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Sequences from Graham, Knuth, Patashnik "Concrete Math"
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- For a long time I (njas) 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 version is only a preliminary draft and needs to be expanded. If you can help, please either edit this page, or contact Neil Sloane, njasloane@gmail.com.
- For the current list of these concordances, see here.
Chapter 1: Recurrent Problems
page | A-number | Description |
? |
Eulerian numbers 2^n - n - 1. (Column 2 of Euler's triangle A008292.) | |
? | Euclid numbers: 1 + product of first n consecutive primes. | |
? | floor(3n/4) | |
? | floor(2n/5) | |
? | floor(3n/5) | |
? | floor(2n/7) | |
? | floor(3n/7) | |
? | floor(4n/7) | |
? | floor(5n/7) | |
? | floor(3n/8) | |
? | floor(5n/8) | |
? | floor(5n/13) | |
? | floor(8n/13) | |
? | floor(8n/21) | |
? | floor(13n/21) | |
? | floor(7n/19) | |
? | floor(11n/30) | |
10 | Josephus problem. | |
18 | Tower of Hanoi with cyclic moves only. | |
18 | Tower of Hanoi with cyclic moves only. |
Chapter 2: Sums
Chapter 3: Integer Functions
page | A-number | Description |
66 | Golomb's sequence: a(n) is the number of times n occurs. | |
66 | Perfect powers: m^k where m is an integer and k >= 2. | |
77 | A Beatty sequence: [ n * sqrt 2 ]. | |
77 | A Beatty sequence: [ n * (2 + sqrt(2)) ]. | |
78 | If n is present so are 2n+1 and 3n+1. | |
78 | Knuth's sequence: a(n+1) = 1 + min ( 2 a[ n/2 ],3 a[ n/3 ] ). | |
97 | n appears n times. | |
99 | Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4). | |
111 | (10^n)-th prime. | |
117 | Numerators of Farey (or Stern-Brocot) tree fractions. | |
122 | Numerators of approximations to e. | |
122 | Denominators of approximations to e. | |
138 |
Number of fractions in Farey series of order n (1 + A002088). | |
147 | Ron Graham's sequence. | |
164 | Triangle of binomial coefficients C(-n,k). | |
194 | Triangle T(n,k) of rencontres numbers (number of permutations of n elements with k fixed points). | |
194 | Triangle of rencontres numbers. | |
231 | Superfactorials: product of first n factorials. | |
244 | Triangle of Stirling numbers of 2nd kind, S2(n,k), n>=1, 1<=k<=n. | |
244 | Triangle of Stirling numbers of 2nd kind, S(n,k), n>=0, 0<=k<=n. | |
245 | Triangle of Stirling numbers of 1st kind, s(n,k), n>=1, 1<=k<=n. | |
245 | Triangle of Stirling numbers of 1st kind, s(n,k), n>=0, 0<=k<=n. | |
254 | Sum of upward diagonals of Eulerian triangle. | |
254 | Multiplicative encoding of the Eulerian number triangle. | |
254 | Triangle of Eulerian numbers. | |
254 | Triangle of Eulerian numbers with rows multiplied by 1+x. | |
254 |
Numbers in the triangle of Eulerian numbers (A008292) that are not 1. | |
254 | Even numbers in the triangle of Eulerian numbers. | |
254 | Odd numbers in the triangle of Eulerian numbers. | |
254 | Odd numbers in the triangle of Eulerian numbers that are not 1. | |
254 | Triangular array formed from elements to right of middle of rows of the triangle of Eulerian numbers. | |
254 | Triangular array formed from elements to right of middle of rows of the triangle of Eulerian numbers that are greater than 1. | |
254 |
Triangular array formed from odd elements to right of middle of rows of the triangle of Eulerian numbers (A008292). | |
254 | Triangular array formed from odd elements to right of middle of rows of the triangle of Eulerian numbers that are greater than 1. | |
254 | Triangular array formed from even elements to right of middle of rows of the triangle of Eulerian numbers. | |
254 | Central Eulerian numbers A(2n-1, n). | |
256 | Second-order Eulerian numbers. | |
256 | Second-order Eulerian numbers: 2^(n+1) - 2n - 2. | |
256 | Second-order Eulerian numbers. | |
256 | Maximal Eulerian numbers of second kind. | |
256 | Second-order Eulerian triangle a(n,k), 1<=k<=n. | |
259 | Wolstenholme numbers: numerator of harmonic number H(n)=Sum_{i=1..n} 1/i. | |
259 | Denominators of harmonic numbers H(n)=Sum 1/i. | |
259 | Books required for n books of overhang. | |
269 | Numerator of n/12. | |
269 | Denominator of n/12. | |
269 | Numerator of n(n-1)(n-2)/720. | |
269 | Denominator of n(n-1)(n-2)/720. | |
316 | Ways of making change for n cents using coins of 1, 2, 5, 10 cents. | |
316 | Ways of making change for n cents using coins of 1, 5, 10, 25 cents. | |
316 | Ways of making change for n cents using coins of 1, 5, 10, 25, 50 cents. | |
316 | Ways of making change for n cents using coins of 1, 2, 5, 10, 25 cents. | |
316 | Ways of making change for n cents using coins of 1, 2, 5, 10, 25, 50 cents. | |
316 | Ways of making change for n cents using coins of 1, 5, 10, 20, 50, 100 cents. | |
316 | Ways of making change for n cents using coins of 1, 2, 4, 10, 20, 40, 100 cents. | |
316 | Ways of making change for n cents using coins of 1, 2, 5, 10, 50, 100 cents. | |
316 | Ways of making change for n cents using coins of 1, 2, 5, 10, 20, 50 cents. | |
316 | Ways of making change for n cents using coins of 2, 5 (two kinds), 10, 20, 50 cents. | |
316 | Ways of making change for n cents using coins of 2, 5, 10, 20, 50 cents. | |
316 | Ways of making change for n cents using coins of 5, 10, 20, 50, 100 cents. | |
316 | Ways of making change for n cents using coins of 1, 2, 4, 10 cents. | |
316 | 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 | 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 | a(n) = a(n-1)+2.a(n-2)+(-1)^n. | |
329 | a(n) = 4a(n-1) - a(n-2). | |
329 | a(n) = 4a(n-1) - a(n-2). | |
329 |
Bisection of A001353. (Indices of square numbers which are also octagonal.) | |
360 | Stacking bricks. | |
477 | Hyperfactorials: Product k^k, k = 1 . . n. | |
481 | From an asymptotic expansion for Pi. | |
528 | Genocchi numbers (of first kind): expansion of tan(x/2). | |
528 | Genocchi numbers (of first kind): expansion of 2x/(exp(x)+1). | |
575 | Central trinomial coefficient: largest coefficient of (1+x+x^2)^n. | |
575 | a_0 = 1, a_{n+2} = 3 * a_{n+1} - F_n*(F_n + 1), where F_n is n-th Fibonacci number. |