A004771 a(n) = 8*n + 7. Or, numbers whose binary expansion ends in 111.

7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87, 95, 103, 111, 119, 127, 135, 143, 151, 159, 167, 175, 183, 191, 199, 207, 215, 223, 231, 239, 247, 255, 263, 271, 279, 287, 295, 303, 311, 319, 327, 335, 343, 351, 359, 367, 375, 383, 391, 399, 407, 415, 423, 431

Offset

0,1

Comments

These numbers cannot be expressed as the sum of 3 squares. - Artur Jasinski, Nov 22 2006

These numbers cannot be perfect squares. - Cino Hilliard, Sep 03 2006

a(n-2), n >= 2, appears in the second column of triangle A239126 related to the Collatz problem. - Wolfdieter Lang, Mar 14 2014

The initial terms 7, 15, 23, 31 are the generating set for the rest of the sequence in the sense that, by Lagrange's Four Square Theorem, any number n of the form 8*k+7 can always be written as a sum of no fewer than four squares, and if n = a^2 + b^2 + c^2 + d^2, then (a mod 4)^2 + (b mod 4)^2 + (c mod 4)^2 + (d mod 4)^2 must be one of 7, 15, 23, 31. - Walter Kehowski, Jul 07 2014

Define a set of consecutive positive odd numbers {1, 3, 5, ..., 12*n + 9} and skip the number 6*n + 5. Then the contraharmonic mean of that set gives this sequence. For example, ContraharmonicMean[{1, 3, 7, 9}] = 7. - Hilko Koning, Aug 27 2018

Jacobi symbol (2, a(n)) = Kronecker symbol (a(n), 2) = 1. - Jianing Song, Aug 28 2018

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Tanya Khovanova, Recursive Sequences

Leo Tavares, Illustration: Twin Square Frames

Leo Tavares, Illustration: Mid-line Hexagons

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 962

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

O.g.f: (7 + x)/(1 - x)^2 = 8/(1 - x)^2 - 1/(1 - x). - R. J. Mathar, Nov 30 2007

a(n) = 2*a(n-1) - a(n-2) for n >= 2. - Vincenzo Librandi, May 28 2011

A056753(a(n)) = 7. - Reinhard Zumkeller, Aug 23 2009

a(n) = t(t(t(n))), where t(i) = 2*i + 1.

a(n) = A004767(2*n+1), for n >= 0. See also A004767(2*n) = A017101(n). - Wolfdieter Lang, Feb 03 2022

From Elmo R. Oliveira, Apr 11 2024: (Start)

E.g.f.: exp(x)*(7 + 8*x).

a(n) = A033954(n+1) - A033954(n). (End)

Maple

A004771:=n->8*n+7; seq(A004771(n), n=0..100); # Wesley Ivan Hurt, Dec 22 2013

Mathematica

8 Range[0, 60] + 7 (* or *) Range[7, 500, 8] (* or *) Table[8 n + 7, {n, 0, 60}] (* Bruno Berselli, Dec 28 2016 *)

Prog

(Magma) [8*n+7: n in [0..60]]; // Vincenzo Librandi, May 28 2011
(PARI) a(n)=8*n+7 \\ Charles R Greathouse IV, Sep 23 2012
(Haskell)
a004771 = (+ 7) . (* 8)
a004771_list = [7, 15 ..]  -- Reinhard Zumkeller, Jan 29 2013
(GAP) List([0..60], n->8*n+7); # Muniru A Asiru, Aug 28 2018

Crossrefs

Cf. A004767, A008590, A017077, A017101, A017137, A033954.

Cf. A007522 (primes), subsequence of A047522.

Cf. A056753, A227144, A227146, A239126.

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved

A059562 Beatty sequence for log(Pi)/(log(Pi)-1).

7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87, 94, 102, 110, 118, 126, 134, 142, 150, 158, 166, 174, 181, 189, 197, 205, 213, 221, 229, 237, 245, 253, 261, 268, 276, 284, 292, 300, 308, 316, 324, 332, 340, 348, 355, 363, 371, 379, 387, 395, 403, 411, 419, 427

Offset

1,1

Links

Harry J. Smith, Table of n, a(n) for n = 1..2000

Aviezri S. Fraenkel, Jonathan Levitt, Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.

Eric Weisstein's World of Mathematics, Beatty Sequence

Index entries for sequences related to Beatty sequences

Prog

(PARI) { default(realprecision, 100); b=log(Pi)/(log(Pi) - 1); for (n = 1, 2000, write("b059562.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009
(PARI) A059562(n, c=1-1/log(Pi))=n\c \\ Use \pXX to set sufficiently large precision. - M. F. Hasler, Oct 06 2014

Crossrefs

Beatty complement is A059561.

Keyword

nonn,easy

Author

Mitch Harris, Jan 22 2001

Status

approved

A017149 Duplicate of A004771.

7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87, 95, 103, 111, 119, 127, 135, 143, 151, 159

Offset

0,1

Links

Table of n, a(n) for n=0..19.

Keyword

dead

Status

approved

A029724 Odd numbers congruent to 7 mod 8 such that (2^(h(-n)+2)-n) is a square, where h(-n) is the class number.

7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87, 103, 127, 135, 151, 175, 183, 199, 207, 223, 247, 255, 271, 295, 343, 367, 375, 463, 487, 511, 583, 727, 751, 823, 847, 967, 1023, 1087, 1255, 1303, 1423, 1527, 2047, 2143, 3063, 3343, 4447, 5503

Offset

1,1

Links

Table of n, a(n) for n=1..48.

Prog

(PARI) forstep(n=7, 1e4, 8, if(issquare(2^(qfbclassno(-n)+2)-n), print1(n", "))) \\ Charles R Greathouse IV, May 08 2011

Keyword

nonn

Author

Eric Rains (rains(AT)caltech.edu)

Status

approved

A133655 a(n) = 2*A016777(n) + A016777(n-1) - (n+1).

1, 7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87, 95, 103, 111, 119, 127, 135, 143, 151, 159, 167, 175, 183, 191, 199, 207, 215, 223, 231, 239, 247, 255, 263, 271, 279, 287, 295, 303, 311, 319, 327, 335, 343, 351, 359, 367, 375, 383, 391, 399, 407, 415

Offset

0,2

Links

Table of n, a(n) for n=0..52.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

Equals "1" followed by A004771.

Binomial transform of [1, 6, 2, -2, 2, -2, 2, ...].

G.f.: (2*x^2+5*x+1)/(x-1)^2. - Harvey P. Dale, Sep 13 2011

Example

a(3) = 23 = 2*A016777(3) + A016777(2) - 4 = 2*10 + 7 - 4.
a(3) = 23 = (1, 3, 3, 1) dot (1, 6, 2, -2) = (1, 18, 6, -2).

Mathematica

CoefficientList[Series[(2 x^2+5 x+1)/(x-1)^2, {x, 0, 60}], x] (* Harvey P. Dale, Sep 13 2011 *)

Crossrefs

Cf. A004771, A016777.

Keyword

nonn,easy

Author

Gary W. Adamson, Sep 20 2007

Extensions

More terms and corrected definition from R. J. Mathar, Jun 08 2008

Status

approved