• For a long time I have had the idea that it would be useful to have a series of concordances which would list the integer sequences to be found in certain classical books (Riordan, Comtet, Harary and Palmer, Stanley, Knuth, etc.). Here is the fourth of these, a concordance to:

  R. L. Graham, D. E. Knuth and O. Patashnik's

  "Concrete Mathematics", Addison-Wesley, 2nd Ed., 1994

  based on a first draft prepared by

  Olivier Gerard, ogerard@ext.jussieu.fr

• The idea is that when you are reading one of these books, these files will give pointers to the On-Line Encyclopedia of Integer Sequences whenever an interesting sequence is mentioned. This will enable you to see the terms of the sequence, recurrences, formulae, other references, links, most recent progress, etc.
• Furthermore, the preparation of these concordances will supply additional sequences for the database, and additional references for existing sequences.
• This page is under construction, Jun 10, 2001

R. L. Graham, D. E. Knuth and O. Patashnik,

Concrete Mathematics, Addison-Wesley, 2nd Ed., 1994

Chapter 1: Recuurent Problems

page	A-number	Description
?	A000295	Eulerian numbers 2^n - n - 1. (Column 2 of Euler's triangle A008292.)
?	A006862	Euclid numbers: 1 + product of first n consecutive primes.
?	A057353	floor(3n/4)
?	A057354	floor(2n/5)
?	A057355	floor(3n/5)
?	A057356	floor(2n/7)
?	A057357	floor(3n/7)
?	A057358	floor(4n/7)
?	A057359	floor(5n/7)
?	A057360	floor(3n/8)
?	A057361	floor(5n/8)
?	A057362	floor(5n/13)
?	A057363	floor(8n/13)
?	A057364	floor(8n/21)
?	A057365	floor(13n/21)
?	A057366	floor(7n/19)
?	A057367	floor(11n/30)
10	A006257	Josephus problem.
18	A005665	Tower of Hanoi with cyclic moves only.
18	A005666	Tower of Hanoi with cyclic moves only.

