A062709 -id:A062709 - cp's OEIS frontend

A254363 a(n) = 4^n + 6*2^n + 3^(n+1) + 10.

20, 35, 77, 203, 605, 1955, 6677, 23723, 86765, 324275, 1231877, 4738043, 18396125, 71940995, 282882677, 1116985163, 4424500685, 17568076115, 69883311077, 278367837083, 1109978272445, 4429440153635, 17686354389077, 70651224045803, 282322365983405, 1128441973997555, 4511225627508677, 18037276107243323, 72126226025905565

Offset: 0

Luciano Ancora, Jan 29 2015

This is the sequence of fourth terms of "third partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas, page 2.
Luciano Ancora, Recurrence relations for partial sums of m-th powers.
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A062709, A254362, A254364.

Table[4^n + 6*2^n + 3^(n + 1) + 10, {n, 0, 28}] (* Michael De Vlieger, Jan 30 2015 *)

vector(30, n, n--; 4^n + 6*2^n + 3^(n+1) + 10) \\ Colin Barker, Jan 30 2015

From Colin Barker, Jan 30 2015: (Start)
G.f.: -(342*x^3-427*x^2+165*x-20)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)).
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4) for n > 3. (End)
E.g.f.: exp(x)*(exp(3*x) + 3*exp(2*x) + 6*exp(x) + 10). - Elmo R. Oliveira, Sep 12 2024

A256994 a(n) = n + 1 when n <= 3, otherwise a(n) = 2^(n-2) + 3; also iterates of A005187 starting from a(1) = 2.

Original entry on oeis.org

2, 3, 4, 7, 11, 19, 35, 67, 131, 259, 515, 1027, 2051, 4099, 8195, 16387, 32771, 65539, 131075, 262147, 524291, 1048579, 2097155, 4194307, 8388611, 16777219, 33554435, 67108867, 134217731, 268435459, 536870915, 1073741827, 2147483651, 4294967299, 8589934595, 17179869187, 34359738371, 68719476739, 137438953475, 274877906947

Offset: 1

Antti Karttunen, Apr 15 2015

nonn

Note that if we instead iterated function b(n) = 1+A005187(n), from b(1) onward, we would get the powers of two, A000079.

Antti Karttunen, Table of n, a(n) for n = 1..128

Topmost row of A256995, leftmost column of A256997.

Cf. A000079, A005187, A062709, A068156.

Table[If[n<4,n+1,2^(n-2)+3],{n,40}] (* Harvey P. Dale, May 14 2019 *)

A256994(n) = if(n < 4, n+1, 2^(n-2) + 3);

\\ By iterating A005187:
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
i=1; k=2; print1(k); while(i <= 40, k = A005187(k); print1(", ", k); i++);

(define (A256994 n) (if (< n 4) (+ n 1) (+ (A000079 (- n 2)) 3)))

;; The following uses memoization-macro definec:
(definec (A256994 n) (if (= 1 n) 2 (A005187 (A256994 (- n 1)))))

If n < 4, a(n) = n + 1, otherwise a(n) = 2^(n-2) + 3 = A062709(n-2).

a(1) = 2; for n > 1, a(n) = A005187(a(n-1)).

A340666 A(n,k) is derived from n by replacing each 0 in its binary representation with a string of k 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 4, 3, 1, 0, 1, 8, 3, 4, 3, 0, 1, 16, 3, 16, 5, 3, 0, 1, 32, 3, 64, 9, 6, 7, 0, 1, 64, 3, 256, 17, 12, 7, 1, 0, 1, 128, 3, 1024, 33, 24, 7, 8, 3, 0, 1, 256, 3, 4096, 65, 48, 7, 64, 9, 3, 0, 1, 512, 3, 16384, 129, 96, 7, 512, 33, 10, 7

