A151723 (i.e., after 2^k >= n generations of the automaton in A151723 have been computed).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 4, 8, 2, 10, 6, 10, 4, 14, 8, 16, 2, 18, 10, 16, 8, 20, 12, 22, 6, 26, 14, 22, 8, 30, 16, 32, 2, 34, 18, 28, 16, 34, 18, 32, 14, 40, 22, 34, 16, 42, 24, 44, 10, 50, 26, 40, 20, 48, 28, 50, 14, 58, 30, 46, 16, 62, 32, 64

Offset: 1

Hartmut F. W. Hoft, Apr 17 2020

nonn

Conjecture 1: Except for a(2^n + 1) = 2, n >= 1, for odd-numbered completed levels a lower bound of the ratio of ON-cells to the length of the level is (2^n + 2)/(3*2^(n+1) + 1) with limit 1/6, determined by the subsequence of levels starting with: 13, 25, 49, 97, 193, 385, 769, 1537, 3073, ..., and the associated ON-cell counts: 4, 6, 10, 18, 34, 66, 130, 258, 514, ..., as listed in the second column of each of the two triangles below.
The ON-cell counts for the indices in each row define a line of slope 1/2.  The formula for the indices of levels in row k >= 2 is  L(k, i) = 1 + Sum_{j = 0, ..., i} 2^(k-j), 0 <= i <= k - 2, and the formula for the associated numbers of ON-cells is C(k, i) = 2 + Sum_{j = 1..i} 2^(k-1-j), 0 <= i <= k - 2:
Index of the level: L(k, i) number of ON-cells: C(k, i)
k\i  0   1   2   3   4   5   6    k/i  0   1   2   3   4   5   6
2:   5                             2:  2
3:   9  13                         3:  2   4
4:  17  25  29                     4:  2   6   8
5:  33  49  57  61                 5:  2  10  14  16
6:  65  97 113 121 125             6:  2  18  26  30  32
7: 129 193 225 241 249 253         7:  2  34  50  58  62  64
8: 257 385 449 481 497 505 509     8:  2  66  98 114 122 126 128
...
The pairs ( L(k, i), C(k, i) ), for 0 <= i <= k-2, define a line of slope 1/2 for each k >= 3.
For triangle L(k, i): column 0 is A000051(n), n >= 2; column 1 is A181565(n), n >= 3; column 2 is A083686(n), n >= 2; columns 3 is A195744(n), n >= 1; column 4 is A206371(n), n >= 2; column 5 is A196657(n), n >= 1; the bounding diagonal is A036563(n), n >= 3.
For triangle C(k, i): column 1 is A052548(n), n >= 1; column 2 is A164094(n), n >= 1.
Conjecture 2: In an even-numbered completed level 2*n the fraction of ON-cells is bounded below by (23 * 2^n - 24)/(2^(n+5) - 36) with limit 23/32, determined by the subsequence of levels starting with: 28, 92, 220, 476, 988, 2012, 4060, ... .
There are 16 numbers less than 1000 that do not occur as the number of ON-cells in a completed level through level 16384: 136, 164, 330, 334, 402, 444, 526, 570, 598, 604, 614, 714, 740, 822, 832, 878.
Sequence A334169 of even-numbered completed levels in which all cells are ON-cells is a subsequence of this sequence.

Hartmut F. W. Hoft, Diagram of ON-cell counts

Cf. A000051, A036563, A052548, A083686, A151723, A164094, A181565, A195744, A196657, A206371, A334169.

(* a169781[] and support functions are defined in A169781 and create the list nTriangle *)
a334164[n_] := Module[{k, levels={}}, a169781[n]; For[k=1, k<=n, k++, AppendTo[levels, Count[nTriangle[[k]], 1] - 2]]; levels]/;(n>=3 && IntegerQ[Log[2,n]])
a334164[64] (* sequence data *)

A366159 Triangle read by rows: T(n, k) = Sum_{i=0..k-2} (-1)^(i+2) * (k-i-1)^n * binomial(k,i).

Original entry on oeis.org

1, 1, 5, 1, 13, 23, 1, 29, 121, 119, 1, 61, 479, 1081, 719, 1, 125, 1681, 6719, 10081, 5039, 1, 253, 5543, 35281, 90719, 100801, 40319, 1, 509, 17641, 168839, 665281, 1239839, 1088641, 362879, 1, 1021, 54959, 763561, 4339439, 12096001, 17539199, 12700801, 3628799

Offset: 2

Michel Marcus, Oct 02 2023

			Triangle begins:
  1;
  1,    5;
  1,   13,    23;
  1,   29,   121,    119;
  1,   61,   479,   1081,     719;
  1,  125,  1681,   6719,   10081,     5039;
  1,  253,  5543,  35281,   90719,   100801,    40319;
  1,  509, 17641, 168839,  665281,  1239839,  1088641,   362879;
  1, 1021, 54959, 763561, 4339439, 12096001, 17539199, 12700801, 3628799;
  ...

Michael De Vlieger, Table of n, a(n) for n = 2..11176 (rows 2..150, flattened)
Dmitry N. Kozlov, Stirling complexes, arXiv:2309.17142 [math.CO], 2023.