Chapter 2: Sums

Chapter 3: Integer Functions

page	A-number	Description
66	A001462	Golomb's sequence: a(n) is the number of times n occurs.
66	A001597	Perfect powers: m^k where m is an integer and k >= 2.
77	A001951	A Beatty sequence: [ n * sqrt 2 ].
77	A001952	A Beatty sequence: [ n * (2 + sqrt(2)) ].
78	A002977	If n is present so are 2n+1 and 3n+1.
78	A007448	Knuth's sequence: a(n+1) = 1 + min ( 2 a[ n/2 ],3 a[ n/3 ] ).
97	A002024	n appears n times.
99	A002620	Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).
111	A006988	(10^n)-th prime.
117	A007305	Numerators of Farey (or Stern-Brocot) tree fractions.
122	A006258	Numerators of approximations to e.
122	A006259	Denominators of approximations to e.
138	A005728	Number of fractions in Farey series of order n (1 + A002088).
147	A006255	Ron Graham's sequence.
164	A027555	Triangle of binomial coefficients C(-n,k).
194	A008290	Triangle T(n,k) of rencontres numbers (number of permutations of n elements with k fixed points).
194	A008291	Triangle of rencontres numbers.
231	A000178	Superfactorials: product of first n factorials.
244	A008277	Triangle of Stirling numbers of 2nd kind, S2(n,k), n>=1, 1<=k<=n.
244	A048993	Triangle of Stirling numbers of 2nd kind, S(n,k), n>=0, 0<=k<=n.
245	A008275	Triangle of Stirling numbers of 1st kind, s(n,k), n>=1, 1<=k<=n.
245	A048994	Triangle of Stirling numbers of 1st kind, s(n,k), n>=0, 0<=k<=n.
254	A000800	Sum of upward diagonals of Eulerian triangle.
254	A007338	Multiplicative encoding of the Eulerian number triangle.
254	A008292	Triangle of Eulerian numbers.
254	A008518	Triangle of Eulerian numbers with rows multiplied by 1+x.
254	A014449	Numbers in the triangle of Eulerian numbers (A008292) that are not 1.
254	A014450	Even numbers in the triangle of Eulerian numbers.
254	A014459	Odd numbers in the triangle of Eulerian numbers.
254	A014461	Odd numbers in the triangle of Eulerian numbers that are not 1.
254	A014467	Triangular array formed from elements to right of middle of rows of the triangle of Eulerian numbers.
254	A014468	Triangular array formed from elements to right of middle of rows of the triangle of Eulerian numbers that are greater than 1.
254	A014469	Triangular array formed from odd elements to right of middle of rows of the triangle of Eulerian numbers (A008292).
254	A014470	Triangular array formed from odd elements to right of middle of rows of the triangle of Eulerian numbers that are greater than 1.
254	A014472	Triangular array formed from even elements to right of middle of rows of the triangle of Eulerian numbers.
254	A025585	Central Eulerian numbers A(2n-1, n).
256	A004301	Second-order Eulerian numbers.
256	A005803	Second-order Eulerian numbers: 2^(n+1) - 2n - 2.
256	A006260	Second-order Eulerian numbers.
256	A007347	Maximal Eulerian numbers of second kind.
256	A008517	Second-order Eulerian triangle a(n,k), 1<=k<=n.
259	A001008	Wolstenholme numbers: numerator of harmonic number H(n)=Sum_{i=1..n} 1/i.
259	A002805	Denominators of harmonic numbers H(n)=Sum 1/i.
259	A014537	Books required for n books of overhang.
269	A051724	Numerator of n/12.
269	A051725	Denominator of n/12.
269	A051726	Numerator of n(n-1)(n-2)/720.
269	A051727	Denominator of n(n-1)(n-2)/720.
316	A000008	Ways of making change for n cents using coins of 1, 2, 5, 10 cents.
316	A001299	Ways of making change for n cents using coins of 1, 5, 10, 25 cents.
316	A001300	Ways of making change for n cents using coins of 1, 5, 10, 25, 50 cents.
316	A001301	Ways of making change for n cents using coins of 1, 2, 5, 10, 25 cents.
316	A001302	Ways of making change for n cents using coins of 1, 2, 5, 10, 25, 50 cents.
316	A001306	Ways of making change for n cents using coins of 1, 5, 10, 20, 50, 100 cents.
316	A001310	Ways of making change for n cents using coins of 1, 2, 4, 10, 20, 40, 100 cents.
316	A001312	Ways of making change for n cents using coins of 1, 2, 5, 10, 50, 100 cents.
316	A001313	Ways of making change for n cents using coins of 1, 2, 5, 10, 20, 50 cents.
316	A001314	Ways of making change for n cents using coins of 2, 5 (two kinds), 10, 20, 50 cents.
316	A001319	Ways of making change for n cents using coins of 2, 5, 10, 20, 50 cents.
316	A001343	Ways of making change for n cents using coins of 5, 10, 20, 50, 100 cents.
316	A001362	Ways of making change for n cents using coins of 1, 2, 4, 10 cents.
316	A001364	Ways of making change for n cents using coins of 1, 2, 4, 12, 24, 48, 96, 120 cents (based on English coinage of 1939).
316	A057537	Ways of making change for n Euro-cents using the Euro currency (coins and bills of size 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000, 2000, 5000, 10000, 20000, 50000 cents).
327	A006904	a(n) = a(n-1)+2.a(n-2)+(-1)^n.
329	A001353	a(n) = 4a(n-1) - a(n-2).
329	A001835	a(n) = 4a(n-1) - a(n-2).
329	A028230	Bisection of A001353. (Indices of square numbers which are also octagonal.)
360	A006253	Stacking bricks.
477	A002109	Hyperfactorials: Product k^k, k = 1 . . n.
481	A019267	From an asymptotic expansion for Pi.
528	A001469	Genocchi numbers (of first kind): expansion of tan(x/2).
528	A036968	Genocchi numbers (of first kind): expansion of 2x/(exp(x)+1).
575	A002426	Central trinomial coefficient: largest coefficient of (1+x+x^2)^n.
575	A011769	a_0 = 1, a_{n+2} = 3 * a_{n+1} - F_n*(F_n + 1), where F_n is n-th Fibonacci number.