Offset: 0

Alois P. Heinz, Jan 15 2021

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,      0,       0,        0, ...
  1, 1,  1,   1,    1,     1,      1,       1,        1, ...
  1, 2,  4,   8,   16,    32,     64,     128,      256, ...
  3, 3,  3,   3,    3,     3,      3,       3,        3, ...
  1, 4, 16,  64,  256,  1024,   4096,   16384,    65536, ...
  3, 5,  9,  17,   33,    65,    129,     257,      513, ...
  3, 6, 12,  24,   48,    96,    192,     384,      768, ...
  7, 7,  7,   7,    7,     7,      7,       7,        7, ...
  1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Columns k=0-2, 4 give: A038573, A001477, A084471, A084473.
Rows n=0..17, 19 give: A000004, A000012, A000079, A010701, A000302, A000051(k+1), A007283, A010727, A001018, A087289, A007582(k+1), A062709(k+2), A164346, A181565(k+1), A005009, A181404(k+3), A001025, A199493, A253208(k+1).
Main diagonal gives A340667.
Cf. A000120, A023416.

A:= (n, k)-> Bits[Join](subs(0=[0$k][], Bits[Split](n))):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= proc(n, k) option remember; `if`(n<2, n,
     `if`(irem(n, 2, 'r')=1, A(r, k)*2+1, A(r, k)*2^k))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := FromDigits[IntegerDigits[n, 2] /. 0 -> Sequence @@ Table[0, {k}], 2];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 02 2021 *)

A000120(A(n,k)) = A000120(n) = log_2(A(n,0)+1).

A023416(A(n,k)) = k * A023416(n) for n >= 1.

A363000 a(n) = numerator(R(n, n, 1)), where R are the rational polynomials R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).

Original entry on oeis.org

1, 5, 19, 188, 1249, 125744, 283517, 303923456, 138604561, 599865008128, 118159023973, 7078040993755136, 155792758736921, 146303841678548271104, 294014633772018349, 64670474732430319157248, 752324747622089633569, 3224753626003393505960919040, 2507759850059601711479669

Offset: 0

Peter Luschny, May 12 2023

R(n, n, 0) are the (0-based) harmonic numbers, R(n, n, -1) are the Bernoulli numbers, and R(n, n, 1) is this sequence in its rational form.

			a(n) are the numerators of the terms on the main diagonal of the triangle:
[0] 1;
[1] 1,  5/2;
[2] 1,  7/2,  19/2;
[3] 1, 11/2, 121/6,  188/3;
[4] 1, 19/2,  95/2,  369/2,   1249/2;
[5] 1, 35/2, 721/6, 1748/3, 35164/15, 125744/15;
[6] 1, 67/2, 639/2, 3877/2,  18533/2,   76317/2, 283517/2;

Cf. A363001 (denominators), A362999 (odd-indexed denominators), A362998.

Cf. A027611, A027612, A062709.

# For better context we put A362998, A362999, A363000, and A363001 together here.
R := (n, k, x) -> add(add(x^j*binomial(u, j)*(j+1)^n, j=0..u)/(u + 1), u=0..k):
### x = 1 -> this sequence
 for n from 0 to 7 do [n], seq(R(n, k, 1), k = 0..n) od;
 seq(R(n, n, 1), n = 0..9);
 A363000 := n -> numer(R(n, n, 1)): seq(A363000(n), n = 0..10);
 A363001 := n -> denom(R(n, n, 1)): seq(A363001(n), n = 0..20);
 A362999 := n -> denom(R(2*n+1, 2*n+1, 1)): seq(A362999(n), n = 0..11);
 A362998 := n -> add(R(2*n, k, 1), k = 0..2*n): seq(A362998(n), n = 0..9);
### x = -1 -> Bernoulli(n, 1)
# for n from 0 to 9 do [n], seq(R(n, k,-1), k = 0..n) od;
# seq(R(n, n, -1), n = 0..12); seq(bernoulli(n, 1), n = 0..12);
### x = 0 -> Harmonic numbers
# for n from 0 to 9 do [n], seq(R(n, k, 0), k = 0..n) od;
# seq(R(n, n, 0), n = 0..9); seq(harmonic(n+1), n = 0..9);

