A010727 -id:A010727 - cp's OEIS frontend

A195151 Square array read by antidiagonals upwards: T(n,k) = n((k-2)(-1)^n+k+2)/4, n >= 0, k >= 0.

0, 1, 0, 0, 1, 0, 3, 1, 1, 0, 0, 3, 2, 1, 0, 5, 2, 3, 3, 1, 0, 0, 5, 4, 3, 4, 1, 0, 7, 3, 5, 6, 3, 5, 1, 0, 0, 7, 6, 5, 8, 3, 6, 1, 0, 9, 4, 7, 9, 5, 10, 3, 7, 1, 0, 0, 9, 8, 7, 12, 5, 12, 3, 8, 1, 0, 11, 5, 9, 12, 7, 15, 5, 14, 3, 9, 1, 0, 0, 11, 10, 9, 16, 7

Offset: 0

Omar E. Pol, Sep 14 2011

Also square array T(n,k) read by antidiagonals in which column k lists the multiples of k and the odd numbers interleaved, n>=0, k>=0. Also square array T(n,k) read by antidiagonals in which if n is even then row n lists the multiples of (n/2), otherwise if n is odd then row n lists a constant sequence: the all n's sequence. Partial sums of the numbers of column k give the column k of A195152. Note that if k >= 1 then partial sums of the numbers of the column k give the generalized m-gonal numbers, where m = k + 4.

All columns are multiplicative. - Andrew Howroyd, Jul 23 2018

			Array begins:
.  0,   0,   0,   0,   0,   0,   0,   0,   0,   0,...
.  1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
.  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,...
.  3,   3,   3,   3,   3,   3,   3,   3,   3,   3,...
.  0,   2,   4,   6,   8,  10,  12,  14,  16,  18,...
.  5,   5,   5,   5,   5,   5,   5,   5,   5,   5,...
.  0,   3,   6,   9,  12,  15,  18,  21,  24,  27,...
.  7,   7,   7,   7,   7,   7,   7,   7,   7,   7,...
.  0,   4,   8,  12,  16,  20,  24,  28,  32,  36,...
.  9,   9,   9,   9,   9,   9,   9,   9,   9,   9,...
.  0,   5,  10,  15,  20,  25,  30,  35,  40,  45,...
...

Rows: A000004, A000012, A001477, A010701, A005843, A010716, A008585, A010727, A008586, A010734, A008587.

Columns k: A026741 (k=1), A001477 (k=2), zero together with A080512 (k=3), A022998 (k=4), A195140 (k=5), zero together with A165998 (k=6), A195159 (k=7), A195161 (k=8), A195312 k=(9), A195817 (k=10), A317311 (k=11), A317312 (k=12), A317313 (k=13), A317314 k=(14), A317315 (k=15), A317316 (k=16), A317317 (k=17), A317318 (k=18), A317319 k=(19), A317320 (k=20), A317321 (k=21), A317322 (k=22), A317323 (k=23), A317324 k=(24), A317325 (k=25), A317326 (k=26).

T(n,k) = n*((k-2)*(-1)^n+k+2)/4 \\ Andrew Howroyd, Jul 23 2018

A186684 Total number of positive integers below 10^n requiring 19 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

0, 1, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Martin Renner, Feb 25 2011

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + A186659(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + a(n) = A002283(n).

J.-M. Deshouillers, K. Kawada, and T. D. Wooley, On sums of sixteen biquadrates, Mem. Soc. Math. Fr. 100 (2005), p. 120.

J.-M. Deshouillers, F. Hennecart and B. Landreau, Waring's Problem for sixteen biquadrates - numerical results, Journal de Théorie des Nombres de Bordeaux 12 (2000), pp. 411-422.
L. E. Dickson, Recent progress on Waring's theorem and its generalizations, Bull. Amer. Math. Soc. 39:10 (1933), pp. 701-727.
Eric Weisstein's World of Mathematics, Waring's Problem.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010727, A046050.

PadRight[{0, 1}, 100, 7] (* Paolo Xausa, Jul 30 2024 *)

a(n) = 7 for n >= 3. - Nathaniel Johnston, May 09 2011
From Elmo R. Oliveira, Aug 05 2024: (Start)
G.f.: x^2*(1 + 6*x)/(1 - x).
E.g.f.: 7*(exp(x) - 1 - x) - 3*x^2. (End)

a(5)-a(6) from Lars Blomberg, May 08 2011

Terms after a(6) from Nathaniel Johnston, May 09 2011

A001081 a(n) = 16*a(n-1) - a(n-2).

