A218723 -id:A218723 - cp's OEIS frontend

A131865 Partial sums of powers of 16.

1, 17, 273, 4369, 69905, 1118481, 17895697, 286331153, 4581298449, 73300775185, 1172812402961, 18764998447377, 300239975158033, 4803839602528529, 76861433640456465, 1229782938247303441, 19676527011956855057, 314824432191309680913, 5037190915060954894609

Offset: 0

Reinhard Zumkeller, Jul 22 2007

16 = 2^4 is the growth measure for the Jacobsthal spiral (compare with phi^4 for the Fibonacci spiral). - Paul Barry, Mar 07 2008
Second quadrisection of A115451. - Paul Curtz, May 21 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=16, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 1, a(n-1) = det(A). - Milan Janjic, Feb 21 2010
Partial sums are in A014899. Also, the sequence is related to A014931 by A014931(n+1) = (n+1)*a(n) - Sum_{i=0..n-1} a(i) for n>0. - Bruno Berselli, Nov 07 2012
a(n) is the total number of holes in a certain box fractal (start with 16 boxes, 1 hole) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 28 2015
Except for 1 and 17, all terms are Brazilian repunits numbers in base 16, and so belong to A125134. All terms >= 273 are composite because a(n) = ((4^(n+1) + 1) * (4^(n+1) - 1))/15. - Bernard Schott, Jun 06 2017
The sequence in binary is 1, 10001, 100010001, 1000100010001, 10001000100010001, ... cf. Plouffe link, A330135. - Frank Ellermann, Mar 05 2020

			a(3) = 1 + 16 + 256 + 4096 = 4369 = in binary: 1000100010001.
a(4) = (16^5 - 1)/15 = (4^5 + 1) * (4^5 - 1)/15 = 1025 * 1023/15 = 205 * 341 = 69905 = 11111_16. - _Bernard Schott_, Jun 06 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..800
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
Kival Ngaokrajang, Illustration of initial terms
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. See Table 1.
Simon Plouffe, Identities and approximations inspired from Ramanujan notebooks, III, 2009.
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (17,-16).

Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723. - M. F. Hasler, Nov 05 2012

[(16^(n+1)-1)/15: n in [0..20]]; // Vincenzo Librandi, Sep 17 2011

A131865:=n->(16^(n+1)-1)/15: seq(A131865(n), n=0..30); # Wesley Ivan Hurt, Apr 29 2017

Table[(2^(4 n) - 1)/15, {n, 16}] (* Robert G. Wilson v, Aug 22 2007 *)
Accumulate[16^Range[0,20]] (* or *) LinearRecurrence[{17,-16},{1,17},20] (* Harvey P. Dale, Jul 19 2019 *)

a[0]:0$
a[n]:=16*a[n-1]+1$
A131865(n):=a[n]$
makelist(A131865(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

A131865(n)=16^n\15  \\ M. F. Hasler, Nov 05 2012

def A131865(n): return (1<<(n+1<<2))//15 # Chai Wah Wu, Nov 10 2022

[gaussian_binomial(n,1,16) for n in range(1,18)] # Zerinvary Lajos, May 28 2009

a(n) = if n=0 then 1 else a(n-1) + A001025(n).
for n > 0: A131851(a(n)) = n and abs(A131851(m)) < n for m < a(n).
a(n) = A098704(n+2)/2.
a(n) = (16^(n+1) - 1)/15. - Bernard Schott, Jun 06 2017
a(n) = (A001025(n+1) - 1)/15.
a(n) = 16*a(n-1) + 1. - Paul Curtz, May 20 2008
G.f.: 1 / ( (16*x-1)*(x-1) ). - R. J. Mathar, Feb 06 2011
E.g.f.: exp(x)*(16*exp(15*x) - 1)/15. - Stefano Spezia, Mar 06 2020

A218722 a(n) = (19^n-1)/18.

Original entry on oeis.org

0, 1, 20, 381, 7240, 137561, 2613660, 49659541, 943531280, 17927094321, 340614792100, 6471681049901, 122961939948120, 2336276859014281, 44389260321271340, 843395946104155461, 16024522975978953760, 304465936543600121441

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 19 (A001029); q-integers for q=19: diagonal k=1 in triangle A022183.

