A158057 -id:A158057 - cp's OEIS frontend

A319281 Numbers of the form 16^i(16j + 1).

1, 16, 17, 33, 49, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, 256, 257, 272, 273, 289, 305, 321, 337, 353, 369, 385, 401, 417, 433, 449, 465, 481, 497, 513, 528, 529, 545, 561, 577, 593, 609, 625, 641, 657, 673, 689, 705, 721, 737, 753, 769, 784

Offset: 1

Jianing Song, Sep 16 2018

nonn

{a(n)} gives all positive fourth powers modulo all powers of 2, that is, positive fourth powers over 2-adic integers. So this sequence is closed under multiplication.

Jianing Song, Table of n, a(n) for n = 1..10002 (all terms <= 150000)

A158057 is a proper subsequence.
Perfect powers over 2-adic integers:
Squares: positive: A234000; negative: A004215 (negated);
Cubes: A191257;
Fourth powers: positive: this sequence; negative: A319282 (negated).

isA319281(n)= n\16^valuation(n, 16)%16==1

def A319281(n):
    if n<3: return 15*n-14
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n-1+x-sum((((x>>(i<<2))-1)>>4)+1 for i in range(x.bit_length()>>2))
    return bisection(f,n,n) # Chai Wah Wu, Feb 17 2025

a(n) = 15*n + O(log(n)).

A249356 8*A200975(n)-7 where A200975 are the numbers on the diagonals in Ulam's spiral.

Original entry on oeis.org

1, 17, 33, 49, 65, 97, 129, 161, 193, 241, 289, 337, 385, 449, 513, 577, 641, 721, 801, 881, 961, 1057, 1153, 1249, 1345, 1457, 1569, 1681, 1793, 1921, 2049, 2177, 2305, 2449, 2593, 2737, 2881, 3041, 3201, 3361, 3521, 3697, 3873, 4049, 4225, 4417, 4609, 4801

Offset: 1

Todd Silvestri, Oct 27 2014

All elements are odd.

The pair (a(n), a(n+1)) is separated by A002265(n-1) elements in A158057.

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

seq(2*n*(n+2)+(-1)^n-4*sin((Pi*n)/2), n=1..100); # Robert Israel, Nov 04 2014

a[n_Integer/;n>0]:=2 n (n+2)+(-1)^n-4 Mod[n^2 (3 n+2),4,-1]
CoefficientList[Series[-(x^5 - x^4 + 15 x + 1) / ((x - 1)^3 (x^3 + x^2 + x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 16 2014 *)
Table[2 n (n + 2) - (1 - (-1)^n) (1 - 2 I^(n + 1)) + 1, {n, 1, 50}] (* Bruno Berselli, Nov 18 2014 *)
LinearRecurrence[{2,-1,0,1,-2,1},{1,17,33,49,65,97},50] (* Harvey P. Dale, Sep 29 2019 *)

a(n) = 2*n*(n+2)+(-1)^n-4*round(sin((Pi*n)/2)) \\ Charles R Greathouse IV, Nov 17 2014

a(n) = 2*n*(n+2)+(-1)^n-4*sin((Pi*n)/2).
G.f.: - x*(x^5-x^4+15*x+1)/((x-1)^3*(x^3+x^2+x+1)).
a(n) = 2*a(n-1) - a(n-2) + 16 if n == 2 mod 4, a(n) = 2*a(n-1) - a(n-2) otherwise. - Robert Israel, Nov 04 2014
a(n) = 2*n*(n+2) - (1-(-1)^n)*(1-2*i^(n+1)) + 1, where i=sqrt(-1). - Bruno Berselli, Nov 18 2014

A215137 a(n) = 17*n + 1.

Original entry on oeis.org

1, 18, 35, 52, 69, 86, 103, 120, 137, 154, 171, 188, 205, 222, 239, 256, 273, 290, 307, 324, 341, 358, 375, 392, 409, 426, 443, 460, 477, 494, 511, 528, 545, 562, 579, 596, 613, 630, 647, 664, 681, 698, 715, 732, 749, 766, 783, 800, 817, 834, 851, 868, 885, 902, 919, 936, 953, 970

Offset: 0

Jeremy Gardiner, Aug 04 2012

Jeremy Gardiner, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A129484, A158057, A161705.

I:=[1,18]; [n le 2 select I[n] else 2*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Apr 19 2018

Range[1, 100, 17]
LinearRecurrence[{2,-1}, {1,18}, 50] (* G. C. Greubel, Apr 19 2018 *)

for(n=0, 50, print1(17*n + 1, ", ")) \\ G. C. Greubel, Apr 19 2018

From G. C. Greubel, Apr 19 2018: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: (1+16*x)/(1-x)^2.
E.g.f.: (17*x + 1)*exp(x). (End)

A005570 Number of walks on cubic lattice.

Original entry on oeis.org

17, 50, 99, 164, 245, 342, 455, 584, 729, 890, 1067, 1260, 1469, 1694, 1935, 2192, 2465, 2754, 3059, 3380, 3717, 4070, 4439, 4824, 5225, 5642, 6075, 6524, 6989, 7470, 7967, 8480, 9009, 9554, 10115, 10692, 11285, 11894, 12519, 13160, 13817, 14490, 15179, 15884, 16605

Offset: 1

N. J. A. Sloane

Partial sums of A158057.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jeremy Gardiner, Table of n, a(n) for n = 1..999
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6 (see figure 7).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A001620, A158057, A250129.