Original entry on oeis.org

1, 8, 127, 2024, 32257, 514088, 8193151, 130576328, 2081028097, 33165873224, 528572943487, 8424001222568, 134255446617601, 2139663144659048, 34100354867927167, 543466014742175624, 8661355881006882817

Offset: 0

N. J. A. Sloane

Chebyshev's polynomials T(n,x) evaluated at x=8.
The a(n) give all (unsigned, integer) solutions of Pell equation a(n)^2 - 63*b(n)^2 = +1 with b(n)= A077412(n-1), n>=1 and b(0)=0.
Also gives solutions to the equation x^2-1=floor(x*r*floor(x/r)) where r=sqrt(7). - Benoit Cloitre, Feb 14 2004
a(7+14k)-1 and a(7+14k)+1 are consecutive odd powerful numbers. The first pair is 130576328+-1. See A076445. - T. D. Noe, May 04 2006
a(n)^2 - 7 * A001080(n)^2 = 1 (this property is equivalent to the second comment). - Vincenzo Librandi, Feb 17 2013
a(n+3)*a(n) - a(n+2)*a(n+1) = 16*63. - Bruno Berselli, Feb 18 2013

Bastida, Julio R. Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163--166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009) - From N. J. A. Sloane, May 30 2012
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
V. Thébault, Les Récréations Mathématiques. Gauthier-Villars, Paris, 1952, p. 281.

T. D. Noe, Table of n, a(n) for n = 0..200
H. Brocard, Notes élémentaires sur le problème de Peel [sic], Nouvelle Correspondance Mathématique, 4 (1878), 337-343.
M. Davis, One equation to rule them all, Trans. New York Acad. Sci. Ser. II, 30 (1968), 766-773.
Tanya Khovanova, Recursive Sequences
Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, Fausto Jarquín-Zárate, The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d)), arXiv:1904.13002 [math.NT], 2019.
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.
Mihai Prunescu, On other two representations of the C-recursive integer sequences by terms in modular arithmetic, arXiv:2406.06436 [math.NT], 2024. See p. 17.
Mihai Prunescu and Lorenzo Sauras-Altuzarra, On the representation of C-recursive integer sequences by arithmetic terms, arXiv:2405.04083 [math.LO], 2024. See p. 16.
Mihai Prunescu and Joseph M. Shunia, On modular representations of C-recursive integer sequences, arXiv:2502.16928 [math.NT], 2025. See p. 5.
N. J. Wildberger, Pell's equation without irrational numbers, J. Int. Seq. 13 (2010), 10.4.3, Section 5.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (16,-1).

Cf. A090727, A001080, A010727.

I:=[1, 8]; [n le 2 select I[n] else 16*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 17 2013

LinearRecurrence[{16, -1}, {1, 8}, 30]
CoefficientList[Series[(1-8*x)/(1-16*x+x^2), {x, 0, 30}], x] (* G. C. Greubel, Dec 20 2017 *)
Table[LucasL[n, 16*I]*(-I)^n/2, {n,0,30}] (* G. C. Greubel, Jun 06 2019 *)

Vec((1-8*x)/(1-16*x+x^2)+O(x^30)) \\ Charles R Greathouse IV, Jul 02 2013

[lucas_number2(n,16,1)/2 for n in range(0,30)] # Zerinvary Lajos, Jun 26 2008

G.f.: (1-8*x)/(1-16*x+x^2). - Simon Plouffe in his 1992 dissertation.
For all members x of the sequence, 7*x^2 - 7 is a square. Limit_{n->infinity} a(n)/a(n-1) = 8 + 3*sqrt(7). - Gregory V. Richardson, Oct 13 2002
a(n) = T(n, 8) = (S(n, 16)-S(n-2, 16))/2, with S(n, x) := U(n, x/2) and T(n), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(-2, x) := -1, S(-1, x) := 0, S(n, 16)= A077412(n).
a(n) = ((8 + 3*sqrt(7))^n + (8 - 3*sqrt(7))^n)/2.
a(n) = sqrt(63*A077412(n-1)^2 + 1), n>=1, (cf. Richardson comment).
a(n) = 16*a(n-1) - a(n-2) with a(1)=1 and a(2)=8. - Sture Sjöstedt, Nov 18 2011
a(n) = A077412(n) - 8*A077412(n-1). - R. J. Mathar, Jul 22 2017
a(n) = (-i)^n*Lucas(n, 16*i)/2, where i = sqrt(-1). - G. C. Greubel, Jun 06 2019

Chebyshev and Pell comments from Wolfdieter Lang, Nov 08 2002

A144263 Number of ways of placing n labeled balls into n unlabeled (but7-colored) boxes.