Partial sums are in A014903. Also, the sequence is related to A014936 by A014936(n) = n*a(n)-sum(a(i), i=0..n-1) for n>0. - Bruno Berselli, Nov 06 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries related to partial sums
Index entries for linear recurrences with constant coefficients, signature (20,-19).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A001029, A014903, A014936, A022183.

[n le 2 select n-1 else 20*Self(n-1)-19*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{20, -19}, {0, 1}, 40] (* Vincenzo Librandi, Nov 07 2012 *)

A218722(n):=(19^n-1)/18$ makelist(A218722(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218722(n)=19^n\18
```

a(n) = floor(19^n/18).
G.f.: x/((1-x)*(1-19*x)). - Bruno Berselli, Nov 06 2012
a(n) = 20*a(n-1) - 19*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(10*x)*sinh(9*x)/9. - Stefano Spezia, Mar 11 2023

A218724 a(n) = (21^n - 1)/20.

Original entry on oeis.org

0, 1, 22, 463, 9724, 204205, 4288306, 90054427, 1891142968, 39714002329, 833994048910, 17513875027111, 367791375569332, 7723618886955973, 162195996626075434, 3406115929147584115, 71528434512099266416, 1502097124754084594737, 31544039619835776489478

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 21 (A009965); q-integers for q=21: diagonal k=1 in triangle A022185.
Partial sums are in A014905. Also, the sequence is related to A014938 by A014938(n) = n*a(n) - Sum_{i=0..n-1} a(i) for n > 0. - Bruno Berselli, Nov 06 2012
For n >= 1, 4*a(n) is the total number of holes in a certain box fractal (start with 21 boxes, 4 holes) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 27 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Kival Ngaokrajang, Illustration of initial terms
Index entries related to partial sums
Index entries for linear recurrences with constant coefficients, signature (22,-21).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218725-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009965, A014905, A014938, A022185.

[n le 2 select n-1 else 22*Self(n-1) - 21*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{22, -21}, {0, 1}, 40] (* Vincenzo Librandi, Nov 07 2012 *)

A218724(n):=(21^n-1)/20$ makelist(A218724(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218724(n)=21^n\20
```

a(n) = floor(21^n/20).
G.f.: x/((1-x)*(1-21*x)). - Bruno Berselli, Nov 06 2012
a(n) = 22*a(n-1) - 21*a(n-2). - Vincenzo Librandi, Nov 07 2012
a(n) = 21*a(n-1) + 1. - Kival Ngaokrajang, Jan 27 2015
a(n) = a(n-1) + 21^(n-1), n >= 1, a(0) = 0.  - Wolfdieter Lang, Feb 02 2015
E.g.f.: exp(11*x)*sinh(10*x)/10. - Elmo R. Oliveira, Aug 29 2024

A218734 a(n) = (31^n - 1)/30.

Original entry on oeis.org

0, 1, 32, 993, 30784, 954305, 29583456, 917087137, 28429701248, 881320738689, 27320942899360, 846949229880161, 26255426126284992, 813918209914834753, 25231464507359877344, 782175399728156197665, 24247437391572842127616, 751670559138758105956097

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 31 (A009975).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See pp. 2, 17.
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (32,-31).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218733, A132469, A218736-A218753, A133853, A094028, A218723.

[n le 2 select n-1 else 32*Self(n-1)-31*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{32, -31}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218734(n):=(31^n-1)/30$
makelist(A218734(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
a(n)=31^n\30
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1 - x)*(1 - 31*x)).
a(n) = 32*a(n-1) - 31*a(n-2) for n > 1.
a(n) = floor(31^n/30). (End)
E.g.f.: exp(16*x)*sinh(15*x)/15. - Stefano Spezia, Mar 11 2023

A132469 a(n) = (2^(5*n) - 1)/31.