Cf. A000012 (col 2), A036563 (col 3), A033312 (right border).

Cf. A105060.

Table[Sum[(-1)^(i + 2)*(k - i - 1)^n*Binomial[k, i], {i, 0, k - 2} ], {n, 2, 10}, {k, 2, n}] // Flatten (* Michael De Vlieger, Oct 02 2023 *)

T(n, k) = sum(i=0, k-2, (-1)^(i+2) * (k-i-1)^n * binomial(k,i));
tabl(nn) = for (n=2, nn, for (k=2, n, print1(T(n,k), ", ")));

A220089 a(n) = 2^n - 243.

Original entry on oeis.org

-242, -241, -239, -235, -227, -211, -179, -115, 13, 269, 781, 1805, 3853, 7949, 16141, 32525, 65293, 130829, 261901, 524045, 1048333, 2096909, 4194061, 8388365, 16776973, 33554189, 67108621, 134217485, 268435213, 536870669, 1073741581, 2147483405, 4294967053

Offset: 0

Andreas Rieber, Dec 04 2012

sign

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

Cf. A000225, A036563, A185346, A220087, A220088.

Table[2^n - 243, {n, 0, 40}] (* T. D. Noe, Dec 04 2012 *)

From Chai Wah Wu, Jan 17 2020: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1.
G.f.: (485*x - 242)/((x - 1)*(2*x - 1)). (End)
From Elmo R. Oliveira, Nov 11 2023: (Start)
a(n) = 2*a(n-1) + 243 with a(0) = -242.
E.g.f.: exp(2*x) - 243*exp(x). (End)

A238339 Square number array read by ascending antidiagonals: T(1,k) = 2k + 1, and T(n,k) = (2n^(k+1)-n-1)/(n-1) otherwise.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 25, 29, 9, 1, 1, 11, 41, 79, 61, 11, 1, 1, 13, 61, 169, 241, 125, 13, 1, 1, 15, 85, 311, 681, 727, 253, 15, 1, 1, 17, 113, 517, 1561, 2729, 2185, 509, 17, 1, 1, 19, 145, 799, 3109, 7811, 10921, 6559, 1021, 19, 1

Offset: 0

Philippe Deléham, Feb 24 2014

			Square array begins:
1..1...1.....1......1.......1........1........1...
1..3...5.....7......9......11.......13.......15...
1..5..13....29.....61.....125......253......509...
1..7..25....79....241.....727.....2185.....6559...
1..9..41...169....681....2729....10921....43689...
1.11..61...311...1561....7811....39061...195311...
1.13..85...517...3109...18661...111973...671845...
1.15.113...799...5601...39215...274513..1921599...
1.17.145..1169...9361...74897...599185..4793489...
1.19.181..1639..14761..132859..1195741.10761679...
1.21.221..2221..22221..222221..2222221.22222221...

Cf. A238303.

T:= proc(n, k); if n=1 then 2*k+1 else (2*n^(k+1)-n-1)/(n-1) fi end:
seq(seq(T(n-k, k), k=0..n), n=0..10); # Georg Fischer, Oct 14 2023

T(0,k) = A000012(k) = 1;
T(1,k) = A005408(k) = 2k+1;
T(2,k) = A036563(k+2);
T(3,k) = A058481(k+1);
T(4,k) = A083584(k);
T(5,k) = A137410(k);
T(6,k) = A233325(k);
T(7,k) = A233326(k);
T(8,k) = A233328(k);
T(9,k) = A211866(k+1);
T(10,k) = A165402(k+1);
T(n,0) = A000012(n) = 1;
T(n,1) = A005408(n) = 2*n+1;
T(n,2) = A001844(n) = 2*n^2 + 2*n + 1.

Definition amended by Georg Fischer, Oct 14 2023

cp's OEIS Frontend

A051464 Number of divisors of 4*(2^n-1) + 1.

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A134349 A007318^(-1) * (A007318*A124929 + A124929*A007318 - A007318).

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A137215 a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

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A185647 Expansion of (1+2x)*(1+2*x^2)/((1-x)*(1+x)*(1-2*x^2)).

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A220088 a(n) = 2^n - 81.

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A225209 a(n) = (392*16^n -1620*8^n +1890*4^n -767)/105.

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A334164 a(n) is the number of ON-cells in the completed n-th level of a triangular wedge in the hexagonal grid of A151723 (i.e., after 2^k >= n generations of the automaton in A151723 have been computed).

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A366159 Triangle read by rows: T(n, k) = Sum_{i=0..k-2} (-1)^(i+2) * (k-i-1)^n * binomial(k,i).

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A134349 A007318^(-1) * (A007318A124929 + A124929A007318 - A007318).

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.

A185647 Expansion of (1+2x)(1+2x^2)/((1-x)(1+x)(1-2*x^2)).

A225209 a(n) = (39216^n -16208^n +1890*4^n -767)/105.

A238339 Square number array read by ascending antidiagonals: T(1,k) = 2k + 1, and T(n,k) = (2n^(k+1)-n-1)/(n-1) otherwise.