[8*n^2 + 9*n : n in [1..40]]; // Vincenzo Librandi, Nov 05 2014

CoefficientList[Series[(17 - x) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 05 2014 *)

Vec((17-x)/(1-x)^3 + O(x^50)) \\ Michel Marcus, Nov 05 2014

a(n) = 8*n^2 + 9*n.
G.f.: (17-x)/(1-x)^3. Simon Plouffe in his 1992 dissertation.
a(n) = 16*A000217(n) + n. - Jon Perry, Nov 05 2014
Sum_{n>=1} 1/a(n) = 80/81 +Psi(1/8)/9+gamma/9 = 0.11973.. see A001620 and A250129. - R. J. Mathar, May 30 2022
Sum_{n>=1} 1/a(n) = 80/81 - (sqrt(2)+1)*Pi/18 - log(1+sqrt(2))*sqrt(2)/9 -4*log(2)/9. - Amiram Eldar, Sep 10 2022
From Elmo R. Oliveira, Jan 28 2025: (Start)
E.g.f.: exp(x)*x*(17 + 8*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Formula and more terms from Jeffrey Shallit, Aug 15 1995

A158056 a(n) = 16n^2 + 2n.

Original entry on oeis.org

18, 68, 150, 264, 410, 588, 798, 1040, 1314, 1620, 1958, 2328, 2730, 3164, 3630, 4128, 4658, 5220, 5814, 6440, 7098, 7788, 8510, 9264, 10050, 10868, 11718, 12600, 13514, 14460, 15438, 16448, 17490, 18564, 19670, 20808, 21978, 23180, 24414, 25680

Offset: 1

Vincenzo Librandi, Mar 12 2009

The identity (16*n + 1)^2 - (16*n^2 + 2*n)*4^2 = 1 can be written as A158057(n)^2 - a(n)*4^2 = 1. - Vincenzo Librandi, Feb 09 2012

Sequence found by reading the line from 18, in the direction 18, 68, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A158057.

I:=[18, 68, 150]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]];

LinearRecurrence[{3,-3,1},{18,68,150},50]
Table[16n^2+2n,{n,40}]  (* Harvey P. Dale, Apr 13 2011 *)

PARI
```
a(n) = 16*n^2 + 2*n.
```

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f.: 2*x*(-9 - 7*x)/(x-1)^3.

A360962 Square array T(n,k) = k((3+6n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

Original entry on oeis.org

0, 0, 1, 0, 4, 5, 0, 7, 17, 12, 0, 10, 29, 39, 22, 0, 13, 41, 66, 70, 35, 0, 16, 53, 93, 118, 110, 51, 0, 19, 65, 120, 166, 185, 159, 70, 0, 22, 77, 147, 214, 260, 267, 217, 92, 0, 25, 89, 174, 262, 335, 375, 364, 284, 117, 0, 28, 101, 201, 310, 410, 483, 511, 476, 360, 145

Offset: 0

Paul Curtz, Feb 27 2023

The main diagonal is A024394.

The antidiagonals sums are A000537.

			The rows are:
  0  1  5  12  22  35  51  70 ... = A000326
  0  4 17  39  70 110 159 217 ... = A022266
  0  7 29  66 118 185 267 364 ... = A022272
  0 10 41  93 166 260 375 511 ... = A022278
  0 13 53 120 214 335 483 658 ... = A022284
  ... .
Columns: A000004, A016777, A017581, A154266=3*A017209, 2*A348845, 5*A161447, 3*A158057(n+1), ... (coefficients from A026741).
Difference between two consecutive rows are: A033428.
This square array read by antidiagonals leads to the triangle
  0
  0  1
  0  4  5
  0  7 17 12
  0 10 29 39  22
  0 13 41 66  70  35
  0 16 53 93 118 110 51
  ... .

Cf. A000326, A000537, A022266, A022272, A022278, A022284, A024394.
Cf. A033428.
Cf. A000004, A016777, A017209, A017581, A026741, A154266, A158057, A161447, A348845.

T:= (n,k)-> k*(k*(3+6*n)-1)/2:
seq(seq(T(d-k,k), k=0..d), d=0..10);  # Alois P. Heinz, Feb 28 2023

T[n_, k_] := ((6*n + 3)*k - 1)*k/2; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 27 2023 *)

T(n,k) = k*((3+6*n)*k-1)/2; \\ Michel Marcus, Feb 27 2023

Take successively sequences n*(3*n-1)/2, n*(9*n-1)/2, n*(15*n-1)/2, n*(21*n-1)/2, ... listed in the EXAMPLE section.
From Stefano Spezia, Feb 21 2024: (Start)
G.f.: y*(1 + 2*y + x*(2 + y))/((1 - x)^2*(1 - y)^3).
E.g.f.: exp(x+y)*y*(2 + 3*y + 6*x*(1 + y))/2. (End)

A367882 Table T(n, k) read by downward antidiagonals: T(n, k) = floor((4T(n, k-1)+3)/3) starting with T(n, 0) = 4n.