Original entry on oeis.org

1, 7, 56, 497, 4809, 50134, 558215, 6593839, 82187658, 1076193867, 14749823893, 210926792244, 3138696242941, 48485723853763, 775929767223352, 12840232627455485, 219355194338036309, 3862794707291567670

Offset: 0

Philippe Deléham, Sep 16 2008

nonn

a(n) is also the exp transform of A010727. - Alois P. Heinz, Oct 09 2008

The number of ways of putting n labeled balls into a set of bags and then putting the bags into 7 labeled boxes. - Peter Bala, Mar 23 2013

Alois P. Heinz, Table of n, a(n) for n = 0..200
N. J. A. Sloane, Transforms

Cf. A000110, A001861, A027710, A078944, A144180, A144223. A189233, A221159, A221176.

a:= proc(n) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, k-1)*a(n-k), k=1..n-1))*7)
    end:
seq(a(n), n=0..25); # Alois P. Heinz, Oct 09 2008

Table[BellB[n,7],{n,0,20}] (* Vaclav Kotesovec, Mar 12 2014 *)

expnums(18, 7) # Zerinvary Lajos, May 15 2009

a(n) = Sum_{k=0..n} 7^k*A048993(n,k); A048993: Stirling2 numbers.
E.g.f.: exp(7*(exp(x)-1)).
G.f.: 7*(x/(1-x))*A(x/(1-x))= A(x)-1; seven times the binomial transform equals this sequence shifted one place left.
a(n) ~ n^n * exp(n/LambertW(n/7)-7-n) / (sqrt(1+LambertW(n/7)) * LambertW(n/7)^n). - Vaclav Kotesovec, Mar 12 2014
G.f.: Sum_{j>=0} 7^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 11 2019

More terms from Alois P. Heinz, Oct 09 2008

A001080 a(n) = 16*a(n-1) - a(n-2) with a(0) = 0, a(1) = 3.

Original entry on oeis.org

0, 3, 48, 765, 12192, 194307, 3096720, 49353213, 786554688, 12535521795, 199781794032, 3183973182717, 50743789129440, 808716652888323, 12888722657083728, 205410845860451325, 3273684811110137472, 52173546131901748227, 831503053299317834160

Offset: 0

N. J. A. Sloane

Also 7*x^2 + 1 is a square; n=7 in PARI script below. - Cino Hilliard, Mar 08 2003
That is, the terms are solutions y of the Pell-Fermat equation x^2 - 7 * y^2 = 1. The corresponding values of x are in A001081. (x,y) = (1,0), (8,3), (127,48), ... - Bernard Schott, Feb 23 2019
The first solution to the equation x^2 - 7*y^2 = 1 is (X(0); Y(0)) = (1; 0) and the other solutions are defined by: (X(n); Y(n))= (8*X(n-1) + 21*Y(n-1); 3*X(n-1) + 8*Y(n-1)), with n >= 1. - Mohamed Bouhamida, Jan 16 2020

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
V. Thébault, Les Récréations Mathématiques. Gauthier-Villars, Paris, 1952, p. 281.

T. D. Noe, Table of n, a(n) for n = 0..200
H. Brocard, Notes élémentaires sur le problème de Peel [sic], Nouvelle Correspondance Mathématique, 4 (1878), 337-343.
M. Davis, One equation to rule them all, Trans. New York Acad. Sci. Ser. II, 30 (1968), 766-773.
Tanya Khovanova, Recursive Sequences
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.
Mihai Prunescu, On other two representations of the C-recursive integer sequences by terms in modular arithmetic, arXiv:2406.06436 [math.NT], 2024. See p. 17.
Mihai Prunescu and Lorenzo Sauras-Altuzarra, On the representation of C-recursive integer sequences by arithmetic terms, arXiv:2405.04083 [math.LO], 2024. See p. 15.
Mihai Prunescu and Joseph M. Shunia, On modular representations of C-recursive integer sequences, arXiv:2502.16928 [math.NT], 2025. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (16,-1).