Sum_{k=0..n} R(n, k, 0) = Sum_{j=0..n} (n-j+1)/(j+1) = (n+2)*Harmonic(n+1)-n-1.
Sum_{k=0..n} R(n, k,-1) = (n + 2 - 0^n) * Bernoulli(n, 1).
Sum_{k=0..2*n} R(2*n, k, 1) = A362998(n).
2*R(n, 1, 1) = A062709(n).

A242475 a(n) = 2^n + 8.

Original entry on oeis.org

9, 10, 12, 16, 24, 40, 72, 136, 264, 520, 1032, 2056, 4104, 8200, 16392, 32776, 65544, 131080, 262152, 524296, 1048584, 2097160, 4194312, 8388616, 16777224, 33554440, 67108872, 134217736, 268435464, 536870920, 1073741832

Offset: 0

Vincenzo Librandi, May 20 2014

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

Cf. A000051, A052548, A062709, A140504, A168614, A153972, A168415, A188165.

Magma
```
[2^n+8: n in [0..40]];
```

Table[2^n + 8, {n, 0, 40}] (* or *) CoefficientList[Series[(9 - 17 x)/((1 - x) (1 - 2 x)),{x, 0, 30}], x]
LinearRecurrence[{3,-2},{9,10},40] (* Harvey P. Dale, May 21 2025 *)

G.f.: (9 - 17*x)/((1 - x)*(1 - 2*x)).
a(n) = 2*a(n-1) - 8 = 3*a(n-1) - 2*a(n-2).
a(n) = A052548(n)+6 = A140504(n)+4 = A153972(n)+2.
E.g.f.: exp(2*x) + 8*exp(x). - Elmo R. Oliveira, Nov 11 2023

A246139 a(n) = 2^n + 10.

Original entry on oeis.org

11, 12, 14, 18, 26, 42, 74, 138, 266, 522, 1034, 2058, 4106, 8202, 16394, 32778, 65546, 131082, 262154, 524298, 1048586, 2097162, 4194314, 8388618, 16777226, 33554442, 67108874, 134217738, 268435466, 536870922, 1073741834, 2147483658, 4294967306

Offset: 0

Vincenzo Librandi, Aug 18 2014

First trisection of A085688. [Bruno Berselli, Aug 19 2014]

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

Cf. Sequences of the form 2^n + k: A000079 (k=0), A000051 (k=1), A052548 (k=2), A062709 (k=3), A140504 (k=4), A168614 (k=5), A153972 (k=6), A168415 (k=7), A242475 (k=8), A188165 (k=9), this sequence (k=10).

Cf. A085688.

Magma
```
[2^n+10: n in [0..40]];
```
Mathematica
```
Table[2^n + 10, {n, 0, 40}]
```

vector(50, n, 2^(n-1)+10) \\ Derek Orr, Aug 18 2014

G.f.: (11 - 21*x)/(1 - 3*x + 2*x^2).
a(n) = A000079(n) + 10.
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1.
E.g.f.: exp(2*x) + 10*exp(x). - Elmo R. Oliveira, Nov 11 2023

A328974 Trajectory of 1496 under repeated application of the map defined in A053392.