Original entry on oeis.org

0, 1, 4, 2, 6, 8, 3, 9, 11, 12, 5, 13, 15, 17, 16, 7, 18, 21, 23, 22, 20, 10, 25, 29, 31, 30, 27, 24, 14, 34, 39, 42, 41, 37, 33, 28, 19, 46, 53, 57, 55, 50, 45, 38, 32, 26, 62, 71, 77, 74, 67, 61, 51, 43, 36, 35, 83, 95, 103, 99, 90, 82, 69, 58, 49, 40

Offset: 0

Philippe Deléham, Dec 04 2023

Permutation of nonnegative numbers.

Let b(m) be the row n in which m appears, this sequence would start: 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 3,... . If we would remove in this sequence the first appearance of each number then we would obtain again the same sequence, hence b(m) is a fractal sequence. - Thomas Scheuerle, Dec 04 2023

			Square array starts:
  0,   1,   2,   3,   5,   7, ...
  4,   6,   9,  13,  18,  25, ...
  8,  11,  15,  21,  29,  39, ...
 12,  17,  23,  31,  42,  57, ...
 16,  22,  30,  41,  55,  74, ...
 ...

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).
Index entries for sequences that are permutations of the nonnegative integers

Cf. A008586, A017101, A106839, A158057.

A367882[n_, k_] := A367882[n, k] = If[k == 0, 4*n, Floor[4*A367882[n, k-1]/3 + 1]];
Table[A367882[k, n-k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Apr 03 2024 *)

T(n, k) = if(k==0, 4*n, (4*T(n, k-1)+3)\3) \\ Thomas Scheuerle, Dec 04 2023

T(n, 0) = 4*n = A008586(n).
T(3*n, 1) = 16*n + 1 = A158057(n).
T(3*n+1, 1) = 16*n + 6 = 2*A017101(n).
T(3*n+2, 1) = 16*n + 11 = A106839(n).
T(3^k+n, k) = 4^(k+1) + T(n, k). - Thomas Scheuerle, Dec 04 2023

More terms from Paolo Xausa, Apr 03 2024

A376102 Array read by ascending antidiagonals: A(n,k) = k*2^(n+1) + 1.

Original entry on oeis.org

1, 1, 3, 1, 5, 5, 1, 9, 9, 7, 1, 17, 17, 13, 9, 1, 33, 33, 25, 17, 11, 1, 65, 65, 49, 33, 21, 13, 1, 129, 129, 97, 65, 41, 25, 15, 1, 257, 257, 193, 129, 81, 49, 29, 17, 1, 513, 513, 385, 257, 161, 97, 57, 33, 19, 1, 1025, 1025, 769, 513, 321, 193, 113, 65, 37, 21

Offset: 0

Stefano Spezia, Sep 14 2024

In 1747, Euler showed that any factor of a Fermat number A000215(n) is of the form k*2^(n+1) + 1. See Wells at p. 148.

			The array begins as:
  1,   3,   5,   7,   9,  11,  13, ...
  1,   5,   9,  13,  17,  21,  25, ...
  1,   9,  17,  25,  33,  41,  49, ...
  1,  17,  33,  49,  65,  81,  97, ...
  1,  33,  65,  97, 129, 161, 193, ...
  1,  65, 129, 193, 257, 321, 385, ...
  1, 129, 257, 385, 513, 641, 769, ...
  ...

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 70-71, 237-242.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 136.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987.

Cf. A000079, A000215, A000295.

Cf. A000012 (k=0), A000051, A000337, A004119, A005408 (n=0), A016813 (n=1), A017077 (n=2), A158057 (n=3).

A[n_,k_]:=k*2^(n+1)+1; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

G.f.: (1 - 2*x + y)/((1 - x)*(1 - 2*x)*(1 - y)^2).
E.g.f.: exp(x+y)*(1 + 2*exp(x)*y).
Sum_{0<=k<=n} A(n-k,k) = A000295(n+2).
A(n,1) = A000051(n+1).
A(n,3) = A004119(n+2).
A(n,n) = A000337(n+1).

cp's OEIS Frontend

A319281 Numbers of the form 16^i*(16*j + 1).

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A249356 8*A200975(n)-7 where A200975 are the numbers on the diagonals in Ulam's spiral.

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A215137 a(n) = 17*n + 1.

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A005570 Number of walks on cubic lattice.

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A158056 a(n) = 16*n^2 + 2*n.

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A360962 Square array T(n,k) = k*((3+6*n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

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A367882 Table T(n, k) read by downward antidiagonals: T(n, k) = floor((4*T(n, k-1)+3)/3) starting with T(n, 0) = 4*n.

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A319281 Numbers of the form 16^i(16j + 1).

A158056 a(n) = 16n^2 + 2n.

A360962 Square array T(n,k) = k((3+6n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

A367882 Table T(n, k) read by downward antidiagonals: T(n, k) = floor((4T(n, k-1)+3)/3) starting with T(n, 0) = 4n.