Equals 3 * A077412. Bisection of A084069.
Cf. A048907.
Cf. A001081, A010727. - Vincenzo Librandi, Feb 16 2009

a:=[0,3];; for n in [3..30] do a[n]:=16*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Feb 23 2019

I:=[0,3]; [n le 2 select I[n] else 16*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 20 2017

A001080:=3*z/(1-16*z+z**2); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

LinearRecurrence[{16,-1},{0,3},30] (* Harvey P. Dale, Nov 01 2011 *)
CoefficientList[Series[3*x/(1-16*x+x^2), {x, 0, 30}], x] (* G. C. Greubel, Dec 20 2017 *)

nxsqp1(m,n) = { for(x=1,m, y = n*x*x+1; if(issquare(y),print1(x" ")) ) }

x='x+O('x^30); concat([0], Vec(3*x/(1-16*x+x^2))) \\ G. C. Greubel, Dec 20 2017

(3*x/(1-16*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 23 2019

G.f.: 3*x/(1-16*x+x^2).
From Mohamed Bouhamida, Sep 20 2006: (Start)
a(n) = 15*(a(n-1) + a(n-2)) - a(n-3).
a(n) = 17*(a(n-1) - a(n-2)) + a(n-3). (End)
a(n) = 16*a(n-1) - a(n-2) with a(1)=0 and a(2)=3. - Sture Sjöstedt, Nov 18 2011
E.g.f.: exp(8*x)*sinh(3*sqrt(7)*x)/sqrt(7). - G. C. Greubel, Feb 23 2019

A176439 Decimal expansion of (7+sqrt(53))/2.

Original entry on oeis.org

7, 1, 4, 0, 0, 5, 4, 9, 4, 4, 6, 4, 0, 2, 5, 9, 1, 3, 5, 5, 4, 8, 6, 5, 1, 2, 4, 5, 7, 6, 3, 5, 1, 6, 3, 9, 6, 8, 8, 8, 8, 3, 4, 8, 4, 1, 2, 8, 8, 2, 3, 8, 7, 1, 9, 1, 8, 9, 0, 9, 0, 8, 9, 5, 6, 4, 2, 0, 5, 7, 8, 6, 9, 3, 1, 2, 4, 5, 2, 5, 9, 1, 6, 6, 4, 7, 8, 9, 7, 0, 4, 5, 4, 0, 4, 6, 3, 3, 7, 6, 0, 9, 6, 3, 1

Offset: 1

Klaus Brockhaus, Apr 19 2010

Continued fraction expansion of (7+sqrt(53))/2 is A010727.
This is the shape of a 7-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 09 2011
c^n = c * A054413(n-1) + A054413(n-2), where c = (7+sqrt(53))/2. - Gary W. Adamson, Apr 14 2024

			(7+sqrt(53))/2 = 7.14005494464025913554...

Wikipedia, Metallic mean
Index entries for algebraic numbers, degree 2

Cf. A010506 (decimal expansion of sqrt(53)), A010727 (all 7's sequence).

Cf. A049310.

r=7; t = (r + (4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
RealDigits[(7+Sqrt[53])/2,10,120][[1]] (* Harvey P. Dale, Nov 03 2024 *)

(7+sqrt(53))/2 \\ Charles R Greathouse IV, Jul 24 2013

Equals lim_{n->oo} S(n, sqrt(53))/S(n-1, sqrt(53)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

Positive solution of x^2 - 7*x - 1 = 0. - Hugo Pfoertner, Apr 14 2024

A023006 Number of partitions of n into parts of 7 kinds.

Original entry on oeis.org

1, 7, 35, 140, 490, 1547, 4522, 12405, 32305, 80465, 192899, 447146, 1006145, 2204475, 4715510, 9869132, 20247710, 40786690, 80782800, 157510780, 302666903, 573720808, 1073720305, 1985506775, 3630307835, 6567206471, 11760658378, 20860415590, 36665885170, 63891010155, 110415782785, 189320804673, 322174588225

Offset: 0

David W. Wilson

nonn

a(n) is Euler transform of A010727. - Alois P. Heinz, Oct 17 2008

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (first 201 terms from Vincenzo Librandi)
Roland Bacher, P. De La Harpe, Conjugacy growth series of some infinitely generated groups. 2016, hal-01285685v2.
P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, A. M. Läuchli, Chiral spin liquids in triangular lattice SU (N) fermionic Mott insulators with artificial gauge fields, arXiv preprint arXiv:1601.00958 [cond-mat.quant-gas], 2016.
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. 7th column of A144064. - Alois P. Heinz, Oct 17 2008

with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*7, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, Oct 17 2008

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^7,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)

Vec(1/eta('x+O('x^66))^7)  /* Joerg Arndt, Jul 30 2011 */

G.f.: Product_{m>=1} 1/(1-x^m)^7.
a(n) ~ 49 * exp(Pi * sqrt(14*n/3)) / (576 * sqrt(2) * n^(5/2)). - Vaclav Kotesovec, Feb 28 2015
a(0) = 1, a(n) = (7/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017
G.f.: exp(7*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

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.