Original entry on oeis.org

1496, 51315, 6446, 10810, 1891, 91710, 10881, 18169, 99715, 181686, 9971414, 18168555, 99714131010, 181685544111, 99714131098522, 18168554419171374, 99714131098510108841011, 1816855441917136111816125112, 99714131098510108849722997737623, 1816855441917136111816121316941118161410101385

Offset: 1

N. J. A. Sloane, Nov 01 2019

1496 is the smallest number whose trajectory under A053392 increases without limit.
More terms than usual are shown in order to display the onset of exponential growth.
Proof that this grows without limit, from Hans Havermann, Nov 01 2019: (Start)
One way to prove that the trajectory of a number under repeated application of the map defined in A053392 increases without limit is to show that there exists a term containing a non-final substring of three adjacent 9's.
If such a substring in term n is followed by a 0, it will grow to a non-final substring of three adjacent 9's followed by a 1 in term n+2: *9990* --> *18189* --> *99917*
If such a substring in term n is followed by a digit d that is neither 0 nor 9, it will grow to a non-final substring of four adjacent 9's in term n+2: *999d* --> *18181(d-1)* --> *9999d*
If such a substring in term n is followed by another 9, then it is a substring of k 9's for k >= 4, which will grow to a substring of (2k-3) 9's in term n+2: *[9]^d* --> *[18]^(d-1)* --> *[9]^(2d-3)*
Putting those together, such a substring in term n will grow to four adjacent 9's by term n+4; to five adjacent 9's by term n+6; to seven adjacent 9's by term n+8; ... to 2^k+3 adjacent 9's (see A062709) by term n+4+2k, regardless of what happens in the rest of the number.
In the present sequence a(41) contains two non-final substrings of three adjacent 9's. QED (End)
[Argument corrected and completed by David J. Seal, Nov 05 2019.]

Paolo Xausa, Table of n, a(n) for n = 1..32
Erich Friedman, Problem of the Month (February 2000).

Cf. A053392.

NestList[FromDigits[Flatten[{IntegerDigits[Total[Partition[IntegerDigits[#], 2, 1], {2}]]}]] &, 1496, 20] (* Paolo Xausa, Jan 10 2025 *)

A134250 Expansion of x(4+9x-7x^2) / ((1-x)(1+3*x-x^2)).

Original entry on oeis.org

4, 1, 7, -14, 55, -173, 580, -1907, 6307, -20822, 68779, -227153, 750244, -2477879, 8183887, -27029534, 89272495, -294847013, 973813540, -3216287627, 10622676427, -35084316902, 115875627139, -382711198313, 1264009222084, -4174738864559, 13788225815767

Offset: 1

Roger L. Bagula, Jan 14 2008

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (-2,4,-1).

Cf. A062709, A052548.

A134250 := proc(n)
        2-17*(-1)^n*A006190(n)+5*(-1)^n*A006190(n+1) ;
end proc:
seq(A134250(n),n=1..10) ; # R. J. Mathar, Dec 06 2011

LinearRecurrence[{-2,4,-1},{4,1,7},30] (* Harvey P. Dale, Aug 15 2015 *)
Rest@ CoefficientList[Series[x (4 + 9 x - 7 x^2)/((1 - x) (1 + 3 x - x^2)), {x, 0, 27}], x] (* Michael De Vlieger, May 16 2017 *)

Vec(x*(4+9*x-7*x^2)/((1-x)*(1+3*x-x^2)) + O(x^30)) \\ Colin Barker, May 16 2017

a(n) = 2-17*(-1)^n*A006190(n) +5*(-1)^n*A006190(n+1). - R. J. Mathar, Dec 06 2011
From Colin Barker, May 16 2017: (Start)
a(n) = 2 + (2^(-1-n)*((-3-sqrt(13))^n*(-19+5*sqrt(13)) + (-3+sqrt(13))^n*(19+5*sqrt(13)))) / sqrt(13).
a(n) = -2*a(n-1) + 4*a(n-2) - a(n-3) for n>3.
(End)

A245179 Numbers of the form 2^k+3 or 3*2^k+1, k >= 2.

Original entry on oeis.org

7, 11, 13, 19, 25, 35, 49, 67, 97, 131, 193, 259, 385, 515, 769, 1027, 1537, 2051, 3073, 4099, 6145, 8195, 12289, 16387, 24577, 32771, 49153, 65539, 98305, 131075, 196609, 262147, 393217, 524291, 786433, 1048579, 1572865, 2097155, 3145729, 4194307, 6291457

Offset: 1

N. J. A. Sloane, Jul 17 2014

Numbers whose binary expansion is 10..011 or 110..01.