Original entry on oeis.org

0, 1, 33, 1057, 33825, 1082401, 34636833, 1108378657, 35468117025, 1134979744801, 36319351833633, 1162219258676257, 37191016277640225, 1190112520884487201, 38083600668303590433, 1218675221385714893857, 38997607084342876603425, 1247923426698972051309601

Offset: 0

A.K. Devaraj, Aug 22 2007

Partial sums of powers of 32 (A009976), a.k.a. q-numbers for q=32. - M. F. Hasler, Nov 05 2012

A. K. Devaraj, "Minimum Universal Exponent Generalisation of Fermat's Theorem", in ISSN #1550-3747, Proceedings of Hawaii Intl Conference on Statistics, Mathematics & Related Fields, 2004.

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. See Table 1.
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (33,-32).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

[n le 2 select n-1 else 33*Self(n-1) - 32*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

Table[(2^(5 n) - 1)/31, {n, 16}] (* Robert G. Wilson v *)
LinearRecurrence[{33, -32}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A132469(n):=(32^n-1)/31$
makelist(A132469(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

A132469(n)=32^n\31  \\ M. F. Hasler, Nov 07 2012

[gaussian_binomial(5*n,1,2)/31 for n in range(1,17)] # Zerinvary Lajos, May 28 2009

a(n) = (32^n - 1)/31 = floor(32^n/31) = Sum_{k=0..n} 32^k. - M. F. Hasler, Nov 05 2012
G.f.: x/((1 - x)*(1 - 32*x)). - Bruno Berselli, Nov 06 2012
E.g.f.: exp(x)*(exp(31*x) - 1)/31. - Stefano Spezia, Mar 23 2023

Edited and extended by Robert G. Wilson v, Aug 22 2007

Edited and extended to offset 0 by M. F. Hasler, Nov 05 2012

A218753 a(n) = (49^n - 1)/48.

Original entry on oeis.org

0, 1, 50, 2451, 120100, 5884901, 288360150, 14129647351, 692352720200, 33925283289801, 1662338881200250, 81454605178812251, 3991275653761800300, 195572507034328214701, 9583052844682082520350, 469569589389422043497151, 23008909880081680131360400

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 49 (A087752).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums
Index entries for linear recurrences with constant coefficients, signature (50,-49)

Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218752, A133853, A094028, A218723.

Cf. A087752.

[n le 2 select n-1 else 50*Self(n-1) - 49*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

LinearRecurrence[{50, -49}, {0, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)
Join[{0},Accumulate[49^Range[0,20]]] (* Harvey P. Dale, Apr 14 2023 *)

A218753(n):=floor(49^n/48)$ makelist(A218753(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218753(n)=49^n\48
```

G.f.: x/((1-x)*(1-49*x)). - Vincenzo Librandi, Nov 08 2012
a(n) = 50*a(n-1) - 49*a(n-2) with a(0)=0, a(1)=1. - Vincenzo Librandi, Nov 08 2012
a(n) = 49*a(n-1) + 1 with a(0)=0. - Vincenzo Librandi, Nov 08 2012
a(n) = floor(49^n/48). - Vincenzo Librandi, Nov 08 2012
E.g.f.: exp(25*x)*sinh(24*x)/24. - Elmo R. Oliveira, Aug 27 2024

A218736 a(n) = (33^n - 1)/32.

Original entry on oeis.org

0, 1, 34, 1123, 37060, 1222981, 40358374, 1331826343, 43950269320, 1450358887561, 47861843289514, 1579440828553963, 52121547342280780, 1720011062295265741, 56760365055743769454, 1873092046839544391983, 61812037545704964935440, 2039797239008263842869521

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 33 (A009977).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (34,-33).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218737-A218753, A133853, A094028, A218723.

[n le 2 select n-1 else 34*Self(n-1)-33*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{34, -33}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218736(n):=(33^n-1)/32$
makelist(A218736(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218736(n)=33^n>>5
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1 - x)*(1 - 33*x)).
a(n) = 34*a(n-1) - 33*a(n-2).
a(n) = floor(33^n/32). (End)
E.g.f.: exp(x)*(exp(32*x) - 1)/32. - Stefano Spezia, Mar 24 2023

A218750 a(n) = (47^n - 1)/46.

Original entry on oeis.org

0, 1, 48, 2257, 106080, 4985761, 234330768, 11013546097, 517636666560, 24328923328321, 1143459396431088, 53742591632261137, 2525901806716273440, 118717384915664851681, 5579717091036248029008, 262246703278703657363377, 12325595054099071896078720

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 47 (A009991).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums
Index entries related to q-numbers
Index entries for linear recurrences with constant coefficients, signature (48,-47)

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009991.

[n le 2 select n-1 else 48*Self(n-1) - 47*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

Table[(47^n - 1)/46, {n, 0, 19}] (* Alonso del Arte, Nov 04 2012 *)
LinearRecurrence[{48, -47}, {0, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)

A218750(n):=(47^n-1)/46$ makelist(A218750(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218750(n)=47^n\46
```