A168309 Period 2: repeat 4,-3.

Original entry on oeis.org

4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3

Offset: 1

Klaus Brockhaus, Nov 22 2009

Interleaving of A010709 and -3*A000012.
Binomial transform of 4 followed by a signed version of A005009.
Inverse binomial transform of 4 followed by A000079.
a(n+1) - a(n) = 7*(-1)^n.
A168230 without initial term 0 gives partial sums.
Nonsimple continued fraction expansion of 2+2*sqrt(2/3) = 3.6329931618... - R. J. Mathar, Mar 08 2012

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010709 (all 4's sequence), A000012 (all 1's sequence), A010727 (all 7's sequence), A168230, A005009 (7*2^n), A000079 (powers of 2).

&cat[ [4, -3]: n in [1..42] ];
[ n eq 1 select 4 else -Self(n-1)+1: n in [1..84] ];

LinearRecurrence[{0,1},{4, -3}, 50] (* or *) Table[(1 - 7*(-1)^n)/2,{n,0,25}] (* G. C. Greubel, Jul 17 2016 *)
PadRight[{},120,{4,-3}] (* Harvey P. Dale, Oct 20 2018 *)

a(n) = (1 - 7*(-1)^n)/2.
a(n) = -a(n-1) + 1 for n > 1; a(1) = 4.
a(n) = a(n-2) for n > 2; a(1) = 4, a(2) = -3.
G.f.: x*(4 - 3*x)/((1-x)*(1+x)).
E.g.f.: (1/2)*(-1 + exp(x))*(7 + exp(x))*exp(-x). - G. C. Greubel, Jul 17 2016

A176415 Periodic sequence: repeat 7,1.

Original entry on oeis.org

7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7

Offset: 0

Klaus Brockhaus, Apr 17 2010

Interleaving of A010727 and A000012.
Also continued fraction expansion of (7+sqrt(77))/2.
Also decimal expansion of 71/99.
Essentially first differences of A047521.
Binomial transform of A176414.
Inverse binomial transform of 2*A020707 preceded by 7.
Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 4*x^2 + 4*x^3 + 10*x^4 + 10*x^5 + ... is the o.g.f. for A058187. - Peter Bala, Mar 13 2015

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010727 (all 7's sequence), A000012 (all 1's sequence), A092290 (decimal expansion of (7+sqrt(77))/2), A010688 (repeat 1, 7), A047521 (congruent to 0 or 7 mod 8), A176414 (expansion of (7+8*x)/(1+2*x)), A020707 (2^(n+2)), A058187.

&cat[ [7, 1]: n in [0..52] ];
[ 4+3*(-1)^n: n in [0..104] ];

PadRight[{},120,{7,1}] (* Harvey P. Dale, Dec 30 2018 *)

a(n)=7-n%2*6 \\ Charles R Greathouse IV, Oct 28 2011

a(n) = 4+3*(-1)^n.
a(n) = a(n-2) for n > 1; a(0) = 7, a(1) = 1.
a(n) = -a(n-1)+8 for n > 0; a(0) = 7.
a(n) = 7*((n+1) mod 2)+(n mod 2).
a(n) = A010688(n+1).
G.f.: (7+x)/(1-x^2).
Dirichglet g.f.: (1+6*2^(-s))*zeta(s). - R. J. Mathar, Apr 06 2011
Multiplicative with a(2^e) = 7, and a(p^e) = 1 for p >= 3. - Amiram Eldar, Jan 01 2023

cp's OEIS Frontend

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A144263 Number of ways of placing n labeled balls into n unlabeled (but7-colored) boxes.

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A001080 a(n) = 16*a(n-1) - a(n-2) with a(0) = 0, a(1) = 3.

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A023006 Number of partitions of n into parts of 7 kinds.

A195151 Square array read by antidiagonals upwards: T(n,k) = n((k-2)(-1)^n+k+2)/4, n >= 0, k >= 0.