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

Essentially the union of A062709 and A181565. Cf. A245178.

&cat [[3*2^i+1,2^(i+2)+3]: i in [1..30]]; // Bruno Berselli, Jul 23 2014

CoefficientList[Series[- (14 x^3 + 8 x^2 - 11 x - 7)/((x - 1) (x + 1) (2 x^2 - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Jul 23 2014 *)
LinearRecurrence[{0,3,0,-2},{7,11,13,19},50] (* Harvey P. Dale, Mar 05 2015 *)

a(2k) = 2^(k+2)+3, a(2k+1) = 3*2^(k+1)+1. - N. J. A. Sloane, Jul 19 2014

a(n) = 3*a(n-2)-2*a(n-4). G.f.: -x*(14*x^3+8*x^2-11*x-7) / ((x-1)*(x+1)*(2*x^2-1)). - Colin Barker, Jul 19 2014

A254463 a(n) = 152^n + 64^n + 103^n + 35^n + 6^n + 21.

Original entry on oeis.org

56, 126, 378, 1386, 5778, 26226, 126378, 636426, 3314178, 17714466, 96660378, 536249466, 3015243378, 17141522706, 98333399178, 568324150506, 3305074833378, 19319850386946, 113420243462778, 668241096915546, 3948892688324178, 23393955029043186, 138880128205091178

Offset: 0

Luciano Ancora, Jan 31 2015

This is the sequence of sixth terms of "third partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas, page 2.
Luciano Ancora, Recurrence relations for partial sums of m-th powers.
Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Cf. A062709, A254362, A254363, A254364, A254464.

Table[15 2^n + 6 4^n + 10 3^n + 3 5^n + 6^n + 21, {n, 0, 25}] (* Michael De Vlieger, Jan 31 2015 *)

vector(30, n, n--; 15*2^n + 6*4^n + 10*3^n + 3*5^n + 6^n + 21) \\ Colin Barker, Jan 31 2015

From Colin Barker, Jan 31 2015: (Start)
G.f.: -2*(12276*x^5 - 20578*x^4 + 12831*x^3 - 3766*x^2 + 525*x - 28)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)).
a(n) = 21*a(n-1) - 175*a(n-2) + 735*a(n-3) - 1624*a(n-4) + 1764*a(n-5) - 720*a(n-6). (End)
E.g.f.: exp(x)*(exp(5*x) + 3*exp(4*x) + 6*exp(3*x) + 10*exp(2*x) + 15*exp(x) + 21). - Elmo R. Oliveira, Sep 16 2024

cp's OEIS Frontend

A254363 a(n) = 4^n + 6*2^n + 3^(n+1) + 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A256994 a(n) = n + 1 when n <= 3, otherwise a(n) = 2^(n-2) + 3; also iterates of A005187 starting from a(1) = 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Scheme

Scheme

Formula

A340666 A(n,k) is derived from n by replacing each 0 in its binary representation with a string of k 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A363000 a(n) = numerator(R(n, n, 1)), where R are the rational polynomials R(n, k, x) = Sum_{u=0..k} ( Sum_{j=0..u} x^j * binomial(u, j) * (j + 1)^n ) / (u + 1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A242475 a(n) = 2^n + 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A246139 a(n) = 2^n + 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A328974 Trajectory of 1496 under repeated application of the map defined in A053392.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A134250 Expansion of x*(4+9*x-7*x^2) / ((1-x)*(1+3*x-x^2)).

Views

A134250 Expansion of x(4+9x-7x^2) / ((1-x)(1+3*x-x^2)).

A254463 a(n) = 152^n + 64^n + 103^n + 35^n + 6^n + 21.