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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

References

• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 246.

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).

Crossrefs

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

Cf. A007522 (primes), subsequence of A047522.

Cf. A056753, A227144, A227146, A239126.

Programs

GAP

List([0..60],n->8*n+7); # Muniru A Asiru, Aug 28 2018

Haskell

a004771 = (+ 7) . (* 8)
a004771_list = [7, 15 ..]  -- Reinhard Zumkeller, Jan 29 2013

Magma

[8*n+7: n in [0..60]]; // Vincenzo Librandi, May 28 2011

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 *)

PARI

a(n)=8*n+7 \\ Charles R Greathouse IV, Sep 23 2012

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)