a(n) = floor(47^n/46).
G.f.: x/(47*x^2-48*x+1) = x/((1-x)*(1-47*x)). [Colin Barker, Nov 06 2012]
a(0)=0, a(n) = 47*a(n-1) + 1. - Vincenzo Librandi, Nov 08 2012
a(n) = 48*a(n-1) - 47*a(n-2). - Wesley Ivan Hurt, Jan 25 2022
E.g.f.: exp(24*x)*sinh(23*x)/23. - Elmo R. Oliveira, Aug 27 2024

A218726 a(n) = (23^n - 1)/22.

Original entry on oeis.org

0, 1, 24, 553, 12720, 292561, 6728904, 154764793, 3559590240, 81870575521, 1883023236984, 43309534450633, 996119292364560, 22910743724384881, 526947105660852264, 12119783430199602073, 278755018894590847680, 6411365434575589496641, 147461404995238558422744

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 23, q-integers for q=23: diagonal k=1 in triangle A022187.

Partial sums are in A014909. Also, the sequence is related to A014941 by A014941(n) = n*a(n) - Sum{a(i), i=0..n-1} for n > 0. - Bruno Berselli, Nov 07 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (24,-23).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A014909, A014941, A022187.

[n le 2 select n-1 else 24*Self(n-1)-23*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{24, -23}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)
(23^Range[0,20]-1)/22 (* Harvey P. Dale, Nov 09 2012 *)

A218726(n):=(23^n-1)/22$
makelist(A218726(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218726(n)=23^n\22
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-23*x)).
a(n) = floor(23^n/22).
a(n) = 24*a(n-1) - 23*a(n-2). (End)
E.g.f.: exp(12*x)*sinh(11*x)/11. - Elmo R. Oliveira, Aug 27 2024

A218732 a(n) = (29^n - 1)/28.

Original entry on oeis.org

0, 1, 30, 871, 25260, 732541, 21243690, 616067011, 17865943320, 518112356281, 15025258332150, 435732491632351, 12636242257338180, 366451025462807221, 10627079738421409410, 308185312414220872891, 8937374060012405313840, 259183847740359754101361

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 29 (A009973).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (30,-29).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009973.

[n le 2 select n-1 else 30*Self(n-1)-29*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{30, -29}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218732(n):=(29^n-1)/28$
makelist(A218732(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
a(n)=29^n\28
```

a(n) = floor(29^n/28).
G.f.: x/((1-x)*(1-29*x)). - Vincenzo Librandi, Nov 07 2012
a(n) = 30*a(n-1) - 29*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(15*x)*sinh(14*x)/14. - Elmo R. Oliveira, Aug 27 2024

cp's OEIS Frontend

A131865 Partial sums of powers of 16.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

Python

Sage

Formula

A218722 a(n) = (19^n-1)/18.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

A218724 a(n) = (21^n - 1)/20.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

A218734 a(n) = (31^n - 1)/30.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

A132469 a(n) = (2^(5*n) - 1)/31.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Sage

Formula

Extensions

A218753 a(n) = (49^n - 1)/48.

Views

Author

Keywords

Comments

Links

Crossrefs